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+{"text":"\\section{Introduction and Preliminary Discussion}\nAs for standard terminology and other terminology used in this paper, we refer to the book by Bondy and Murty, \\cite{Bon}, and to the papers quoted in the references. Let $G$ be a connected graph. A \\emph{2-block} is a 2-connected graph or a block of $G$ containing more than two vertices. The square of a graph $G$, denoted $G^2$, is the graph obtained from $G$ by joining any two nonadjacent vertices which have a common neighbor, by an edge.\n\nIt was shown in 1970 and published in 1974 that the square of every 2-block contains a       hamiltonian cycle, \\cite{Fle2}. Key in proving this was the existence of EPS-graphs $S$ in connected bridgeless graphs $G$, where $S$ is the edge-disjoint union of a not necessarily connected eulerian subgraph $E$ and a linear forest $P$, and $S$ is connected and spans $G$, \\cite{Fle1}. In subsequent papers \\cite{Fle}, \\cite{FleHob} the existence of various types of EPS-graphs was established. Their relevance was based on the fact that the total graph $T(G)$ of any connected graph $G$ other than $K_1$ is hamiltonian if and only if $G$ has an EPS-graph, \\cite{FleHob}. This and the theory of EPS-graphs led to a description of the most general block-cutvertex graph $\\mbox{bc}(G)$ of a graph $G$ may have such that $T(G)$ is hamiltonian and if $\\mbox{bc}(G)$ does not have the corresponding structure, then exchanging certain 2-blocks in $G$ with some special 2-blocks yields a graph $G^*$ such that $\\mbox{bc}(G)$ and $\\mbox{bc}(G^*)$ are isomorphic but $T(G^*)$ is not hamiltonian, \\cite{FleHob}. In dealing with hamiltonian cycles and hamiltonian paths by methods developed up to that point, it was shown in \\cite{Fle} that in the square of graphs hamiltonicity and vertex-pancyclicity are equivalent concepts, and so are hamiltonian connectedness and panconnectedness. In this context Theorem~\\ref{2-blockcycle} stated below was established as a tool needed to prove the equivalences just mentioned.\n\nHowever, in the course of time much shorter proofs of Fleischner's Theorem were developed \\cite{Geo}, \\cite{Riha}; the same applies to Theorem \\ref{2-blockcycle} below, \\cite{MutRau}. More recently, an algorithm yielding a hamiltonian cycle in the square of a 2-block in linear time, was developed, \\cite{AlsGeo}. The methods developed in these much shorter proofs (including the algorithm just mentioned) do not seem to yield short proofs of Theorems \\ref{H_4} and \\ref{F_4} below, \\cite{EkFle}, \\cite{FleChia}. These latter theorems are, on the other hand, instrumental in proving the central results of this paper, i.e., Theorems \\ref{hamiltonian} and \\ref{hamconnected}, and related algorithms. \n\nLet $\\mbox{bc}(G)$ denote the \\emph{block-cutvertex graph} of $G$. Blocks corresponding to leaves of $\\mbox{bc}(G)$ are called \\emph{endblocks}, otherwise \\emph{innerblocks}. Note that a block in a graph $G$ is either a 2-block or a bridge of $G$. For each cutvertex $i$ of $G$, let $k_{i}$ be the number of 2-blocks of $G$ which include vertex $i$ and let $\\mbox{bn}(i)$ be the number of nontrivial bridges of $G$ which are incident with vertex $i$. In what follows a bridge is called nontrivial if it is not incident to a leaf.\n  \nIn Theorem \\ref{hamiltonian}, we introduce an array $m_{i}(B)$ of numbers with an entry for each pair consisting of a cutvertex $i$ and a 2-block $B$ of $G$. We may think of this number $m_{i}(B)$ as the number of edges of $B$ incident with $i$ which are possibly contained in a hamiltonian cycle in $G^{2}$. \n\nStatement of Theorem \\ref{hamiltonian} describes the most general block-cutvertex structure a graph $G$ may have in order to guarantee that $G^2$ is hamiltonian using parameters $m_{i}(B)$ as in \\cite{FleHob}.\n\n\\begin{theorem}\n \\label{hamiltonian}\n Let $G$ be a connected graph with at least three vertices. Let the 2-blocks of $G$ be labelled $B_{1},B_{2},...,B_{n}$. Let the cutvertices of $G$ be labelled $1,2,...,s$. Suppose there is a labelling $m_{i}(B_{t})$ for each $i\\in\\{1,2,...,s\\}$ and each $t\\in\\{1,2,...,n\\}$ such that the following conditions are fulfilled.\n \\begin{mathitem}\n  \\item[1)] $0\\leq m_{i}(B_t)\\leq2$ for all $i$ and all 2-blocks $B_t$;\n  \\item[2)] for 2-block $B_t$ $m_{i}(B_t)=0$ if and only if cutvertex $i$ is not in $V(B_t)$;\n  \\item[3)] for 2-block $B_t$, $m_{i}(B_t)\\geq\\mbox{bn}(i)$, if cutvertex $i\\in V(B_t)$;\n  \\item[4)] $\\mbox{bn}(i)\\leq2$ for all $i\\in\\{1,2,...,s\\}$; \n  \\item[5)] $\\sum_{i=1}^{s}m_{i}(B_t)\\leq4$ for each 2-block $B_t$ of $G$ and, if $m_i(B_t)=2$ \n            for some $i$, then $\\sum_{i=1}^sm_{i}(B_t)\\leq3$; and\n  \\item[6)] $\\sum_{t=1}^{n}m_i(B_t)\\geq2k_{i}+\\mbox{bn}(i)-2$ for each $i\\in\\{1,2,...,s\\}$.\n \\end{mathitem}\n Then $G^{2}$ is hamiltonian. \n\nMoreover, if the labelling $m_{i}(B_{t})$  satisfying conditions 1), 2) and 3) is given and at least one of conditions  4), 5), 6) is violated by some $G$, then there exists a class of graphs $G'$ with non-hamiltonian square but $\\mbox{bc}(G')$ and $\\mbox{bc}(G)$ are isomorphic.\n\\end{theorem}\n\nAlso, we obtain a similar result for hamiltonian connectedness (Theorem~\\ref{hamconnected}). Quite surprisingly, its formulation is much simpler than that of Theorem \\ref{hamiltonian}.\n\n\\begin{theorem}\n \\label{hamconnected}\n Let $G$ be a connected graph such that the following conditions are fulfilled:\n \\begin{itemize}\n  \\item[1)] there is no nontrivial bridge of $G$;\n  \\item[2)] every block contains at most 2 cutvertices.\n \\end{itemize}\n Then $G^2$ is hamiltonian connected.\n\nMoreover,\n\\begin{itemize}\n  \\item[$\\cdot$] if a graph $G$ contains a nontrivial bridge, then $G^2$ is not hamiltonian connected;  \n  \\item[$\\cdot$] if $G$ contains a block containing more than 2 cutvertices, then there is a graph $G'$ such that $\\mbox{bc}(G)$ and $\\mbox{bc}(G')$ are isomorphic but $(G')^2$ is not hamiltonian connected. \n\\end{itemize}\n\\end{theorem}\n\nA fundamental result regarding hamiltonicity in the square of a 2-block is the following theorem.\n\n\\begin{theorem}\\emph{\\textbf{\\cite{Fle}}}\n\\label{2-blockcycle}\nSuppose $v$ and $w$ are two arbitrarily chosen vertices of a $2$-block $G$.\nThen $G^2$ contains a hamiltonian cycle $C$ such that the edges of $C$ incident to $v$ are in $G$ and at least one of the edges of $C$ incident to $w$ is in $G$. Furthermore, if $v$ and $w$ are adjacent in $G$, then these are three different edges.  \n\\end{theorem}\n\nThe hamiltonian theme in the square of a 2-block has been recently revisited (\\cite{EkChiaFle}, \\cite{EkFle}, \\cite{FleChia}), yielding the following results which are essential for this paper. \n\nA graph $G$ is said to have the \\emph{$\\mathcal{H}_{k}$ property} if for any given vertices $x_1,...,x_k$ there is a hamiltonian cycle in $G^2$ containing distinct edges $x_1y_1,...,x_ky_k$ of~$G$.\n \n\\begin{theorem}\\emph{\\textbf{\\cite{EkFle}}}\n \\label{H_4}\n Given a 2-block $G$ on at least 4 vertices, then $G$ has the $\\mathcal{H}_{4}$ property, and there are 2-blocks of arbitrary order greater than 4 without the $\\mathcal{H}_{5}$ property.\n\\end{theorem}\n\nBy a \\emph{$uv$-path} we mean a path from $u$ to $v$ in $G$. If a $uv$-path is hamiltonian, we call it a \\emph{$uv$-hamiltonian path}. Let $A=\\{x_{1},x_{2},...,x_{k}\\}$ be a set of $k\\geq 3$ distinct vertices in $G$. An $x_{1}x_{2}$-hamiltonian path in $G^{2}$ which contains $k-2$ distinct edges $x_{i}y_{i}\\in E(G), i=3,...,k$, is said to be $\\mathcal{F}_{k}$. A graph $G$ is said to have the \\emph{$\\mathcal{F}_{k}$ property} if, for any set $A=\\{x_{1},x_{2},...,x_{k}\\}\\subseteq V(G)$, there is an $\\mathcal{F}_{k}$ $x_{1}x_{2}$-hamiltonian path in $G^{2}$.\n\n\\begin{theorem}\\emph{\\textbf{\\cite{FleChia}}}\n \\label{F_4}\n  Every 2-block on at least 4 vertices has the $\\mathcal{F}_{4}$ property. \n\\end{theorem}\n\nA graph $G$ is said to have the \\emph{strong $\\mathcal{F}_{3}$ property} if, for any set of 3 vertices $\\{x_{1},x_{2},x_{3}\\}$ in $G$, there is an $x_{1}x_{2}$-hamiltonian path in $G^{2}$ containing distinct edges $x_{3}z_{3},x_{i}z_{i}\\in E(G)$ for a given $i\\in\\{1,2\\}$. Such an $x_{1}x_{2}$-hamiltonian path in $G^{2}$ is called a strong          $\\mathcal{F}_{3}$ $x_{1}x_{2}$-hamiltonian path.\n\n\\begin{theorem}\\emph{\\textbf{\\cite{FleChia}}}\n \\label{strongF_3}\n  Every 2-block has the strong $\\mathcal{F}_{3}$ property.\n\\end{theorem}\n\n\\begin{theorem}\\emph{\\textbf{\\cite{FleChia}}}\n\t\\label{strongF_3ends}\nLet $G$ be a $2$-connected graph and let $x, y$ be two vertices in $G$. Then $G^2$ has an $xy$-hamiltonian path $P(x, y)$ such that\n\n(i) $xz \\in E(G) \\cap E(P(x,y))$  for some $z \\in V(G)$, \nand\n\n(ii) either $yw \\in  E(G) \\cap E(P(x,y))$ for some $w\\in V(G)$,  or  else $P(x, y)$ contains an edge $uv$ for some vertices $u, v \\in N(y)$.\n\\end{theorem} \n\n\\section{Proofs and algorithms}\n\nPROOF OF THEOREM \\ref{hamiltonian}\n\n\\begin{proof}\n Set $P_0=G-\\cup_{t=1}^{n}B_t$. Then every component of $P_0$ is a tree. Since by 4) $\\mbox{bn}(i)\\leq 2$ every component of $P_0$ is even a caterpillar. \n \n For every caterpillar $T$ of $P_0$ except $T=K_2$ we have the following observation which can be proved easily. \n\n\\medskip\n\n\\noindent\n\\emph{Observation: Let $T$ be a caterpillar with at least three vertices and $P=x_1x_2...x_m$ be some longest path in $T$. Then $T^2$ contains a hamiltonian cycle containing edges $x_1x_2, x_{m-1}x_m$  and different edges $u_jv_j$, where $u_j,v_j\\in N_G(x_j)$ for $j=2,3,...m-1$.}\n\nSee Figure \\ref{Catter} for illustration in which for $x_3$ we have $u_3=x_2$ and $v_3=x_4$.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\end{center}\\caption{Hamiltonian cycle in a caterpillar for $m=7$ (bold edges)}\\label{Catter}\n\\end{figure} \n\n\\medskip\n\n Every 2-block $B_t$ contains a hamitonian cycle in $(B_t)^2$ which is one of two types depending on labellings $m_{i}(B_{t})$:\n \nIf $m_i(B_t)\\neq 2$ for every $i=1,2,...,s$, then for at most 4 cutvertices $a,b,c,d$ it holds that $m_j(B_t)=1$ for $j=a,b,c,d$ by condition 5). By Theorem \\ref{H_4}, $(B_t)^2$ has a hamiltonian cycle $C_t$ containing 4 different edges $aa',bb',cc',dd'$ of~$B_t$. \n \nIf $m_i(B_t)=2$ for some $i\\in\\{1,2,...,s\\}$, then at most one cutvertex $a$ has $m_a(B_t)=1$ by condition 5). By Theorem \\ref{2-blockcycle}, $(B_t)^2$ has a hamiltonian cycle $C_t$ containing 3 different edges $ii',ii'',aa'$ of $B_t$.   \n\nThe union of hamiltonian cycles $C_t$ in $(B_t)^2$, for $t=1,2,...n$, hamiltonian cycles in the square of each catepillar (nontrivial component of $P_0$) and trivial components of $P_0$  is a connected spanning subgraph $S$ of $G^2$. \n\nWe construct a hamiltonian cycle $C$ in $G^2$ from $S$ repeating step by step the following procedure for every cutvertex $i$ of $G$ with $m_{i}(B)\\geq 1$ for some 2-block $B$. \n\nIf $i$ does not exist, then $n=0$ and $G=P_0$ is a caterpillar. Hence $S$ is a hamiltonian cycle in $G^2$. Otherwise we join all hamiltonian cycles from $S$ containing $i$ together with trivial components of $P_0$ containing $i$ to one cycle in the following way.\n\n\\medskip\n\n\\noindent\nFirst assume that $\\mbox{bn}(i)=0$.\n\nBy condition 6) we have  $\\sum_{t=1}^{n}m_i(B_t)\\geq 2k_i-2$. Without loss of ge\\-nerality for $k_i>1$ we may assume that $m_i(B_1)\\geq 1$, $m_i(B_2)\\geq 1$ and $m_i(B_3)=m_i(B_4)=...=m_i(B_{k_i})=2$, where $m_i(B_t)$  corresponds to the number of edges of $B_t$ incident to $i$ in $C_t$. If $k_i=1$, then by condition 2) we have $m_i(B_1)\\geq 1$.\n\nWe find a cycle $C^i$ on $\\cup_{r=1}^{k_i}V(C_r)\\cup L$, where $L$ is the set of all leaves incident to $i$, by appropriately replacing edges of $C_r\\cap B_r$, $r=1,2,...,k_i$, incident to $i$ (guaranteed by definition of $m_i(B_t)$) with edges of $G^2$ joining vertices in different $C_r$ adjacent to $i$ and leaves adjacent to $i$. Note that we preserve properties given by $m_j(B_t)$ for all $j\\neq i$.\n\n\\noindent\nNow assume that $\\mbox{bn}(i)=1$.\n\nBy condition 6) we have  $\\sum_{t=1}^{n}m_i(B_t)\\geq 2k_i+1-2=2k_i-1$. Without loss of generality we may assume that $m_i(B_1)\\geq 1$ and $m_i(B_2)=m_i(B_3)=...=m_i(B_{k_i})=2$, where $m_i(B_t)$  corresponds to the number of edges of $B_t$ incident to $i$ in $C_t$.\nLet $T$ be the component of $P_0$ containing $i$.\n\nIf $T=K_2=ii'$, where $i'$ is also a cutvertex of $G$ with $m_{i'}(B)\\geq 1$  (T is a trivial component of $P_0$), then we find a cycle $C^i$ on $\\cup_{r=1}^{k_i}V(C_r)\\cup T$ containing the edge $ii'$ by appropriately replacing edges of $C_r\\cap B_r$, $r=1,2,...,k_i$, incident to $i$ (guaranteed by definition of $m_i(B_t)$) with edges of $G^2$ joining $i'$ and vertices in different $C_r$ adjacent to $i$. Also here we preserve properties given by $m_j(B_t)$ for all $j\\neq i$.\n\nIf $T$ is a nontrivial component of $P_0$, then $T^2$ contains a hamiltonian cycle $C_T$ containing end-edges of any fixed longest paths $P$ in $T$ (we choose end-edges containing cutvertices of $G$ with $m_{i}(B_t)\\geq 1$) - see Observation above. Again we find a cycle $C^i$ on $\\cup_{r=1}^{k_i}V(C_r)\\cup V(C_T)$ by appropriately replacing edges of $C_r\\cap B_r$, $r=1,2,...,k_i$, incident to $i$ (guaranteed by definition of $m_i(B_t)$) and the end-edge $ii^*$ of $P$ with edges of $G^2$ joining $i^*$ and vertices in different $C_r$ adjacent to $i$. Again we preserve properties given by $m_j(B_t)$ for all $j\\neq i$ and by $C_T$.\n\n\\noindent\nFinally assume that $\\mbox{bn}(i)=2$.\n\nBy condition 6) we have  $\\sum_{t=1}^{n}m_i(B_t)\\geq 2k_i+2-2=2k_i$. It follows necessarily that  $m_i(B_1)=m_i(B_2)=...=m_i(B_{k_i})=2$, where $m_i(B_t)$  corresponds to the number of edges of $B_t$ incident to $i$ in $C_t$.\n\nLet $T$ be the nontrivial component of $P_0$ containing $i$. Note that $i$ is not an endvertex of $T$ because of $\\mbox{bn}(i)=2$. Then $T^2$ contains a hamiltonian cycle $C_T$ containing end-edges of any fixed longest paths in $T$ (we choose end-edges containing cutvertices of $G$ with $m_{i}(B_t)\\geq 1$) and an edge $u_iv_i$ of $G^2$ where $u_i,v_i\\in N_G(i)$  (see Observation above). We find a cycle $C^i$ on $\\cup_{r=1}^{k_i}V(C_r)\\cup V(C_T)$ by appropriately replacing edges of $C_r\\cap B_r$, $r=1,2,...,k_i$, incident to $i$ (guaranteed by definition of $m_i(B_t)$) and the edge $u_iv_i$ of $P$ with edges of $G^2$ joining $u_i, v_i$ and vertices in different $C_r$ adjacent to $i$ if $k_i>1$. If, however, $k_i=1$, then $u_i$ and $v_i$ are joined to the neighbors of $C_r\\cap B_r$ in $N_G(i)$. Also here we preserve properties given by $m_j(B_t)$ for all $j\\neq i$ and by $C_T$.\n\nNow we choose next cutvertex $i$ with $m_{i}(B)\\geq 1$ for some 2-block $B$ successively and we use all cycles  formed in the previous steps instead of previously formed cycles. Note that we preserve all properties given by $m_j(B)$ for all $j\\neq i$ in every case. We stop with the hamiltonian cycle in $G^2$ as required.\n\n\\vskip 8mm\n\nNow assume that there is no labelling satisfying conditions 1) - 6), that is, the labelling $m_i(B_t)$ satisfying conditions 1), 2) and 3) is given and at least one of conditions  4), 5), 6) is violated. We show that there exists a class of graphs $G'$ with non-hamiltonian square but $\\mbox{bc}(G')$ and $\\mbox{bc}(G)$ are isomorphic.\n\n\\bigskip\n\n\\begin{figure}[ht]\n\\begin{center}\n\\end{center}\\caption{Graphs without hamiltonian square}\\label{Counter}\n\\end{figure} \n\n\\noindent\n\\emph{Condition 4) does not hold.} \n\nHence $\\mbox{bn}(i)\\geq 3$ for at least one $i\\in\\{1,2,...,s\\}$. Clearly this is a class of graphs $G'$ such that the square of every such graph $G'$ does not contain a hamiltonian cycle (if we try to construct a hamiltonian cycle in the square, then the degree of the cutvertex $i$ is at least 3, a contradiction), e.g. see graphs in Figure \\ref{Counter} a), where $H_1$ is arbitrary connected graphs, $H_2$, $H_3$, $H_4$ are arbitrary connected graphs with at least one edge each and $\\mbox{bn}(i)=3$. Note that conditions 5) and 6) may hold.\n\n\\vskip 0.9cm\n\n\\noindent\n\\emph{Condition 5) does not hold.}\n\nHence $\\sum_{i=1}^{s}m_{i}(B)\\geq 5$ for some 2-block $B$ and $m_i(B)<2$ for all $i$ or $\\sum_{j=1}^{s}m_{j}(B)\\geq 4$ for some 2-block $B$ and $m_i(B)=2$ for some $i\\in\\{1,2,...,s\\}$.\n\nFirst suppose that $k=\\sum_{i=1}^{s}m_{i}(B)\\geq 5$ for some 2-block $B$ of $G$ and $m_i(B)<2$ for all $i$. Clearly $B$ has exactly $k$ cutvertices by condition 2). Then we exhange $B$ with $K_{2,k}$ where $k$ 2-valent vertices are cutvertices of $G$ and all other blocks with arbitrary blocks to get a class of graphs $G'$ such that $\\mbox{bc}(G')$ and $\\mbox{bc}(G)$ are isomorphic. The square of every such graph $G'$ does not contain a hamiltonian cycle (if we try to construct a hamiltonian cycle in the square, then the degree of at least one of the two $k$-valent vertices of $K_{2,k}$ is at least 3, a contradiction), e.g. see graphs in Figure \\ref{Counter} b), where $k=5$ and $H_1,...,H_5$ are arbitrary connected graphs with at least one edge each. Note that conditions 4), 6) and the second part of condition 5) may hold.\n\nNow suppose that $\\sum_{j=1}^{s}m_{j}(B)\\geq 4$ for some 2-block $B$ and $m_i(B)=2$ for some $i$. If $B$ contains at least 5 cutvertices of $G$, then we continue similarly as above. If $B$ contains $k$ cutvertices of $G$ where $2\\leq k \\leq 4$, then without loss of generality we may assume that we tried to set the labelling $m_i(B_t)$ satisfying firstly conditions 5) and subsequently condition 6). Hence $\\mbox{bn}(i)\\geq 2$ and $\\mbox{bn}(j)\\geq 2$ where $j$ is the second cutvertex of $G$ in $B$ if $k=2$, otherwise we find a labelling $m_i(B_t)$ satisfying condition 5), a contradiction (see Algorithm~1 cases e) and f) below).\n\nFor $k=3,4$ we exhange $B$ with a cycle $C_k$ to get a class of graphs $G'$ such that $\\mbox{bc}(G')$ and $\\mbox{bc}(G)$ are isomorphic. The square of every such graph $G'$ does not contain a hamiltonian cycle (if we try to construct a hamiltonian cycle in the square, then the degree of the cutvertex $i$ is at least 3, a contradiction), e.g. see graphs in Figure \\ref{Counter} c), where $k=3$ and $H_1,...,H_4$ are arbitrary connected graphs with at least one edge each. Note that conditions 4), 6) and the first part of condition 5) may hold.\n\nFor $k=2$, we exchange $B$ with $K_{2,3}$, where two of the three 2-valent vertices are $i$ and $j$, to get a class of graphs $G'$ such that $\\mbox{bc}(G')$ and $\\mbox{bc}(G)$ are isomorphic. The square of every such graph $G'$ does not contain a hamiltonian cycle (it is not possible to find a hamiltonian cycle in the square containing the third 2-valent vertex different from $i$, $j$, a contradiction), e.g. see graphs in Figure \\ref{Counter} d), where $H_1,...,H_4$ are arbitrary connected graphs with at least one edge each. Note that conditions 4), 6) and the first part of condition 5) may hold.\n\n\\medskip\n\n\\noindent\n\\emph{Condition 6) does not hold.}\n  \nHence $\\sum_{t=1}^{n}m_{i}(B_t)<2k_i+\\mbox{bn}(i)-2$ for some $i$ and consequently $m_i(B_t)=1$ for at least $3-\\mbox{bn}(i)$ 2-blocks containing $i$. Note that, clearly, $\\mbox{bn}(i)<2$ with respect to condition 3).\n\nLet $r$ be the number of 2-blocks with $m_i(B_t)=1$. Each of these 2-blocks contains either exactly 2 cutvertices of $G$ or at least 3 cutvertices of $G$. Note that for 2-blocks containing only cutvertex $i$ we have $m_i(B_t)=2$ (see Algorithm 1 case d) below). We exchange every 2-block containing exactly 2 cutvertices of $G$ with a cycle $C_3$ and every 2-blocks containing $k$ cutvertices of $G$, $k\\geq 3$, with a cycle $C_k$. In the first case note that we assume without loss of generality that there is no labelling such that we switch values 1 and 2 for both cutvertices of this 2-block to get a permissible labelling (again see Algorithm~1 case e) below).\n\nSince $r\\geq 3-\\mbox{bn}(i)$, by the exchanging 2-block mentioned above we get a class of graphs $G'$ such that $\\mbox{bc}(G')$ and $\\mbox{bc}(G)$ are isomorphic. The square of every such graph $G'$ does not contain a hamiltonian cycle (if we try to construct a hamiltonian cycle in the square, then the degree of the cutvertex $i$ is at least 3, a contradiction), e.g. see graphs in Figure \\ref{Counter} $e_1$) and $e_2$). For the graph in Figure \\ref{Counter} $e_1$) it holds that $r=3-\\mbox{bn}(i)=3-1=2$, the 2-block $B_1$ has exactly 2 cutvertices of $G$, the 2-block $B_2$ has $k=3$ cutvertices of $G$ (and hence $B_1$, $B_2$ are isomorphic to $C_3$) and $H_1,...,H_5$ are arbitrary connected graphs with at least one edge. For the graph in Figure \\ref{Counter} $e_2$) it holds that $r=3-\\mbox{bn}(i)=3-0=3$, the 2-block $B_1$ has exactly 2 cutvertices of $G$, the 2-block $B_2$ has $k=3$ cutvertices of $G$, the 2-block $B_3$ has $k=4$ cutvertices of $G$ (hence $B_1$, $B_2$ are isomorphic to $C_3$ and $B_3$ is isomorphic to $C_4$) and $H_1,...,H_7$ are arbitrary connected graphs with at least one edge. Note that conditions 4) and 5) may hold.\n\nThis finished the proof of Theorem \\ref{hamiltonian}.\n\\end{proof}\n\nIf there is a graph $G$ such that every labelling $m_i(B_t)$ violates at least one of the conditions 4) - 6) of Theorem \\ref{hamiltonian}, then there is a graph $G'$ with $\\mbox{bc}(G')=\\mbox{bc}(G)$ such that $(G')^2$ is not hamiltonian  as it has been shown in the proof of Theorem~\\ref{hamiltonian}. On the other hand, if we are able to construct a labelling $m_i(B_t)$ satisfying conditions 1) - 6) using the following algorithm, then $G^2$ is hamiltonian as it has been shown in the proof of Theorem~\\ref{hamiltonian}.\n\n\\bigskip\n\n\\noindent\n\\emph{ALGORITHM 1:} \n\nSet $P_0=G-\\cup_{t=1}^nB_t$. If any component of $P_0$ is not a caterpillar, then $\\mbox{bn}(i)\\geq 3$ for some $i\\in\\{1,2,...,s\\}$ contradicting condition 4) in Theorem \\ref{hamiltonian} and $G^2$ is not hamiltonian (e.g. see Figure \\ref{Counter} a)). STOP.\n\nIf $G=P_0$, then $G$ is a caterpillar,  $n=0$ and $G^2$ is hamiltonian (see Observation in the proof of Theorem \\ref{hamiltonian}) and $m_i(B_t)$ is not defined ($n=0$). STOP.\n\nIf $G$ is a 2-block, $G^2$ is hamiltonian by Theorem \\ref{2-blockcycle} and $m_i(B_t)$ is not defined ($s=0$ and $n=1$). STOP.\n\nWe set $G_0=G-P_0$ and $m_i(B_t)=0$ if $i\\notin V(B_t)$ for $i\\in\\{1,2,...,s\\}$ and $t\\in\\{1,2,...,n\\}$.\n\n\\bigskip\n\n\\noindent\nSTART\n\nWe choose a 2-block $B$ containing at most 1 cutvertex of $G_0$. Note that $B$ is either a component of $G_0$ or an endblock of some component of $G_0$. If such endblock does not exist, we choose 2-block $B$ as a component of $G_0-H$ or an endblock of $G_0-H$ where $H$ is the union of all 2-blocks for which the labelling $m_i(B_t)$ is already set.  Let $c_1,c_2,...,c_k$ be all cutvertices of $G$ contained in $B$, $k\\geq 1$. \n\n\\medskip\n\n\\begin{itemize}\n \\item[a)] If $k\\geq 5$, then by condition 2) $m_{c_i}(B)\\geq 1$ for $i=1,2,...,k$. Hence condition 5) in Theorem \\ref{hamiltonian} does not hold and $G^2$ may not be hamiltonian (e.g. see Figure \\ref{Counter} b)). STOP.\n\n \\item[b)] If $k\\geq 3$ and $\\mbox{bn}(c_i)=2$ for some $i\\in\\{1,2,...,k\\}$ , then by condition 3) $m_{c_i}(B)=2$ and by 2) $m_{c_j}(B)\\geq 1$ for $j=1,2,...,k$. Hence condition~5) in Theorem \\ref{hamiltonian} does not hold and $G^2$ may not be hamiltonian (e.g. see Figure \\ref{Counter} c)). STOP.\n\n \\item[c)] If $k=2$ and $\\mbox{bn}(c_1)=\\mbox{bn}(c_2)=2$, then by condition 3) $m_{c_1}(B)=2$ and $m_{c_2}(B)=2$. Hence condition 5) in Theorem \\ref{hamiltonian} does not hold and $G^2$ may not be hamiltonian (e.g. see Figure \\ref{Counter} d)). STOP.\n\n \\item[d)] If $k=1$, then we set $m_{c_1}(B)=2$ (we maximize values $m_i(B_t)$ with respect to condition 6) in Theorem \\ref{hamiltonian}). Note that, if the labelling $m_{i}(B_t)$ is set for all 2-blocks incident with $c_1$, then condition 6) holds for cutvertex $c_1$ with respect to the choice of $B$. \n \n If the labelling $m_{i}(B_t)$ is set for all 2-blocks of $G$, then the labelling $m_i(B_t)$ satisfies the conditions of Theorem \\ref{hamiltonian} and $G^2$ is hamiltonian. STOP.\n \n Otherwise we go to START. \n\n \\item[e)] If $k=2$ and $\\mbox{bn}(c_i)\\leq 1$ for $i\\in\\{1,2\\}$, then we set $m_{c_1}(B)$ and $m_{c_2}(B)$ in the following way (without loss of generality $i=1$).\n \n Let $\\mbox{bn}(c_2)=2$. Then we set $m_{c_1}(B)=1$ and $m_{c_2}(B)=2$ with respect to conditions 2), 3) and 5). \n \n Let $\\mbox{bn}(c_2)\\leq 1$. Then for at least one of $c_1$, $c_2$ it holds that $m_{c_j}(B_t)$ for $j\\in\\{1,2\\}$ is set for all 2-blocks $B_t$ except $B$ with respect to the choice of $B$ (again without loss of generality $j=1$). We set $m_{c_1}(B)=1$ and we verify condition 6) for $c_1$. If it holds, then we set $m_{c_2}(B)=2$ (again we maximize values $m_i(B_t)$ with respect to condition 6)). If condition~6) for $c_1$ does not hold for $m_{c_1}(B)=1$, then we set $m_{c_1}(B)=2$ and $m_{c_2}(B)=1$. \n \n Now in both cases we verify condition 6) for $c_1$ and $c_2$ if the labelling $m_{c_1}(B_t)$ and $m_{c_2}(B_t)$ is set for all 2-blocks $B_t$. \n \n If condition 6) does not hold in at least one case, then $G^2$ may not be hamiltonian (e.g. see Figure \\ref{Counter} $e_1)$). STOP. \n \n Hence suppose that condition 6) holds for $c_1$, $c_2$ if $m_{c_1}(B_t)$, $m_{c_2}(B_t)$ is set for all $B_t$, respectively.\n \n If the labelling $m_{i}(B_t)$ is set for all 2-blocks, then the labelling $m_i(B_t)$ satisfies the conditions of Theorem \\ref{hamiltonian} and $G^2$ is hamiltonian. STOP.\n  \n Otherwise we go to START. \n \n  \\item[f)] If $k\\in\\{3,4\\}$ and $\\mbox{bn}(c_i)\\leq 1$, then we set $m_{c_i}(B)=1$ for $i=1,2,...,k$. We verify condition 6) for all $c_i$ if the labelling $m_{c_i}(B_t)$ is set for all 2-blocks $B_t$. \n \n If condition 6) does not hold in at least one case, then $G^2$ may not be hamiltonian (e.g. see Figure \\ref{Counter} $e_2)$). STOP.\n \n Hence suppose that condition 6) holds for all $c_i$, $i=1,2,...,k$, for which $m_{c_i}(B_t)$ is set for all $B_t$.\n \n If the labelling $m_{i}(B_t)$ is set for all 2-blocks, then the labelling $m_i(B_t)$ satisfies the conditions of Theorem \\ref{hamiltonian} and $G^2$ is hamiltonian. STOP.\n \n Otherwise we go to START. \n\\end{itemize}\n\n\\noindent\nPROOF OF THEOREM \\ref{hamconnected}\n\n\\begin{proof}\n Let $x,y\\in V(G)$. First we prove that there exists an $xy$-hamiltonian path $P$ in $G^2$ if there is no nontrivial bridge of $G$ and every block contains at most 2 cutvertices.\n\n\\bigskip\n \n (A) Suppose that $x$ and $y$ are in the same block $B$ of $G$. We proceed by induction on $n$, where $n$ is the number of blocks of $G$, $n\\geq 1$.  \n \n For $n=1$, clearly $G=B$. If $B=K_2=xy$, then $G$ is also the $xy$-hamiltonian path in $G^2$ as required. If $B$ is a 2-block, then by Theorem \\ref{strongF_3}, $G^2=B^2$ contains an $xy$-hamiltonian path $P$ as required.\n\n\n\\bigskip\n\nNow suppose that the statement of Theorem \\ref{hamconnected} is true for every graph with $n$ blocks and $G$ is a graph with $n+1$ blocks, $n\\geq 1$. We distinguish 2 cases.\n\n\\begin{itemize}\n \\item $B$ has exactly one cutvertex $c$.\n \n Without loss of generality we assume that $x\\neq c$. If $B$ is a 2-block, then by Theorem \\ref{strongF_3}, $B^2$ contains an $xy$-hamiltonian path $P_B$ containing an edge $cy'$ where $y'$ is a neighbor of $c$ in $B$. Note that $y'=x$ or $c=y$ is possible. If $B=K_2$, then $B=xy=y'c$ and $P_B=xy$ is an $xc$-hamiltonian path in $B^2$. By the induction hypothesis $(G-B)^2$ contains a $cc'$-hamiltonian path $P_G$ where $c'$ is a neighbor of $c$ in $G-B$. Then $P=P_B\\cup P_G-cy'+y'c'$ is an $xy$-hamiltonian path in $G^2$ as required.\n \n \\item $B$ has two cutvertices $c_1$, $c_2$.\n \n We denote by $G_1$, $G_2$ the two components of $G-B$ such that $c_i\\in V(G_i)$ and let $c'_i$ be a neighbor of $c_i$ in $G_i$, $i=1,2$. By the induction hypothesis $(G_i)^2$ contains a $c_ic'_i$-hamiltonian path $P_{G_i}$, $i=1,2$.\n \n \\begin{itemize}\n  \\item[a)] $c_i\\notin \\{x,y\\}$ ($x$ and $y$ are not cutvertices).\n\n By Theorem \\ref{F_4}, $B^2$ contains an $xy$-hamiltonian path $P_B$ containing the edges $c_iz_i$ where $z_i$ is a neighbor of $c_i$ in $B$, $i=1,2$. Note that $z_i\\in\\{x,y\\}$ is possible. \n \n  \\item[b)] Up to symmetry $c_1=x$ and $c_2\\neq y$ (either $x$ or $y$ is a cutvertex of $G$).\n  \n  By Theorem \\ref{strongF_3}, $B^2$ contains an $xy$-hamiltonian path $P_B$ containing the edges $c_iz_i$ where $z_i$ is a neighbor of $c_i$ in $B$, $i=1,2$. Note that $z_1=c_2$ or $z_2=y$ is possible. \n \n \\item[c)] $c_1=x$ and $c_2=y$ (similarly $c_1=y$ and $c_2=x$).\n \n By Theorem \\ref{strongF_3ends}, $B^2$ contains an $xy$-hamiltonian path $P_B$ containing either the edges $c_iz_i$ where $z_i$ is a neighbor of $c_i$ in $B$, $i=1,2$, or the edges $c_1z_1$, $uv$ where $z_1$ is a neighbor of $c_1$ in $B$ and $u,v$ are neighbors of $c_2$ in $B$. \n \\end{itemize}\n \n In all cases except the case c), if $uv$ is the edge of $P_B$, $$P=P_{G_1}\\cup P_B\\cup P_{G_2}-\\{c_1z_1,c_2z_2\\}\\cup\\{c'_1z_1,c'_2z_2\\}$$ is an $xy$-hamiltonian path in $G^2$ as required. \n \n It remains to find an $xy$-hamiltonian path in $G^2$ if $uv$ is the edge of $P_B$. \n \n If $G_2=K_2=c_2u_2$, then $$P=P_{G_1}\\cup P_B\\cup -\\{c_1z_1,uv,c_2u_2\\}\\cup\\{c'_1z_1,u_2u, u_2v\\}$$ is an $xy$-hamiltonian path in $G^2$ as required.    \n \n If $G_2\\neq K_2$, then we prove that $(G_2)^2$ contains a hamiltonian cycle $C$ containing edges $c_2u_2$, $c_2v_2$ of $G_2$. Let $B_1,B_2,...,B_k$ be all 2-blocks of $G_2$ containing $c_2$. By Theorem \\ref{2-blockcycle}, for  $i=1,2,...,k$, $(B_i)^2$ contains a hamiltonian cycle $C'_i$ containing three different edges $c_2u_2^i$, $c_2v_2^i$, $y_iy'_i$ of $B_i$ where $y_i$ is the second cutvertex of $G_2$ in $B_i$ if it exists.\n \n If $y_i$ exists, then we denote by $H_i$ a component of $G_2-(B_i-y_i)$ containing $y_i$. By the induction hypothesis $(H_i)^2$ contains a $y_id_i$-hamiltonian path $P_i$ where $d_i$ is a neighbor of $y_i$ in $H_i$. Then we set $C_i=C'_i\\cup P_i-y_iy'_i+y'_id_i$. If $y_i$ does not exist, then we set $C_i=C'_i$.\n \n Let $T$ be the set of all leaves of $G_2$ adjacent to $c_2$. Then we find a cycle $C$ on $\\cup_{i=1}^{k}V(C_i)\\cup T$ by appropriately replacing edges $c_2u_2^i$, $c_2v_2^i$ with edges of $G^2$ joining $u_2^i$, $v_2^i$ in different $C_i$ and leaves adjacent to $c_2$ (similarly as in the proof of Theorem \\ref{hamiltonian}) such that we preserve two edges ($c_2u_2^i$, $c_2v_2^i$ or $c_2l_1$, $c_2,l_2$ where $l_1,l_2$ are two leaves of $G_2$ adjacent to $c_2$) as $c_2u_2$, $c_2v_2$. \n  \n Now $$P=P_{G_1}\\cup P_B\\cup C-\\{c_1z_1,uv,c_2u_2,c_2v_2\\}\\cup\\{c'_1z_1,u_2u, v_2v\\}$$ is an $xy$-hamiltonian path in $G^2$ as required.    \n\n\n\\end{itemize}\n\n\\bigskip \n \n (B) Suppose that $x$ and $y$ are in different blocks of $G$.\n\n Let $P_G$ be any $xy$-path in $G$ and $c\\in V(P_G)\\setminus\\{x,y\\}$ be a cutvertex of $G$. Let $K$ be the component of $G-c$ containing $x$, $G_y=G-V(K)$ and $G_x=G-G_y$. Clearly $G_x\\cup G_y=G$ and $G_x\\cap G_y=c$.  If $G_x$, $G_y$ are isomorphic to $K_2$, then we set $P_x=G_x$, $P_y=G_y$, respectively. If $G_x$, $G_y$ are 2-blocks, then $(G_x)^2$, $(G_y)^2$ contains an $xc$-hamiltonian path $P_x$, a $cy$-hamiltonian path $P_y$ by Theorem \\ref{strongF_3}, respectively. We proceed by induction on $n$, where $n$ is the number of blocks of $G$, $n\\geq 2$.\n  \n First assume that $G$ has exactly 2 blocks. Hence $G_x$, $G_y$ are isomorphic to $K_2$ or 2-blocks and $P=P_x\\cup P_y$ is an $xy$-hamiltonian path in $G^2$ as required.\n \n Now suppose that the statement of Theorem \\ref{hamconnected} is true for every graph with $n$ blocks and $G$ is a graph with $n+1$ blocks, $n\\geq 2$. If $G_x$, $G_y$ is not a block, then by the induction hypothesis $(G_x)^2$, $(G_y)^2$ contains an $xc$-hamiltonian path $P_x$, a $cy$-hamiltonian path $P_y$, respectively. Then $P=P_x\\cup P_y$ is an $xy$-hamiltonian path in $G^2$ as required.\n  \n\\bigskip \n\n Now it remains to prove that if there is a nontrivial bridge of $G$, then  $G^2$ is not hamiltonian connected and if $G$ contains a block containing more than 2 cutvertices, then there is a graph $G'$ such that $\\mbox{bc}(G)$ and $\\mbox{bc}(G')$ are isomorphic but $(G')^2$ is not hamiltonian connected. \n \n Clearly, if there exists a nontrivial bridge $xy$ in $G$, then there is no $xy$-hamiltonian path in $G^2$ and $G^2$ is not hamiltonian connected.\n \n \\begin{figure}[ht]\n\\begin{center}\n\\end{center}\\caption{Graphs without $xy$-hamiltonian path in the square}\\label{Counter2}\n\\end{figure} \n\n Finally assume that $G$ contains a block $B$ containing $r$ cutvertices, where $r>2$. Then we exhange $B$ with a cycle $C_r$ and all other blocks with arbitrary blocks to get a class of graphs $G'$ such that $\\mbox{bc}(G')$ and $\\mbox{bc}(G)$ are isomorphic. Clearly the square of every such graph $G'$ does not contain a hamiltonian path between arbitrary two cutvertices of $G'$ in $C_r$ and hence $(G')^2$ is not hamiltonian connected, e.g. with Figure \\ref{Counter2}, where $r=3$ and $H_1,...,H_3$ are arbitrary connected graphs with at least one edge.\n\\end{proof}\n\nSimilarly as for Theorem \\ref{hamiltonian} we state the following algorithm to verify conditions of Theorem \\ref{hamconnected}.\n\n\\bigskip\n\n\\noindent\n\\emph{ALGORITHM 2:} \n\nLet $G'=G-S$ where $S$ is the set of all endblocks of $G$. Let $\\mbox{cvn}_G(B)$ be the number of cutvertices of $G$ in $B$.\n\n\\noindent\nSTART\n\nFind an endblock $B$ of $G'$.\n\n\\begin{itemize}\n \\item If $B$ is a bridge of $G'$, then $B$ is a nontrivial bridge of $G$ and $G^2$ is not hamiltonian connected. STOP.\n \\item Let $B$ be a 2-block.  \n   \\begin{itemize}\n     \\item if $\\mbox{cvn}_G(B)>2$, then $G^2$ may not be hamiltonian connected (e.g. see Figure \\ref{Counter2}). STOP.\n     \\item if $\\mbox{cvn}_G(B)\\leq 2$, then $G'=G'-B$.\n        \\begin{itemize}\n         \\item if $G'=\\emptyset$, then $G^2$ is hamiltonian connected. STOP.\n         \\item if $G'\\neq\\emptyset$, then go to START.\n        \\end{itemize}\n   \\end{itemize}\n\\end{itemize}\n\nIn both algorithms in this paper, determining blocks and especially endblocks and bridges, cutvertices, block-cutvertex graphs, and the parameters $\\mbox{bn}(i)$, $\\mbox{cvn}_G(B)$ can be determined in polynomial time. \n\nAs a consequence, polynomial running time in Algorithm 2 is guaranteed. For, determining (potentially) not being Hamiltonian connected, can be determined instantly once a nontrivial bridge, a block with more than 2 cutvertices has been found. And deleting an endblock reduces the size of $G'$ linearly. \n\nNow consider the running time of Algorithm 1. The first decision to be made is whether $P_0$  is a forest of caterpillars \u2013 this can be done in linear time. After that, at every step 'one chooses a 2-block $B$ as a component of $G_0-H$ or an endblock of $G_0-H$ where $H$ is the union of all 2-blocks for which the labelling $m_i(B_t)$ is already set'. Clearly, identifying such $B$ can be done in linear time. The same applies to working through the cases for defining the various values of $m_i(B)$. \n\nSummarizing, it follows that both algorithms run in polynomial time. We note however, that these algorithms can only decide the existence or potential non-existence of hamiltonian cycles or hamiltonian paths in the square of graphs under consideration; they do not construct any such cycle or path.\n\n\n\\section{Conclusion}\nThe main results of this paper are Theorem \\ref{hamiltonian} and Theorem \\ref{hamconnected}.\nAs we mention in Introduction Fleischner in \\cite{Fle} proved that in the square of graphs hamiltonicity and vertex-pancyclicity are equivalent concepts, and so are hamiltonian connectedness and panconnectedness. Hence we proved in fact that for graphs satisfying assumptions of Theorem \\ref{hamiltonian}, Theorem \\ref{hamconnected} the square of these graphs is vertex-pancyclic, panconnected, respectively.\n\\medskip\n\n As easy corollary of Theorem \\ref{hamconnected}  we get the following result.\n\n\\begin{corollary}\n\\label{Block-chain}\n Let $G$ be a block-chain. Then $G^2$ is panconnected if and only if every innerblock of $G$ is a 2-block. \n\\end{corollary}\n\nMoreover Corollary \\ref{Block-chain} is also the answer to Problem 1 stated by Chia et al. in \\cite{ChiaOngTan} that for a graph $G$ with only two cutvertices it is true that $G^2$ is pannconnected if and only if the unique block containing the two cutvertices is not the complete graph on two vertices.\n\n\\medskip\n\n\\noindent {\\bf Acknowledgements}.\nThis work was partly supported by the European Regional Development Fund (ERDF), project NTIS - New Technologies for Information Society, European Centre of Excellence, CZ.1.05\/1.1.00\/02.0090.\n\nThe first author was partly supported by project GA20-09525S of the Czech Science Foundation. The second author was supported in part by FWF grant P27615.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Basic Motivation and Structure for the Generalized Uncertainty~Principle}\n\n In this opening section, we review the ideas and motivations underlying the generalized uncertainty principle (GUP) approach to quantum gravity. GUP is a phenomenological approach to quantum gravity which introduces an absolute minimal length in the theory. Many different approaches to combining quantum mechanics and gravity are thought to require a minimal length~\\cite{vene,amati,amati2,gross,maggiore,garay,KMM,adler,scardigli}. There is a simple physical argument for this. From~the Heisenberg uncertainty relationship (i.e.,~$\\Delta x \\Delta p \\ge \\frac{\\hbar}{2}$), one sees that quantum mechanics gives the following relationship between uncertainty in position and momentum $\\Delta x \\sim \\frac{Const.}{\\Delta p}$. From~the gravity side, one argues that as one tries to probe smaller distances, one needs to go to higher center of mass energies\/momenta. At~some point, the energy\/momentum will be large enough that one will form a micro-black hole whose event horizon size can be estimated by the Schwarzschild radius $r_{Sch} = 2 G \\Delta E \/ c^2$, whereas in this expression, $\\Delta E$ has replaced the conventional mass of the black hole, $M$. Now, further setting $c=1$, replacing $r_{Sch}$ with $\\Delta x$ and $\\Delta E$ with $\\Delta p$, this relationship becomes $\\Delta x = 2 G \\Delta p$, i.e.,~there is now a linear relationship between $\\Delta x$ and $\\Delta p$. It should be noted that since we have a set $c=1$ mass, the energy and momentum are interchangeable. If~one combines this linear relationship from gravity with the inverse relationship from quantum mechanics, one finds $\\Delta x \\sim \\frac{\\hbar}{\\Delta p} + G \\Delta p$, (ignoring the factors of 2). The~interplay between the linear term and the inverse term lead to a minimum in $\\Delta x$ at $\\Delta p_m \\sim \\sqrt {\\hbar \/G}$ of $\\Delta x_m \\sim \\sqrt{\\hbar G}$.\n \n The minimal $\\Delta x$ that comes from the GUP, as~described above, is one way to avoid the point singularities that occur in certain solutions in general relativity such as black hole spacetimes. If~one cannot resolve distances smaller than $\\Delta x_m$, this may lead to the avoidance of the singularities of general relativity. This is the hope for theories of quantum gravity---that they will allow one to avoid the singularities of classical general relativity. Another approach to avoid these singularities of classical black hole solutions is non-commutative geometry~\\cite{nicolini}. In~this approach, one proposes that coordinates do not commute with one another. For~example, $[X,Y] \\ne 0$ or $[Y,Z]\\ne 0$. We will later show that the types of GUP models favored by our analysis are also connected to non-commutative geometry~theories. \n \n One of the strengths of the phenomenological GUP approach to quantum gravity is that it offers the possibility to  make experimental tests of quantum gravity---to experimentally check whether there is a minimal distance resolution, $\\Delta x_m$, as~implied by the above arguments, and if so, what is the size of this minimal distance resolution? Some tests of the GUP scenario rely on astrophysical phenomena. For~example, reference~\\cite{AC-nature} proposed a test of minimal lengths based on the dispersion of high-energy photons coming from short gamma ray bursts. The~idea of~\\cite{AC-nature} was that having a minimal distance scale in ones' theory would alter the standard energy--momentum relationship of special relativity, $E^2 = p^2 c^2 + m^2 c^4$. This altered energy--momentum relationship would then lead to an energy-dependent speed of light in the vacuum, which in turn would lead to photons of different energies dispersing or spreading out as they travel long distances through the vacuum. In~2009~\\cite{abdo}, the Fermi gamma ray satellite detected high energy photons coming from a distant gamma ray burst. Using the analysis of~\\cite{AC-nature}, the observation by the Fermi satellite was able to place bounds on the deviations from $E^2 = p^2 c^2 + m^2 c^4$ due to a minimal distance scale. Surprisingly, if the deviations from the special relativistic photon energy and momentum relationship were linear in energy (i.e.,~$p^2 c^2= E^2 [1 + \\zeta (E\/E_{QG})+\\ldots]$ with $E_{QG}$ being the quantum gravity scale and $\\zeta$ is a parameter of order $1$) then the observations of~\\cite{abdo} implied the bound $E_{QG} > E_{Planck}$, i.e.,~that there was no deviation to energies beyond the Planck energy scale. Or~putting this in terms of length $l_{QG} < l_{Planck}$, which is counter to the expectation that hints of quantum gravity should occur before reaching the~Planck-scale.\n \n There are also tabletop and small-scale laboratory test proposals for testing for effects connected with the minimal distance scale coming from GUP. In~the works~\\cite{vagenas,vagenas2}, the proposal was made to use the Lamb shift, Landau levels, quantum tunneling in scanning tunneling microscopes. This work showed than using these tabletop experiments, one could put a bound on the parameter $\\beta$ which were, in~units of Planck momentum squared, of~$\\beta < 10^{36}$, $\\beta < 10^{50}$ and $\\beta < 10^{21}$ for the Lamb shift, Landau levels and tunneling , respectively. There is also a proposal~\\cite{bekenstein} to test Planck scale physics with a tabletop cryogenic optical step-up where the optical photon's momentum is coupled to the center of mass motion of a macroscopic transparent block in such a way that the block is displaced in space by approximately a Planck length. In~the works~\\cite{sujoy,sujoy2,sujoy3}, it was shown that one could use detailed studies of wavefunctions of large molecule to probe Planck-scale physics. Finally there is recent work~\\cite{bosso} which looked at using gravitational waves to put bounds on the parameters coming from GUP models. All of these various experimental approaches are welcome since they hold out the hope that one may experimentally probe Planck-scale~physics.\n\nWe now briefly review some of the basic background behind GUP models and particularly focus on the role that modified operators have in determining whether or not there is a minimal length. The~uncertainty relationship between two physical quantities is closely tied to the commutation relationship between the operators which represent this quantities. In~general, for two operators ${\\hat A}$ and ${\\hat B}$, one has the following relationship between the uncertainties and the commutator:\n\\begin{equation}\n    \\label{dAdB}\n    \\Delta A \\Delta B \\ge \\frac{1}{2i} \\langle [ {\\hat A} , {\\hat B}] \\rangle ~.\n\\end{equation}\nwhere the uncertainties are defined as $\\Delta A = \\sqrt{\\langle {\\hat A} ^2 \\rangle - \\langle {\\hat A} \\rangle ^2 }$ and similarly for $\\langle {\\hat B} \\rangle $.\nFor standard position and momentum operators in position space ${\\hat x} = x$ and ${\\hat p} = -i \\hbar \\partial _x$,  one obtains the usual commutator $[{\\hat x} , {\\hat p}] = i \\hbar$ which then implies the standard uncertainty~principle, \n$$\\Delta x \\Delta p \\ge \\frac{\\hbar}{2}$$ \nor the operators are ${\\hat x} = i \\hbar \\partial _p$ and ${\\hat p} = p$ in momentum~space. \n\nTo obtain a GUP characterized by $\\Delta x \\sim \\frac{\\hbar}{\\Delta p} + \\beta \\Delta p$, \\cite{KMM} proposed the following modified commutator:\n\\begin{equation}\n    \\label{KMMxp}\n    [ {\\hat X} , {\\hat p}] = i \\hbar (1 + \\beta {\\hat p}^2)  ~.\n    \\end{equation}\n \nNote that following~\\cite{KMM}, we replace $G$ by a phenomenological parameter $\\beta$ which characterizes the scale where quantum gravity is important. Naively, the~quantum gravity scale would be set by $G, \\hbar$ and $c$, but~in large extra dimension models~\\cite{ADD,ADD2} or brane world models~\\cite{RS,RS2,Gog,Gog2,Gog3}, the quantum gravity scale can be lower than the typical Planck scale. In~these brane world models, $\\beta$ would be set by the value of the higher dimensional Newton's constant and the size of the extra dimensions. Furthermore, in \\eqref{KMMxp}, the position operator is capitalized while the momentum operators, on~both the left and right sides of \\eqref{KMMxp}, are not. This is because in reference~\\cite{KMM}, the choice was made that in order to obtain the modified commutator in \\eqref{KMMxp}, they would modify the position and momentum operators as\n\\begin{equation}\n\\label{xp1}\n{\\hat X} =i \\hbar (1 + \\beta p^2) \\partial_p ~~~{\\rm and}~~~ {\\hat p} = p  \n\\end{equation}\nwhere only the position operator is modified. This choice of operators in \\eqref{xp1}\nis absolutely crucial to obtaining a minimal length. Using \\eqref{KMMxp} in  \\eqref{dAdB} gives:\n\\begin{equation}\n    \\label{min-Xp}\n    \\Delta X \\Delta p \\ge \\frac{\\hbar}{2}(1+ \\beta \\Delta p ^2)\n\\end{equation}\nIn arriving at \\eqref{min-Xp}, we are taking the center of mass coordinates with $\\langle {\\hat p} \\rangle =0$. Dividing both sides by $\\Delta p$ gives:\n\\begin{equation}\n    \\label{min-x}\n    \\Delta X \\ge \\frac{\\hbar}{2}\\left( \\frac{1}{\\Delta p} + \\beta \\Delta p \\right) ~,\n\\end{equation}\nso that one arrives at the type of GUP described in the opening paragraph which definitely leads to a minimal length. Minimizing $\\Delta X$ in \\eqref{min-x} shows it has a minimal length at \\mbox{$\\Delta p = \\sqrt{1\/\\beta}$} which then yields $\\Delta X_{min} = \\hbar \\sqrt{\\beta}$.\n\n\nIn contrast to the work on GUP from reference~\\cite{KMM}, as outlined above, most recent works have followed a different approach, such as that found in  \\cite{pedram} where the {\\it same} modified commutator of \\eqref{KMMxp} was obtained by modifying the momentum operator {\\it but} not the position operator. By~taking the operators to be of the form:\n\\begin{equation}\n\\label{xp2}\n    {\\hat x}= i \\hbar \\partial _p ~~~{\\rm and}~~~ {\\hat P} = p\\left( 1 + \\frac{\\beta}{3}p^2 \\right)~, \n\\end{equation}\none can see that plugging these into the commutator immediately leads to the same right hand side of the commutator as in \\eqref{KMMxp} where capitals indicate the operator is modified. However, the~uncertainty between the two operators in \\eqref{xp2} is subtly different from the two operators in \\eqref{xp1}. Using the operators in \\eqref{xp2} to calculate the uncertainty relationship via \\eqref{dAdB}~gives:\n\\begin{equation}\n\\label{min-xP}    \n\\Delta x \\Delta P \\ge \\frac{\\hbar}{2}(1+ \\beta \\Delta p ^2).\n\\end{equation}\nAlthough  this relationship superficially appears to be the same as before, now the left hand side has $\\Delta P$ for the modified momentum, while the right hand side has $\\Delta p$ for the standard momentum. Thus, instead of \\eqref{min-x} one obtains:\n\\begin{equation}\n    \\label{min-x2}\n    \\Delta x \\ge \\frac{\\hbar}{2} \\left( \\frac{1}{\\Delta P} + \\frac{\\beta \\Delta p ^2}{ \\Delta P} \\right). \n\\end{equation}\nIn contrast to \\eqref{min-x}, it is not clear whether the interplay between $\\Delta p$ and $\\Delta P$ on the right hand side of \\eqref{min-x2} will lead to a minimum. If~the second term  $\\beta \\frac{\\Delta p^2}{\\Delta P}$ is increasing with $\\Delta p$, then there is a minimum; but if this term decreases with $\\Delta p$, then there is no minimum. Since the highest power of $p$ in ${\\hat P} =p (1 + \\frac{1}{3} \\beta p^2)$ is $p^3$, one can show that $\\Delta P \\approx \\beta \\Delta p ^3$, which implies that the second term $\\beta \\frac{\\Delta p^2}{\\Delta P}$ will go like $\\frac{1}{\\Delta p}$, i.e.,~decreasing with $\\Delta p$ and thus there is no minimum in~position. \n\nTo explicitly see the difference between the GUP from \\eqref{min-x} versus the GUP from \\eqref{min-x2}, one can plot the uncertainty in position versus the uncertainty in momentum in a family of test functions for each operator pair. In~Figure~\\ref{fig1}, we plot $\\Delta X$ versus $\\Delta p$ from \\eqref{min-x} for $\\beta = 0.01$. One can see that there is a minimum in $\\Delta X$ around $\\Delta p = 1\/\\sqrt{\\beta}$ as~expected.\n\\begin{figure}[H]\n    \\includegraphics[scale=0.8]{kmm.jpg}\n    \\caption{The relationship between $\\Delta X$ and $\\Delta p$ for the GUP from \\eqref{min-x} using $\\beta =0.01$. As~expected, a minimum length occurs at approximately $\\Delta p = 1\\sqrt{\\beta}$, and~after this minimum, $\\Delta X$ increases with $\\Delta p$.}\\label{fig1}\n\\end{figure}\nIn Figure~\\ref{fig2}, we plot $\\Delta x$ versus $\\Delta p$ from \\eqref{min-x2} again with $\\beta = 0.01$. In~contrast to Figure~\\ref{fig1}, Figure~\\ref{fig2} has no minimum $\\Delta x$ and in fact looks like the standard relationship between $\\Delta x$ and $\\Delta p$ from quantum~mechanics.  \n\\begin{figure}[H]\n    \\includegraphics[scale=0.8]{pendram.jpg}\n    \\caption{The relationship between $\\Delta x$ and $\\Delta p$ for the GUP from \\eqref{min-x2} using $\\beta =0.01$. This GUP has no minimal length and essentially behaves in a manner resembling the standard uncertainty~principle. }\\label{fig2}\n\\end{figure}\nThere are some subtleties in how Figures~\\ref{fig1} and \\ref{fig2} were obtained. First, for~both figures, we used the lower bound (i.e.,~the equal sign) of Equations \\eqref{min-x} and \\eqref{min-x2}. Second, in~calculating $\\Delta P$ for use with \\eqref{min-x2}, we used a test Gaussian wavefunction in momentum space, $\\Psi (p) \\propto e^{p^2\/2 \\sigma}$. The~reason for using a specific test wave function, rather than making a general argument that would apply regardless of the form of the wavefunction, was the complicated relationship between ${\\hat P}$ and ${\\hat p}$. For~a GUP like that in \\eqref{min-Xp}, one has the same $\\Delta p$ on both the left and right hand side of the equation, regardless of the wavefunction. This is the result of not modifying the momentum operator in this GUP model. On~the other hand for a GUP such as \\eqref{min-xP}, the momentum uncertainty is different between the left and right hand sides of the equation, and~there is not a simple relationship between $\\Delta P$ and $\\Delta p$. Thus, for this case, we picked a specific wavefunction to calculate $\\Delta P$ and $\\Delta p$ and check how the uncertainty principle worked out. One could choose a different test wavefunction instead of the Gaussian, but~the results would be qualitatively similar to that  shown in Figure~\\ref{fig2}. Note that the effects for the type of GUP shown in Figure~\\ref{fig2} are most pronounced in the $\\Delta p \\to 0$ limit. The~above calculations are a prelude and check for the examples of GUPs given in the following~section.  \n\nFinally, even though the momentum operator associated with \\eqref{min-x} is the standard one from quantum mechanics, the~change in the position operator forces one to change the measure of integration in $p$ in order to ensure that ${\\hat X}$ and ${\\hat p}$ are symmetric~\\cite{KMM}. This change in the measure of the momentum integration amounts to $\\int \\ldots dp \\to \\int \\ldots \\frac{dp}{1+ \\beta p^2}$. This modifies the normalization constants but does not qualitatively affect the behavior $\\Delta X$ in the limit as $\\Delta p \\to \\infty$.  All of these issues are more fully examined in~\\cite{BLS}.\n\nThe main point of this section is to show that many works on GUP following the seminal work of KMM~\\cite{KMM} and make a different choice for how the operators are modified, which does not necessarily result in a minimum length scale. KMM~\\cite{KMM} modified the position operator, but~left the momentum operator unchanged, whereas many other works chose to modify the momentum but leave the position operator unchanged. Thus, certain choices of modifying the operators are wrong in the sense that they do not lead to a minimal length. From~the above, we conclude that, when it comes to determining whether a theory has a minimum length scale, {\\bf how the operators are modified is more important than how the commutator is modified}. In~short, a modified commutator is insufficient to guarantee a minimum length scale; in fact, a~modified commutator is not necessary at all, as shown in the next~section.\n \n\n\\section{A GUP with an  Unmodified~Commutator}\n\nWe now move on to give details of how a modified commutator is not necessary to achieve a minimum length scale. If~we do not modify the commutator nor modify the operators, we would obviously end up with ordinary quantum mechanics, which does not have a minimal length. Thus, we want modified position and momentum operators ${\\hat X}$ and ${\\hat P}$ which give rise to the usual commutator $[{\\hat X} , {\\hat P}] = i \\hbar$,  \\cite{BJLS,mlake}. This in turn leads to a standard looking uncertainty relationship but now in terms of the modified operators $\\Delta X \\Delta P \\ge \\frac{\\hbar}{2}$. In~order to have a minimum $\\Delta X$, one needs to modify ${\\hat P}$ so that $\\Delta P$ is either capped at some constant value or decreases. In~reference~\\cite{BJLS}, a~GUP of this kind was constructed by defining a modified momentum by\n\\begin{equation}\n    \\label{p-tanh}\n    {\\hat P} = p_M \\tanh \\left( \\frac{{\\hat p}}{p_M} \\right)~,\n\\end{equation}\nwhere $p_M$ is the maximum cap on the modified momentum. This maximum momentum, $p_M$, is an upper limit to the momentum in the relationship $E^2 = p^2 + m^2$. This cap in momentum also implies a cap in the energy $E$. This connection between a cap on momentum and a cap on the energy of a single particle, as~well as how this relates to modified position and time operators, is discussed in greater detail in~\\cite{BJLS}. Picking the modified position to have the form:\n\\begin{equation}\n\\label{x-tanh}\n   {\\hat X} = i\\hbar\\cosh^2{\\left(\\frac{{\\hat p}}{p_M}\\right)}\\partial_p \n\\end{equation}\nthen leads to the standard looking commutator $[{\\hat X} , {\\hat P}] = i \\hbar$ as can be verified by the substitution of \\eqref{p-tanh} and \\eqref{x-tanh} into the commutator. Another variant with a capped momentum is given in~\\cite{BJLS} by\n\\begin{equation}\n    \\label{p-tan}\n    {\\hat P}  = \\frac{2 p_M}{\\pi } \\arctan \\left( \\frac{\\pi p}{2 p_M} \\right)~,\n\\end{equation}\nand for the modified position operator, one has:\n\\begin{equation}\n    \\label{x-tan}\n    {\\hat X} = i \\hbar \\left[ 1+ \\left( \\frac{\\pi p}{2 p_M}\\right)^2 \\right] \\partial _p ~.\n\\end{equation}\nBoth forms of the modified momenta given in \\eqref{p-tanh} and \\eqref{p-tan} lead to the result $\\Delta P \\le p_M$ which then gives $\\Delta X \\ge \\frac{\\hbar}{2 p_M}$. Despite modified operators \\eqref{p-tanh} and \\eqref{x-tanh} or \\eqref{p-tan} and \\eqref{x-tan} leading to a minimum $\\Delta X$, there is a difference with how this is achieved compared to the KMM modified operators of \\eqref{xp1}. Plotting $\\Delta X$ versus $\\Delta p$ coming from \\eqref{p-tanh}, \\eqref{x-tanh}, the~associated uncertainty principle gives the result shown in Figure~\\ref{fig3}; the graph of $\\Delta X$ versus $\\Delta p$ for the operators \\eqref{p-tan} and \\eqref{x-tan} looks very similar. In~Figure~\\ref{fig1}, which shows the plot of the Kempf--Mangano--Mann GUP~\\cite{KMM}, the~minimum in $\\Delta X$ is reached at some $\\Delta p$ and thereafter $\\Delta X$ linearly increases with $\\Delta p$. In~contrast, the~GUP shown in Figure~\\ref{fig3} asymptotically approaches the minimum $\\Delta X$. \n\\begin{figure}[H]\n    \\includegraphics[scale=0.8]{mup.jpg}\n    \\caption{The relationship between $\\Delta X$ and $\\Delta p$ for the modified operators \\eqref{p-tanh} and \\eqref{x-tanh} and the associated GUP using $p_M =0.25$}\\label{fig3}\n\\end{figure}\nThe GUP model given by Figure~\\ref{fig1} has a physical motivation for its form: $\\Delta x$ inversely proportional to $\\Delta p$ for small momentum where quantum mechanics dominates and has $\\Delta x$ proportional to $\\Delta p$ for a large momentum where quantum gravity is thought to dominate. The~motivation for this behavior was laid out in the introduction and relied on the formation of micro black holes at large momentum\/small distances. In~contrast the GUP model given by Figure~\\ref{fig3}, which has  $\\Delta x$ inversely proportional to $\\Delta p$ for small momentum, but~at a large momentum has $\\Delta x$ approach a constant value asymptotically, does not have a simple, physical~motivation. \n\nThe main take-away message of this section was to emphasize that the most important factor in whether or not a given GUP leads to a minimum length is how the operators are~modified.  \n\n\\section{Connection to Non-Commutative Geometry and Running of \\boldmath$G$}\n\nIn this section, we tie the above considerations about GUPs to another approach to quantizing gravity and a potential physical consequence of quantum gravity. The~other approach to quantizing gravity is non-commutative geometry and the physical consequence is the running of the coupling constant of a theory---in this case, the running of Newton's $G$.\n\nIn general, for non-commutative spacetimes, one has a non-trivial commutation relationship between coordinates of the form $[X_i, X_j] = i \\theta _{ij}$ where $\\theta _{ij}$ is an anti-symmetric matrix~\\cite{nicolini}. This non-commutativity between coordinates implies an uncertainty relationship $\\Delta X_i \\Delta X_j \\ge \\frac{1}{2} |\\theta _{ij}|$, which in turn implies that there is a minimal area and volume---one cannot make $\\Delta X \\Delta Y$ or $\\Delta X \\Delta Y \\Delta Z$ (for example) arbitrarily small due to the non-commutativity between the coordinates. This has the effect, similar to GUP models, of~preventing the formation of point singularities that occur in a black hole and other solutions of classical general relativity. It has previously been noted~\\cite{KMM} that certain GUP models lead to modified position operators in three dimensions (3D)  which naturally result in the non-commutativity of the modified position operators. In~one spatial dimension (1D), one does not encounter this non-commutativity since a single operator will always commute with itself. However, in~three dimensions, different coordinates may not commute with one another, e.g.,~$[{\\hat X} , {\\hat Z}] \\ne 0$. This is the reason behind the matrix $\\theta _{ij}$ being~antisymmetric. \n\nAs a specific example of the GUP-non-commutative geometry connection, one can look at the 3D version of the modified position operators of~\\cite{KMM}. Letting the position and momentum in \\eqref{xp1} go from 1D to 3D (i.e.,~${\\hat X} \\to {\\hat X}_i$, ${\\hat p} \\to {\\hat p}_i$ and $\\partial_p \\to \\partial _{p_i}$ gives a 3D version of \\eqref{xp1} of the form:\n\\begin{equation}\n\\label{x13d}\n{\\hat X}_i =i \\hbar (1 + \\beta |{\\vec p}|^2) \\partial_{p_i} ~.\n\\end{equation}\nWith this 3D version of \\eqref{xp1}, the coordinate commutator becomes:\n\\begin{eqnarray}\n\\label{x13d-a}\n[{\\hat X}_i , {\\hat X}_j ] &=& -2 \\beta \\hbar ^2 (1 + \\beta |{\\vec p}|^2 \\partial_{p_j} ) (p_i \\partial _{p_j} - p_j \\partial _{p_i} )\\nonumber \\\\\n&=& 2 i \\hbar \\beta ({\\hat p}_i {\\hat X}_j  -{\\hat p}_j {\\hat X}_i )  ~,\n\\end{eqnarray}\nwhich implies a connection to the non-commutative parameter of $\\theta_{ij} =  2 \\hbar \\beta ({\\hat p}_i {\\hat X}_j  -{\\hat p}_j {\\hat X}_i )$. As~required, this $\\theta_{ij}$ is antisymmetric. In~the second line in \\eqref{x13d-a}, we used \\eqref{x13d} to turn this back into an expression in terms of the (modified) position operator and (unmodified) momentum operator. The~result in \\eqref{x13d-a} was previously derived in~\\cite{KMM}.\n\nOne can also create 3D versions of the modified position operators from \\eqref{x-tanh} and \\eqref{x-tan} which take the form:\n\\begin{equation}\n\\label{x-tanh3d}\n   {\\hat X}_i = i\\hbar\\cosh^2{\\left(\\frac{|{\\vec p}|}{p_M}\\right)}\\partial_{p_i} ~,\n\\end{equation}\nand:\n\\begin{equation}\n    \\label{x-tan3d}\n    {\\hat X}_i = i \\hbar \\left[ 1+ \\left( \\frac{\\pi |{\\vec p}|}{2 p_M}\\right)^2 \\right] \\partial _{p_i} ~.\n\\end{equation}\nJust as in the case of the 3D modified position operators in \\eqref{x13d}, the~modified 3D position operators in \\eqref{x-tanh3d} and \\eqref{x-tan3d} also lead to a non-commutativity of the coordinates,  $[{\\hat X}_i, {\\hat X}_j] \\ne 0$. We do not give the specific form of $\\theta_{ij}$ for the modified 3D position operators from \\eqref{x-tanh3d} and \\eqref{x-tan3d} since we only want to make the point here that some GUPs (particularly those studied in this work) lead to non-commutative~geometry.\n\nIn quantum field theory, when interactions are quantized, one encounters the phenomenon that the coupling ``constants'' of the interaction become dependent on the energy\/momentum scale at which the effects of the interaction are measure. Colloquially one talks about  coupling constant ``running'' with the energy\/momentum scale. For~example, in~quantum electrodynamics, the fine structure constant, $\\alpha = \\frac{e^2}{\\hbar c}$, which measures the strength of the electromagnetic interaction, depends on the energy\/momentum scale, $E\/p$, at~which it is measured. Similarly, the weak and strong nuclear interactions have couplings which scale with energy\/momentum. \nOne can view the GUP approach to quantum gravity to be as running from Newton's constant $G$.  The~commutator from \\eqref{KMMxp} implies that the GUP parameter  $\\beta$ depends on the momentum of the interaction as $\\beta (p) = \\beta p^2$, i.e.,~the parameter $\\beta$ scales quadratically with momentum in this~case. \n\nTo translate this running of the phenomenological parameter $\\beta$ into a running of Newton's constant $G$. We simply recall that from the heuristic arguments $\\Delta x_m \\sim \\sqrt{\\hbar G}$ as well as in terms of $\\beta$, one has $\\Delta x_m \\sim \\hbar \\sqrt{\\beta}$. Thus, we have the connection between $G$ and $\\beta$ of $\\beta \\sim \\frac{G}{\\hbar}$. Thus, a $\\beta$ which ``runs'' with the momentum directly implies a running $G$. There are two things to note: (i) the usual running of a coupling like the fine structure parameter $\\alpha = \\frac{e^2}{\\hbar c}$ is usually logarithmic in perturbative quantum fields theory; (ii) there are differences between gravity and the other interactions that make it unclear that one can consistently define a running gravitational coupling, at~least in the usual approach of quantum fields theory, as detailed in the arguments of reference~\\cite{anber}. \n\n\\section{Summary and Conclusions}\n\nIn this short note, we examined the role that modifying the position and momentum operators plays in determining a minimum length and that focusing on modifying the commutator is insufficient. We examined models that had modified commutators with the same right hand sides, as~in \\eqref{KMMxp}, but~where the operators on the left hand side were different. We considered the case where the position was modified and the momentum remained unchanged (see \\eqref{xp1}) and we considered the case where the position remained the same and the momentum was modified (see \\eqref{xp2}). The~former case led to a minimum length while the latter case did not. This can be explicitly  seen by comparing \\mbox{Figures~\\ref{fig1} and \\ref{fig2}}. \nFinally, we presented a case where the position and momentum were modified, but~the commutator remained the same as the standard one from quantum mechanics---\\mbox{Equations \\eqref{p-tanh} and \\eqref{x-tanh}}. This system show that there can be a minimum length scale without modifying the commutation~relation.\n\nA general conclusion that can be distilled from the work in this paper is the following. {\\bf In order for a GUP to have a minimal length, the key factor is the modification of the operators rather than the modification of the commutator.} \n\nThe thrust of this paper was to argue that there are constraints on the specifics of how one can formulate a GUP to obtain a minimal length. This is welcome since while the phenomenological approach of GUPs has strengths (e.g.,~the possibility to confront ideas about quantum gravity with experiments and observations), one would also like a way to constrain and focus on the range of variations in the operators and commutator. Other recent works~\\cite{mlake1,mlake2} have also looked at ways to constrain the form of~GUPs. \n\nFinally, in the last section, we tied the type of GUP models discussed here with other models of quantum gravity and with other potential consequences of quantum gravity, namely non-commutative geometry and the ``running'' of the coupling strengths of interactions. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nQuantum computers have attracted much attention recently, mainly due to the rapid development of actual hardware \\cite{Barends2014, Bernien2017, Wright2019}.\nThe quantum computer that is to appear shortly is called noisy intermediate scale quantum devices, or in short, NISQ devices \\cite{Preskill2018}.\nWe expect NISQ devices to have $\\sim$100 of qubits with non-negligible noise in the near future.\nSuch devices are believed to be not simulatable by classical computers when the control precision of the qubits is sufficiently high \\cite{Harrow2017, Boixo2018, Neill2018, Bravyi2018}.\nIn this sense, NISQ devices have computational power that exceeds classical computers.\nMany researchers are actively developing ways to exploit their power for practical applications \\cite{Peruzzo2014, Kandala2017, Nam2019, Farhi2014, Otterbach2017, Mitarai2018, Havlicek2019}.\nHowever, we still suffer from the limited number of qubits available on actual devices and the limited depth of circuits that can be run while maintaining the resultant quantum state meaningful.\n\nIf techniques to decompose a quantum circuit to a smaller one are developed, they can extend the applicability of such devices.\nSmaller quantum circuits may refer to ones with the smaller number of qubits or gates.\nPeng \\textit{et al.} recently proposed a clustering approach based on a tensor network representation of a quantum circuit~\\cite{Peng2019}, which greatly progressed the technical development.\nThey showed that we can ``cut'' an identity gate, by sampling measure-and-prepare channels on a qubit according to a certain quasi-probability distribution.\nIn Ref.~\\cite{Mitarai2019}, we proposed methods to construct quantum circuits equivalent to the Hadamard test, which successfully reduces the depth of certain quantum circuits.\nThese techniques share a same idea in that they reconstruct a result of a coherent quantum operation from certain incoherent operations by combining the results obtained from them.\n\nAn approach which has the same flavor as the above have been utilized in the context of memory-efficient classical simulation of quantum circuits.\nSince the direct simulation of a quantum circuit with over 50 qubits breaks down due to the need of storing $2^{50}$ complex numbers in memory, the classical simulator must decompose the given quantum circuit to smaller ones, especially in the number of qubits.\nRefs. \\cite{Chen2018, Pednault2017} have provided one way for such decomposition, which ``cuts'' controlled-Z gates by separately simulating two cases where the control qubit is $\\ket{0}$ or $\\ket{1}$ and then combining them, and they performed classical simulation of over 50-qubit quantum circuits.\nA similar technique has been utilized by Bravyi \\textit{et al.} in Ref. \\cite{Bravyi2016} to remove a relatively small number of qubits from a large quantum circuit by replacing the qubits with a classical simulator.\nTheir approach can be viewed as ``space-like'' cut rather than the ``time-like'' cut proposed by Peng \\textit{et al} \\cite{Peng2019}.\nHowever, their techniques are intended to run on a classical computer and cannot be utilized for simulating a large quantum circuit with a small quantum computer.\n\nIn this work, we present a technique to perform ``space-like'' cut on a quantum computer.\nMore specifically, we present a way to decompose a controlled gate into a sequence of single-qubit operations which consists of projective measurements of Pauli $X$, $Y$, and $Z$ operators, and single-qubit rotations around x, y, and z-axes.\nWe note that our method does not generate any entanglement between the qubits as it is impossible to do so with such single-qubit operations.\nOur method only ``simulates'' effects of entanglement using classical post-processing and sampling.\nMore concretely, although entangling gates cannot be performed with local operations and classical communications in single-shot experiments as widely known \\cite{PhysRevA.52.3457}, we show that it is possible to perform a computational task of evaluating expectation values of the output of entangling circuits by sampling certain sets of gates and applying classical post-processing.\nThe overhead required for our proposed technique, which scales exponentially to the number of decomposition performed, gives a characterization of the entangling gates from a computational viewpoint, which is different from the existing theories of entanglement quantification in e.g. \\cite{Vidal_2000}.\n\nThe method proposed here can be considered as a generalization of our previous work \\cite{Mitarai2019} and a variant of the quantum circuit decomposition presented in Ref.~\\cite{Peng2019}.\nIt can also be viewed as a fully quantum version of the technique utilized in efficient classical simulation schemes \\cite{Chen2018, Pednault2017, Bravyi2016}. \nIn some cases, our method provides a better scaling against Ref. \\cite{Peng2019} when simulating a large quantum circuit with smaller ones.\nThe proposed technique is also useful when we want to apply two-qubit gates between a distant pair of qubits, which otherwise would require many swap operations to perform.\nThis work extends the applicability of NISQ devices whose circuit depth and connectivity are limited.\n\n\\section{Gate decomposition}\n\\subsection{Tensor network representation of quantum circuits}\nQuantum computation is completely specified with a quantum circuit, $U$, an initial state with its density matrix representation, $\\rho$, and an observable, $O$, measured at the output.\nGiven $U$, $\\rho$, and $O$, Any quantum computation can be represented by a tensor network \\cite{Shi2006,Markov2008,Vidal2003}.\nWe define the tensor representation of $U$, $\\rho$, and $O$ in the following manner.\n\nSuppose that our quantum computer has $n$ qubits.\nWe define a complete set of basis in the space of $2 \\times 2$ complex matrix and its dual as $\\{\\kket{e_i}\\}_{i=1}^{4}$ and $\\{\\bbra{e_i}\\}_{i=1}^{4}$ respectively, and assume orthonormality under the trace inner product; $\\bbraket{e_i|e_j}=\\delta_{ij}$.\nWe use the trace inner product, that is, for matrices $A$ and $B$, $\\bbraket{A|B} = \\mathrm{Tr}(A^\\dagger B)$.\nA density matrix $\\rho$ can be decomposed into the sum of $\\kket{e_{j_1}}\\otimes \\kket{e_{j_2}} \\otimes \\cdots \\otimes \\kket{e_{j_n}} = \\kket{e_{j_1}e_{j_2}\\cdots e_{j_n}}$ as \n\\begin{align}\n    \\kket{\\rho} = \\sum_{j_1,\\cdots, j_n} \\rho_{\\bm{j}} \\kket{e_{j_1}e_{j_2}\\cdots e_{j_n}},\n\\end{align}\nwhere $\\bm{j} = (j_1,j_2,\\cdots,j_n)$. \nWe refer to the elements $\\rho_{\\bm{j}} = \\bbraket{e_{j_1}e_{j_2}\\cdots e_{j_n}|\\rho}$ as the tensor representation of $\\rho$.\nAn observable $O$ can also be decomposed into the same form.\nNote that we can naturally assume tensor representations of observables and density matrices consist of real numbers because they are always Hermitian and we can choose the basis $\\{\\kket{e_i}\\}_{i=1}^{4}$ as Hermitian, e.g., we can use the Pauli matrices $\\{I, X, Y, Z\\}$ as the basis.\nTherefore, we assume $\\rho_i$ and $O_i$ are real henceforth.\nThe quantum circuit, $U$, transforms $\\rho$ into $U\\rho U^\\dagger$.\nWe define a corresponding superoperator $\\mathcal{S}(U)$ whose action is defined by $\\mathcal{S}(U) \\rho  = U\\rho U^\\dagger$.\nSuperoperator can be decomposed as,\n\\begin{align}\n    \\mathcal{S}(U) = \\sum_{j_1,\\cdots, j_n}\\sum_{k_1,\\cdots, k_n} \\mathcal{S}(U)_{\\bm{j}, \\bm{k}} \\kket{e_{j_1}\\cdots e_{j_n}}\\bbra{e_{k_1}\\cdots e_{k_n}}.\n\\end{align}\nNote that this decomposition is not limited to superoperators of unitary matrices, but also is applicable for any linear operator that acts on a density matrix.\nWe call $\\mathcal{S}(U)_{\\bm{j}, \\bm{k}} = \\bbra{e_{j_1}\\cdots e_{j_n}}\\mathcal{S}(U)\\kket{e_{k_1}\\cdots e_{k_n}}$ tensor representation of $\\mathcal{S}(U)$. \nWhen we use the Pauli operators as basis set, $\\mathcal{S}(U)_{\\bm{j}, \\bm{k}}$ is refered as Pauli transfer matrix.\n\nQuantum computation ends with measuring the observable $O$.\nThis output can be written down as,\n\\begin{align}\n    \\bbra{O}\\mathcal{S}(U)\\kket{\\rho} &= \\mathrm{Tr}(O U\\rho U^\\dagger) \\\\\n    &= \\sum_{j_1,\\cdots, j_n}\\sum_{k_1,\\cdots, k_n} O_{\\bm{j}}\\mathcal{S}(U)_{\\bm{j}, \\bm{k}} \\rho_{\\bm{k}},\n\\end{align}\nIn many cases, $U$ is a product of elementary gates $\\{U_i\\}_{i=1}^L$, that is, $U=U_L\\cdots U_1$.\nThe tensor representation of the overall gate, $\\mathcal{S}(U)$, is also a product of $\\mathcal{S}(U_i)$; $\\mathcal{S}(U) = \\mathcal{S}(U_L)\\cdots\\mathcal{S}(U_1)$.\nAn important note is that as long as the tensor representation of each element is unchanged, the result of the overall computation is also unchanged.\nIf $\\mathcal{S}(U)$ can be represented by a sum of some simple operations as $\\mathcal{S}(U) = \\sum_i c_i \\mathcal{S}(V_i)$ with coefficients $\\{c_i\\}$, the expectation value of an observable $O$ can be computed with the following equality,\n\\begin{equation}\\label{eq:superop_sum_decomposition}\n    \\bbra{O}\\mathcal{S}(U)\\kket{\\rho} = \\sum_i c_i \\bbra{O}\\mathcal{S}(V_i)\\kket{\\rho}.\n\\end{equation}\nNote that $c_i$ can, in general, depend on the state $\\kket{\\rho}$.\nWe use this scheme to perform the ``decomposition'' of a circuit in this work.\n\nIt is noteworthy that as we perform decompositions of a superoperator rather than an operator such as $U$ itself, the method becomes friendly for a realistic quantum device.\nA direct decomposition of $U$ into some simple operators $\\{V_i\\}$, i.e. $U=\\sum_{i} c_i V_i$, can also be utilized for the same task; however, as expectation values are calculated as $\\bra{0}U^\\dagger O U\\ket{0}$ where $\\ket{0}$ is an initial state, this approach requires us to evaluate $\\sum_{i,j} c_ic_j^* \\bra{0}V_j^\\dagger O V_i\\ket{0}$ which are rather hard for the NISQ devices.\nThis fact demonstrates the advantage of using the above formalism.\nThe tensor network representation of the superoperator formalism allows us to graphically understand the decompositions.\n\n\n\\subsection{Virtual two-qubit gate}\nWe can show the following, which can then be utilized to decompose any two-qubit gate into a sequence of single-qubit operations.\n\\begin{lemma}\\label{thm:two_to_single}\n    For operators $A_1$ and $A_2$ such that $A_1^2=I$ and $A_2^2=I$,\n    \\begin{align}\n        &\\mathcal{S}(e^{i\\theta A_1\\otimes A_2}) = \\cos^2\\theta \\mathcal{S}(I\\otimes I) + \\sin^2\\theta \\mathcal{S}(A_1\\otimes A_2)+ \\nonumber\\\\\n        &\\frac{1}{8}\\cos\\theta\\sin\\theta\\sum_{(\\alpha_1,\\alpha_2)\\in\\{\\pm 1\\}^2} \\alpha_1 \\alpha_2\\left[\\mathcal{S}((I+ \\alpha_1 A_1)\\otimes (I+ i\\alpha_2  A_2)) \\right. \\nonumber\\\\\n        &\\qquad\\qquad\\qquad\\qquad\\qquad \\left.+ \\mathcal{S}((I+ i\\alpha_1 A_1)\\otimes (I+ \\alpha_2  A_2))\\right]\n    \\end{align}\n\\end{lemma}\n\n\\begin{figure*}\n    \\includegraphics[width=\\linewidth]{two_qubit_operations_cut_wide_crop.pdf}\n    \\caption{\\label{fig:two_qubit_gate_cut} Decomposition of (a) a non-local gate and (b) a non-local non-destructive measurement into a sequence of local operations. $A_1$ and $A_2$ are operators such that $A_1^2 = I$ and $A_2^2 = I$.\n    }\n\\end{figure*}\n\nTo prove this, we can directly check the tensor representation of both hand side is equivalent. For detailed calculation, see Appendix \\ref{app:two_to_single}.\nThis theorem is schematically depicted in Fig. \\ref{fig:two_qubit_gate_cut} (a).\nNotice that the operation that is proportional to $I\\pm A$ and $I \\pm iA$ for $A \\in \\{X, Y, Z\\}$ can respectively be performed by a projective measurement and a single-qubit rotation.\n\nThe correspondence with a single-qubit rotation is clear from the formula, $e^{\\pm i\\pi A\/4} = \\frac{1}{\\sqrt{2}}(I\\pm iA)$, which is the rotation of angle $\\pi\/2$ around the $A$ axis.\nLet $\\mathcal{M}_A$ be the projective measurement on the $A$ basis ($A\\in \\{X,Y,Z\\}$), that is, $\\mathcal{M}_A$ acts on a density matrix $\\rho$ as,\n\\begin{align}\n    \\mathcal{M}_A\\rho &= \\frac{1}{\\mathrm{Tr}\\left(\\rho \\frac{I + \\alpha A}{2}\\right)} \\left(\\frac{I+\\alpha A}{2}\\right)\\rho \\left(\\frac{I+\\alpha A}{2}\\right),\n\\end{align}\ndepending on the result of the measurement $\\alpha \\in \\{1,-1\\}$.\nThis is equivalent to $\\mathcal{S}(I\\pm A)$ up to the factor of  $4\\mathrm{Tr}\\left(\\rho \\frac{I + \\alpha A}{2}\\right)$, that is,\n\\begin{align}\\label{eq:corresp_measurement}\n    \\mathcal{S}(I + \\alpha A) &= 4\\mathrm{Tr}\\left(\\rho \\frac{I + \\alpha A}{2}\\right) \\mathcal{M}_{A,\\alpha},\n\\end{align}\nwhere $\\mathcal{M}_{A,\\alpha}$ is a measurement operation postselected with the measurement outcome $\\alpha$.\n$\\mathrm{Tr}\\left(\\rho \\frac{I + \\alpha A}{2}\\right)$ is the probability of getting the result $\\alpha$ by measuring $\\rho$ on the $A$ basis.\nLemma \\ref{thm:two_to_single} with this fact implies that the gate $e^{i\\theta A_1\\otimes A_2}$ can be decomposed, in a sense of Eq.~(\\ref{eq:superop_sum_decomposition}), into a sum of $I\\otimes I$, $A_1\\otimes A_2$, $\\mathcal{M}_{A_1}\\otimes e^{\\pm i\\pi A_2\/4}$, and $e^{\\pm i\\pi A_1\/4}\\otimes \\mathcal{M}_{A_2}$, which can be stated as Lemma below.\nNotably, this technique can be applied for any $\\theta$, which enables us to perform continuous two-qubit gates.\n\\begin{lemma}\\label{thm:two_qubit_gate_decomposition}\n    A quantum gate $e^{i\\theta A_1\\otimes A_2}$ with operators $A_1$ and $A_2$ such that $A_1^2=I$ and $A_2^2=I$ can be decomposed into 6 single-qubit operations.\n    For any quantum state $\\kket{\\rho}$, to achieve the error $\\epsilon$ of the decomposition with respect to the trace distance with probability at least $1-\\delta$, the required number of circuit runs is $O(\\log (1\/\\delta)\/\\epsilon^2)$.\n\\end{lemma}\nThe detailed proof is given in Appendix \\ref{appsec:proof_of_two_qubit_gate_decomp}.\nIntuitively, since the error comes from the probabilistic part of the decomposition, that is the renomalization factor in Eq. (\\ref{eq:corresp_measurement}) $\\mathrm{Tr}\\left(\\rho \\frac{I + \\alpha A}{2}\\right)$,  if we want to estimate $\\mathrm{Tr}\\left(\\rho \\frac{I + \\alpha A}{2}\\right)$ within error $\\epsilon$, $O(1\/\\epsilon^2)$ repetition would suffice.\n\nLet us finally mention the case of the controlled-Z gate, which we denote by $CZ$.\n$CZ$ can be decomposed into\n\\begin{equation}\n    CZ = e^{i\\pi I\\otimes Z\/4}e^{i\\pi Z\\otimes I\/4}e^{-i\\pi Z\\otimes Z\/4},\n\\end{equation}\nignoring the global phase.\nThis means we can decompose a CZ gate using Lemma \\ref{thm:two_qubit_gate_decomposition}.\nThe decomposition is shown in Fig.~\\ref{fig:cz_gate_cut}.\nSimilar decompositions can be performed on some basic two-qubit gates such as CNOT.\nEndo \\textit{et al.} \\cite{Endo2018} also provides such decomposition (Ref. \\cite{Endo2018}, Appendix B).\nHowever, our protocol above is slightly advantageous in that the number of single-qubit operations required is 6 compared to theirs which requires 9 of them.\n\n\\begin{figure*}\n    \\includegraphics[width=0.8\\linewidth]{cz_gate_cut_wide_crop.pdf}\n    \\caption{\\label{fig:cz_gate_cut} Decomposition of controlled-Z gate into a sequence of single-qubit operations.\n    }\n\\end{figure*}\n\n\n\n\n\\subsection{Virtual non-destructive measurement of two-qubit operators}\nIn the previous subsection, we showed that any two-qubit rotation can be decomposed into a sum of single-qubit operations.\nHere, we extend the strategy to construct virtual non-destructive measurement\nof two-qubit operators.\nSimilar to the previous section, we can show the following.\nThis theorem is schematically shown in Fig.~\\ref{fig:two_qubit_gate_cut}~(b).\n\\begin{lemma}\\label{thm:two_to_single_meas}\n    For operators $A_1$ and $A_2$ such that $A_1^2=I$ and $A_2^2=I$,\n    \\begin{align}\n        &\\mathcal{S}(I + A_1\\otimes A_2) = \\mathcal{S}(I\\otimes I) + \\mathcal{S}(A_1\\otimes A_2)+ \\nonumber\\\\\n        &\\frac{1}{8}\\sum_{(\\alpha_1, \\alpha_2)\\in\\{\\pm 1\\}^2} \\alpha_1 \\alpha_2\\left[\\mathcal{S}((I+ \\alpha_1 A_1)\\otimes (I+ \\alpha_2  A_2)) \\right. \\nonumber\\\\\n        &\\qquad\\qquad\\qquad \\left.- \\mathcal{S}((I+ i\\alpha_1 A_1)\\otimes (I+ i\\alpha_2  A_2))\\right]\n    \\end{align}\n\\end{lemma}\nThis can also be shown by the direct calculation of both hand side. See Appendix \\ref{app:two_to_single_meas} for detailed calculation.\n\nThe above Lemma can be utilized to show the following.\n\\begin{lemma}\\label{thm:twoq_meas_decomp}\n    A non-local projection $\\frac{\\red{I}+A_1\\otimes A_2}{2}$ with operators $A_1$ and $A_2$ such that $A_1^2=1$ and $A_2^2=2$ can be decomposed into 6 single-qubit operations.\n    For any quantum state $\\kket{\\rho}$, to achieve the error $\\epsilon$ of the decomposition with respect to the trace distance with probability at least $1-\\delta$, the required number of circuit runs is $O(\\log (1\/\\delta)\/\\epsilon^2)$.\n\\end{lemma}\nThis can be shown with exactly the same approach taken to prove Lemma \\ref{thm:two_qubit_gate_decomposition}, which is provided in Appendix \\ref{appsec:proof_of_two_qubit_gate_decomp}.\n\n\n\n\n\\section{Application}\n\\subsection{Simulation of large quantum circuits}\\label{sec:simulation}\nThe idea of simulating a large quantum circuit by a small quantum computer has been put forward in Ref.~\\cite{Peng2019}.\nPeng \\textit{et al.} utilized the equivalence shown in Fig. \\ref{fig:identitygatecut}.\nIn the figure,\n\\begin{equation}\\label{eq:Pengcut_pairs}\n\\begin{array}{lll}\n    O_1=I, & \\rho_1 = \\ket{0}\\bra{0}, & c_1 = +1\/2, \\\\\n    O_2=I, & \\rho_2 = \\ket{1}\\bra{1}, & c_2 = +1\/2, \\\\\n    O_3=X, & \\rho_3 = \\ket{+}\\bra{+}, & c_3 = +1\/2, \\\\\n    O_4=X, & \\rho_4 = \\ket{-}\\bra{-}, & c_4 = -1\/2, \\\\\n    O_5=Y, & \\rho_5 = \\ket{+i}\\bra{+i}, & c_5 = +1\/2, \\\\\n    O_6=Y, & \\rho_6 = \\ket{-i}\\bra{-i}, & c_6 = -1\/2, \\\\\n    O_7=Z, & \\rho_7 = \\ket{0}\\bra{0}, & c_7 = +1\/2, \\\\\n    O_8=Z, & \\rho_5 = \\ket{1}\\bra{1}, & c_8 = -1\/2, \n\\end{array}\n\\end{equation}\nwhere $\\ket{\\pm}=(\\ket{0}\\pm \\ket{1})\/\\sqrt{2}$ and $\\ket{\\pm i}=(\\ket{0}\\pm i\\ket{1})\/\\sqrt{2}$. \nThe symbols $\\triangleright$ and $\\triangleleft$ denotes the measurement of a certain observable and the preparation of a certain state, respectively.\nContrasting this technique and ours, we refer to the former and the latter as ``time-like'' and ``space-like'' cut, respectively.\nMore concretely, \\textit{a time-like cut} of a quantum channel can be defined as a decomposition  of the channel in the sense of Eq. (\\ref{eq:superop_sum_decomposition}) using measure-and-prepare channels only.\nIn contrast, \\textit{a space-like cut} of a non-local quantum channel is a decomposition of the channel using local quantum channels only.\n\n\n\\begin{figure}\n    \\includegraphics[width=0.7\\linewidth]{identitygatecut_crop.pdf}\n    \\caption{\\label{fig:identitygatecut} Time-like cut employed in Ref.~\\cite{Peng2019}.}\n\\end{figure}\n\n\\begin{figure}\n    \\includegraphics[width=\\linewidth]{simpleclustering_crop.pdf}\n    \\caption{\\label{fig:simpleclustering} Two decomposition approach compared in main text. The top-right approach is the presented, and the bottom-right approach is of Ref.~\\cite{Peng2019}.}\n\\end{figure}\n\nThe decomposition presented in the previous section can also be used in this direction.\nLet us compare the scaling of cost of our decomposition scheme and that of Peng \\textit{et al.} by a simple example.\nWe consider the case where we have an $n$-qubit quantum computer to simulate a $2n$-qubit quantum circuit of Fig.~\\ref{fig:simpleclustering}, which has only one CZ gate between n-qubit ``cluster''.\nThe task is to estimate the expectation value of a final observable $O_f$ by measuring it in the computational basis.\nTo simplify the discussion, we assume $O_f$ is a string of Pauli $Z$'s.\n\nLet $v$ be a desired variance of the estimation of the expectation value of $O_f$.\nWe can show a naive algorithm, which runs the equal number of circuits for each terms appearing in the decomposition, to perform the decomposition with time-like cuts, in the worst case, requires $2048\/v$ runs of $n$-qubit circuit, while the space-like cut approach takes $\\frac{15}{2v}$ runs.\nThe analysis of this simple example is given in Appendix \\ref{app:simpleexample}.\nAlthough the analysis given here is based on a naive algorithm and there are possibilities to improve it, this analysis somewhat shows the enhancement provided by our space-like cut protocol.\n\n\n\n\\subsubsection*{General case}\nWe can consider a general case where we perform the time-like and space-like cuts simultaneously to make a given $m$-qubit quantum circuit runnable on an $n$-qubit quantum computer.\nLet the number of time-like and space-like cuts be $M_t$ and $M_s$, respectively.\nFor space-like cuts, we assume they are performed only on CZ gates.\nThe input state $\\rho$ is initialized in $\\ket{0}\\bra{0}^{\\otimes m}$ and $O_f$ is an output (diagonal) observable calculated from some output function $f:\\{0,1\\}^m\\to [-1,1]$.\nOur task here is to estimate the expectation $\\mathbb{E}[f(y)]$ for a random bitstring $y\\in\\{0,1\\}^m$ sampled from the original circuit.\nThis model is adopted from Ref. \\cite{Peng2019} which originates in Ref. \\cite{Bravyi2016}.\nWith this definition, we can get the following.\n\n\\begin{theorem}\\label{thm:multiple_twoqgate_cut}\n    The number of $n$-qubit circuit runs required to estimate $\\mathbb{E}[f(y)]$ within accuracy $\\epsilon$ with some high probability $1-\\delta$ is $O\\left(\\frac{9^{M_s} 16^{M_t}}{\\epsilon^2}\\log\\left(\\frac{1}{2\\delta}\\right)\\right)$.\n\\end{theorem}\nThis implies that the decomposition of the circuit should be performed to minimize $9^{M_s} 16^{M_t}$.\nA detailed proof is given in Appendix \\ref{appsec:proof_of_multiple_twoqgate_cut}, however, the above can roughly be explained as follows.\nAt each space-like cut, we get 6 different sets of single-qubit operations, so $M_s$ cuts induce $6^{M_s}$ terms.\nLikewise, $M_t$ time-like cuts induce $8^{M_t}$ terms, which makes the total number of circuits in decomposition $6^{M_s}8^{M_t}$.\nWith this decomposition, we can take a Monte-Carlo approach to estimate the sum, that is, we randomly choose circuits to run and average them.\nHoeffding's inequality can be used to bound the error of such protocol, which states that if a magnitude of a random variable is always bounded by some constant $a$, then $O(a^2\/\\epsilon^2)$ samples would suffice to obtain an accuracy of $\\epsilon$.\nIn this case, we are to estimate $\\mathbb{E}[f(y)] = \\sum_{i=1}^{6^{M_s}8^{M_t}} c_i \\bbra{O_f}\\mathcal{S}(V_i)\\kket{\\rho}$ with $i$ randomly drawn from $\\{1,\\cdots,6^{M_s}8^{M_t}\\}$ and $|c_i|=1\/2^{M_s+M_t}$, that is, $\\mathbb{E}[f(y)]$ is estimated by $\\mathbb{E}_i[6^{M_s}8^{M_t} c_i\\bbra{O_f}\\mathcal{S}(V_i)\\kket{\\rho}]$.\nThe magnitude of random variable $6^{M_s}8^{M_t} c_i\\bbra{O_f}\\mathcal{S}(V_i)\\kket{\\rho}$ is roughly $3^{M_s}4^{M_t}$, thus we can apply the Hoeffding bound to get the result.\n\n\\begin{figure}\n    \\includegraphics[width=\\linewidth]{combination_crop.pdf}\n    \\caption{\\label{fig:combination}Schematic illustration of performing the space-like cut and the time-like cut simultaneously.}\n\\end{figure}\n\n\\subsection{Distant two-qubit gates}\nThe theorem introduced above can be utilized to ``virtually'' perform a two-qubit gate between qubits at distance.\nFigure \\ref{fig:distant_2qubit_gate} shows an example of such a virtual two-qubit gate.\nNotice that this protocol works irrespective of the distance between the qubits.\nMany swap gates are otherwise necessary for performing such gates, which makes them impractical on NISQ devices due to the non-negligible amount of decoherence and gate error of such devices.\n\n\\begin{figure}\n    \\includegraphics[width=\\linewidth]{distant_2qubit_gate_crop.pdf}\n    \\caption{\\label{fig:distant_2qubit_gate} Decomposition of distant two-qubit gate on a square lattice. \n    Each vertex of the graph represents a qubit and the edge represents the connectivity of the qubits.\n    $S$ is the set of pairs of single-qubit operations which appears in the formula in Lemma \\ref{thm:two_to_single}, and $c_s$ is the corresponding coefficient for each pair.}\n\\end{figure}\n\nThis protocol might be useful for the variational algorithms such as the variational quantum eigensolver (VQE) \\cite{Peruzzo2014} and the quantum approximate optimization algorithms (QAOA) \\cite{Farhi2014}.\nHere, we describe an example in the QAOA.\nIn the QAOA, we seek to find a ground state of a Hamiltonian $H$ on $n$-qubit which is a sum of Pauli $Z$'s and its products.\nFor example, a Hamiltonian may have the form of,\n\\begin{align}\n    H = \\sum_{ij} J_{ij} Z_iZ_j.\n\\end{align}\nThe QAOA tries to solve the problem by converting it to a optimization problem of a continuous variable $\\bm{\\beta}$ and $\\bm{\\gamma}$.\nThe optimization of $\\bm{\\beta}$ and $\\bm{\\gamma}$ are performed so as to minimize the function,\n\\begin{align}\n    \\expect{H(\\bm{\\beta}, \\bm{\\gamma})} =\\bra{+}^{\\otimes n} U^\\dagger(\\bm{\\beta}, \\bm{\\gamma}) H U(\\bm{\\beta}, \\bm{\\gamma})\\ket{+}^{\\otimes n},\n\\end{align}\nwhere,\n\\begin{align}\\label{eq:qaoa-circuit}\n    U(\\bm{\\beta}, \\bm{\\gamma}) = e^{i\\beta_p \\sum_i X_i}e^{i\\gamma_p H} \\cdots e^{i\\gamma_2 H}e^{i\\beta_1 \\sum_i X_i}e^{i\\gamma_1 H}.\n\\end{align}\nThis algorithm has been experimentally demonstrated \\cite{Otterbach2017} with the connectivity of the target Hamiltonian being equivalent to the connectivity of the actual device.\n\nThe equivalence of the connectivity is almost necessary from the requirement to perform $e^{i\\gamma H}$.\nThis requirement can somewhat be relaxed by our protocol which enables qubits to virtually interact irrespective of the distance between them.\nLet us now assume that an available device has a square-lattice connectivity of Fig. \\ref{fig:distant_2qubit_gate}, and a Hamiltonian of the QAOA which we aim to solve has a interaction between one pair of qubits that is not included in the hardware connectivity graph.\nIn this case, to execute the QAOA circuit (Eq. (\\ref{eq:qaoa-circuit})), we can use our space-like technique $p$ times to virtually apply the unitary.\nThe scaling of the cost can be bounded by setting $M_t=0$ and $M_s=p$ in Theorem \\ref{thm:multiple_twoqgate_cut} which gives us a scaling of $9^p\\epsilon^{-2}\\log[1\/(2\\delta)]$.\nThe time-like cut approach of Peng \\textit{et al.} \\cite{Peng2019} can also be utilized in this direction.\nHowever, as this approach would require 4 cuts per gate, the cost scaling is bounded by $16^{4p}\\epsilon^{-2}\\log[1\/(2\\delta)]$ by setting $M_t=4p$ and $M_s=0$ in Theorem \\ref{thm:multiple_twoqgate_cut}.\nThis demonstrates an advantage, albeit in this special settings, of our technique over the previous result.\n\nIn the context of the VQE, which is also an algorithm to find a ground state of a Hamiltonian but mainly targets a concrete physical system such as molecules, it has been proposed to use the same kind of quantum circuits as the QAOA \\cite{Wecker2015, Mitarai2019g}.\nOur result may also be applicable in constructing such circuits.\n\n\\section{discussion and conclusion}\nWe described a technique to decompose a non-local operations into a sequence of local operations.\n\\red{As the single-qubit operations are generally more accurate on NISQ devices, the proposed technique can be used to enhance their capability. We believe intrinsic noise on single-qubit operations can be compensated by recent sophisticated error mitigation techniques \\cite{Endo2018}.}\nIn particular, our technique of the space-like cut of two-qubit gates can improve the simulation of a large quantum circuit with a small quantum computer in some cases.\nIt would be interesting to investigate the best strategy to perform ``cuts'' to reduce the number of qubits compatible with an available device.\nAlso, the algorithm we have given to bound the cost scaling is rather straight forward and we believe it can be improved with a more sophisticated strategy.\n\n\\red{The proposed algorithm can also be compared to the classical simulation strategy that splits a large circuit by decomposing two-qubit gates. For example, a controlled-NOT gate can be splitted using a tensor network based technique \\cite{biamonte2017tensor}. However, such techniques generally does not focus on decompositions of $\\mathcal{S}(U)$ considered in this work but rather the two-qubit unitary $U$ itself, which takes makes them difficult to be used on NISQ devices as Eq. (\\ref{eq:superop_sum_decomposition}) cannot be utilized anymore. }\n\nOur technique can induce a entanglement-like effect without performing any two-qubit gate with the cost mentioned in Lemmas \\ref{thm:two_qubit_gate_decomposition} and \\ref{thm:twoq_meas_decomp}.\nThis connects this work to areas like quantum communication.\nThis ``virtual'' entanglement creation could be done with the time-like cut proposed by Peng \\textit{et al.}, but our work lowered the cost to perform the task.\nIt is interesting to know whether ours is the optimal protocol or there is a more efficient way.\n\nTo summarize, our technique allows qubits to virtually interact irrespective of physical distances between them.\nThe result is useful for applying a two-qubit gate to a distant pair of qubits.\nIn particular, when applied to the NISQ devices, this may be employed to enhance the power of them.\nFuture direction can be to explore if we can lower the resource to perform such virtual operations.\n\n\\begin{acknowledgements}\n    KM thanks the METI and IPA for their support through the MITOU Target program.\n    KM is also supported by JSPS KAKENHI No. 19J10978 and No. 20K22330, and JST PRESTO JPMJPR2019.\n\tKF is supported by KAKENHI No.16H02211, JST PRESTO JPMJPR1668, JST ERATO JPMJER1601, and JST CREST JPMJCR1673.\n\tThe authors thank Suguru Endo for fruitful discussions and letting us become aware of Ref.  \\cite{Endo2018}.\n    This work is supported by MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) Grant Number JPMXS0118067394.\n\\end{acknowledgements}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sect:intro} \nThe exclusion process is a paradigm for non-equilibrium behaviour \\cite{bib:M1}. \nIn the one-dimensional totally asymmetric simple exclusion process (TASEP),\neach site $i$ $ ( i=1,2,\\dots, L) $ is either occupied by one particle ($ \\tau_i =1 $) or empty ($ \\tau_i =0$),\nand a particle at site $ i$ hops to $i+1 $ with rate 1, if the target site is empty. Investigation of shocks is one of central issues in the TASEP\\cite{bib:BCFG,bib:FF,bib:DJLS,bib:DEM,bib:M2,bib:KSKS,bib:CHA}. Let us recall a known result in the TASEP with open boundaries (shortly, open TASEP), where a particle is injected at site $i=1$ with rate $\\alpha$, and extracted at site $i=L$ with rate $ \\beta $ \\cite{bib:DEHP}. These rates are regarded as reservoir densities $\\alpha$ and $ 1-\\beta $, respectively. The case $ \\alpha = \\beta < 1\/2 $ is called the co-existence line. Two plateaus with densities $ \\alpha$ and $ 1-\\alpha$ co-exist in the domains\n $ 1 < x < S(t) $ and $ S(t) < x < L $, respectively. The position $ S (t) $ of the shock (domain wall) is \\textit{dynamical}, and its  behaviour is diffusive \\cite{bib:DEM,bib:KSKS}, \\textit{i.e.}  \n\\begin{align} \\label{eq:D=} \n    \\big\\langle \\big( S (t) - S (0) \\big)^2 \\big\\rangle_{\\mathrm E} \\simeq   2 D (\\alpha ) t,   \\    \n     D (\\alpha ) = \\frac{ \\alpha(1-\\alpha) }{ 1 - 2\\alpha } , \n\\end{align} \nas $t\\to \\infty$. Here $ \\big\\langle \\cdot \\big\\rangle_{\\mathrm E} $ denotes the ensemble average. \nThis asymptotic diffusion coefficient is also true on $ \\mathbb Z $ \\cite{bib:FF}.\n\n\nThe so-called second-class particle ($ \\tau_i =2 $)  microscopically \\textit{defines}  \n the positions of shocks \\cite{bib:BCFG}. It behaves as a hole for particles, and as a particle for holes, \\textit{i.e.}  the  hops $ 20  \\to  02 $ and  $ 12 \\to  21 $ occur between sites $i$ and $ i+1 $, with the same rate   as for $ 10 \\to 01 $. We confine only one second-class particle into the system in the initial state. It is not extracted from the system and we do not inject another second-class particle, \\textit{i.e.}  \n``semi-permeable boundary condition'' \\cite{bib:A1,bib:A2,bib:U,bib:ALS}. Figure \\ref{fig:openTASEP} (a,b) shows   simulation results of the mean-squared displacements (MSD) of the second-class particle, which agree with the formula  \\eqref{eq:D=}. We notice that finite-time effect becomes strong, when $\\alpha$ approaches $1\/2$. The density profile averaged over a large time interval \n\\begin{align}\n  \\rho_i := \\langle \\tau_i  ( 2-\\tau_i ) \\rangle_{\\mathrm T}    \n  \\label{eq:rho_i:=}\n\\end{align}\nlooks different from snapshots in simulations. \nSince the shock position moves evenly in the chain,  $ \\rho_i $ is given by \n\\begin{align} \\rho_i \\simeq ( 1 - 2\\alpha ) \\frac i L  + \\alpha, \\label{eq:rho_i=open} \\end{align}\nwhich was shown by the exact stationary state  \\cite{bib:DEHP}.\n\n\n\\begin{figure}\\begin{center}\n     \\includegraphics[width=0.24\\textwidth]{MSD-open.pdf} \\  \n     \\includegraphics[width=0.24\\textwidth]{D-vs-rho.pdf} \\end{center}\n\\caption{ \n  (a) MSD of the second-class particle vs time, and\n  (b) diffusion coefficient vs boundary rate \n      on the co-existence line of the open TASEP with $ L=10^4 $. \n The markers in (a) are simulation results averaged over  $ 10^3 $ runs.\n  We used simulation data  $  ( S(t) -S(0) )^2  \/ (2t ) $ in $  5\\times 10^3 \\le  t \\le 10^4  $ to plot the markers in (b).   The lines in (a,b) correspond to eq.~\\eqref{eq:D=}.   \n \\label{fig:openTASEP}} \n\\end{figure}\n\n\nOn the other hand,  static shocks are realized in the TASEP, by imposing  attachment and detachment of particles in the bulk of the chain (the Langmuir kinetics \\cite{bib:PFF}).   Static shocks  were also found in  TASEPs with inhomogeneous hopping rates on a ring.     One of the simplest cases    is  the Janowsky-Lebowitz (JL) model \\cite{bib:JL}: \n \\begin{align}    p_i =     1  \\  ( 1 \\le i < L )   , \\  r<1 \\  ( i=L )  ,  \\end{align} \n where  $ p_i $  denotes the hopping  rate from site $i$ to $i+1$ ($ L+1 := 1 $). \n Due the inhomogeneity of the bond between sites $ L $ and 1,  the JL model exhibits a shock profile in a certain parameter region.  A mean-field theory qualitatively explains a phase transition between shock and flat density profiles, which  is still a fascinating problem \\cite{bib:CLST,bib:SPS}. \n \nIn \\cite{bib:TB},  another inhomogeneous TASEP   was introduced: \n\\begin{align}\\label{eq:two_pi=}  p_i =   1  \\  ( 1 \\le i \\le \\ell    )   , \\  r  \\  ( \\ell <  i \\le 2 \\ell=L   )  .\\end{align}\n We refer to this model as 2-segment TASEP.   In a certain region of the parameter space $ (r,\\rho) $ (where $ \\rho$ is the global density \\textit{i.e.} the number of particles $\/L$), this model also exhibits a static shock.  Recently  the authors of  \\cite{bib:BSB} investigated TASEPs with three and four parts, generalizing \\eqref{eq:two_pi=}.  Here, we mainly study a specific  4-segment TASEP \n\\begin{align}\\label{eq:four_pi=}\n  p_i = \n  \\begin{cases}  \n     1 & ( 1 \\le i \\le \\ell   \\ \\vee \\   2\\ell  < i \\le  3\\ell  )   , \\\\ \n     r & (  \\ell < i \\le 2\\ell \\  \\vee\\   3\\ell < i \\le 4\\ell =L) .    \\\\ \n  \\end{cases}\n\\end{align}\nThere can exist two shocks in  the 1st and 3rd segments. The positions of the shocks cannot be fixed even in the macroscopic level, but they are related to each other by an equation derived by the particle number conservation  \\cite{bib:BSB}. \n The purpose of this work is performing more detailed Monte Carlo simulations (in continuous time),  in order to deeply understand this synchronization phenomenon. \n\nBefore investigating the 4-segment TASEP, we reconsider the case of 2 segments via the second-class particle.  As an evidence that the second-class particle indicates the shock position, we check that its spatial distribution becomes gaussian, corresponding to the density profile written in the error function.  We also examine properties of the standard deviation of the shock position. \nThen we turn to  the 4-segment TASEP. We find that the MSD exhibits various behaviours, depending on the time scale that we focus. The open boundary condition that we have already reviewed is the reference case. We quantify the interaction between the  shocks by a correlation function.\n Furthermore we introduce a crossover time distinguishing between time scales  of   independency and synchronization of shocks. We also  numerically  estimate the dynamical exponent of the crossover time. Finally we give the conclusions of this work and some remarks, including possible future studies.  Overall in this work, we use the following  definition  for  macroscopic density profiles  with  \\textit{mesoscopic} length  of the lattice  $ 2 \\delta +1 $ ($ 1 \\ll \\delta \\ll  \\ell $),  which is in general different from  the microscopic density profile \\eqref{eq:rho_i:=}:\n\\begin{align}\n   \\rho (x)  = \\rho ( i\/\\ell   )  = \n    \\frac{1}{ 2 \\delta + 1 } \\sum_{ i' = i - \\delta }^{ i + \\delta }  \\tau_{i'}  ( 2 - \\tau_{i'}  )  .   \\label{eq:macro-density}\n\\end{align}\n \n\n\n\n\n\n\\section{2-segment TASEP}\\label{sect:2-seg}\n Let us consider the model \\eqref{eq:two_pi=}.  \n We begin with the assumption that the global density $ \\rho $ is enough small, and \n each segment has a flat density profile       \n\\begin{align}\\label{eq:rho(x)=12}\n   \\rho(x) =      \\alpha_1   (  0  < x < 1   )   , \\       \\alpha_2   (  1 < x < 2  )   .    \n\\end{align}\nThe conservation laws of the number of  particles  and the stationary current $ J $ are written as \n\\begin{align}\n  \\label{eq:2rho=a1+a2}\n  2 \\rho &=     \\alpha_1   +     \\alpha_2  ,  \\\\  \n        J &=  \\alpha_1 (1-\\alpha_1) = r  \\alpha_2 (1-\\alpha_2)  , \n  \\label{eq:J=J1=J2}\n\\end{align}\nrespectively. \nThese are easy to solve \\cite{bib:TB}: \n\\begin{align}\n\\label{eq:alpha1=,alpha2=}\n  \\alpha_1  &= \\frac{ 1 + r -4r\\rho - R }{  2 (1-r)  } , \\ \n  \\alpha_2  = \\frac{ 4\\rho - (1+r)  + R}{  2 (1-r)  } , \\\\\n  J &=          \\frac{ r(1-2\\rho)(R-(1+r)(1-2\\rho)  ) }{   (1-r)^2  } \n    \\label{eq:J:LDLD}  \n\\end{align}  \n with $ R  = \\sqrt{(1+r)^2-16r\\rho (1-\\rho)}  $. \nThe density profile \\eqref{eq:rho(x)=12} is realized as long as  $ \\alpha_2 <  1\/2  $\nwith \\eqref{eq:alpha1=,alpha2=}, \\textit{i.e.}  \n\\begin{align}  \\label{eq:rho_c=} \\rho <   \\rho_c : =   \\frac { 2-\\sqrt{1- r} }{4}     .\n   \\end{align}\nSince both $ \\alpha_1 $ and $  \\alpha_2 $ are less than $ 1\/2 $, \n the case $ \\rho < \\rho_c $ is called LD-LD phase (LD$=$low density). \n\n\n\\begin{figure}\\begin{center}\n  \\includegraphics[width=0.24\\textwidth]{Phase-diagram.pdf} \\ \n  \\includegraphics[width=0.24\\textwidth]{Fundamental-diagram.pdf} \\end{center} \n\\caption{  \n (a) Phase diagram  and\n (b) fundamental diagram \n    of the 2- and 4-segment TASEPs \\cite{bib:TB, bib:BSB}. \nThe phase boundaries in (a) are given by $ \\rho = \\rho_c ,1 - \\rho_c$  \n\\textit{i.e.}  the parabola  $ r =  1  -  1 6  ( \\rho - 1\/2   )^2    $.  \nFor  (b),  we have set    $ \\ell = 10^3 $ and $ r =1\/2$.  To obtain each marker, we counted the number of flowing particles in a single simulation run, and took average over $ 10^6 \\le t \\le 10^7 $. The line in (b)  corresponds to the   predictions  \\eqref{eq:J:LDLD}, \\eqref{eq:J:SMC}. \n  \\label{fig:diagrams}} \n\\end{figure}\n\n\n\n\nWhen  the total density $  \\rho $ exceeds the critical density, \nthe 2nd segment maintains  the density $1\/2 $ and\na shock appears on the 1st segment: \n\\begin{align}\\label{eq:rho(x)=1lh2}\n   \\rho(x) = \n  \\begin{cases}  \n     \\alpha_1  \\  (  0  < x < s )   , \\  \n     1- \\alpha_1 \\  (  s < x < 1   )   , \\\\\n     \\alpha_2 =   \\frac{1}{2}   \\  (  1 < x < 2  )  .    \n  \\end{cases}\n\\end{align}\nWe refer to this case as S-MC (shock-maximal current) phase. The shock position $s$ is macroscopically static, \\textit{i.e.}  localized.  By solving the conservation laws of the number of particles and the stationary current \n\\begin{align}\n 2 \\rho =&  s \\alpha_1  + (1-s)(1 - \\alpha_1 ) +  \\  \\alpha_2  ,  \\\\\n J  ( \\alpha_1 ) =& \\alpha_1 (1-\\alpha_1) =  r \\alpha_2 (1-\\alpha_2 )  =   r \/ 4   \\label{eq:J:SMC} , \n\\end{align}\nthe densities and the shock position in the 1st segment  are determined as \n\\begin{align}\n  \\alpha_1 = \\frac{1-\\sqrt{1-r}}{2}  ,  \\    s  = \\frac{1}{2} + \\frac{1-2\\rho}{ \\sqrt{1-r}}  . \n  \\label{eq:alpha1=,s=}\n\\end{align}\nThe parameters $ ( r , \\rho ) $ are inversely specified by $ (\\alpha_1,s) $ in the S-MC phase. We have emphasized the current as a function of the density $ \\alpha_1 $. \n \nWhen the global density $\\rho$ exceeds  $1-\\rho_c $, the shock position reaches site $i=1 $. The densities in both segments are flat, and larger than $ 1\/2 $, \\textit{i.e.}  the HD-HD phase (HD$=$high density). Figure \\ref{fig:diagrams} (a) summarizes the three phases. In fig.~\\ref{fig:diagrams} (b), we check that the predicted currents \\eqref{eq:J:LDLD} and \\eqref{eq:J:SMC} are realized by simulations. Because of the particle-hole symmetry, we restrict our consideration to $ \\rho \\le 1\/2 $.  \n\n\nLet us investigate properties of the shock in the S-MC phase in more detail. There is a microscopic deviation of the density profile near the shock position. It obeys a gaussian distribution, \\textit{i.e.}  the probability distribution $ P ( S ) := \\frac 1 2 \\langle \\tau_S (\\tau_S-1) \\rangle_{\\mathrm T} $ of the position $S$ of the second-class particle is given as \n\\begin{align}\n\\label{eq:P(S)=Gauss}\n P (S)  =   \\frac{1}{ \\sqrt{2 \\pi  \\sigma^2  }   }  \n  \\exp  \\bigg[  - \\frac{1}{ 2\\sigma^2 }  ( S - \\langle S \\rangle_{\\mathrm T}  ) ^2  \\bigg]     \n\\end{align}\nwith $ \\langle S \\rangle_{\\mathrm T}  \\simeq  \\ell s $ $ ( \\ell \\to \\infty ) $. We see good agreement of simulations to this distribution in fig.~\\ref{fig:2-seg} (a).  This corresponds to the fact that the (microscopic) density profile \\eqref{eq:rho_i:=} in the 1st segment  is well described in terms of  the error function  $\\mathrm{erf}[\\cdot]$  \\cite{bib:BSB}, see  fig.~\\ref{fig:2-seg} (b): \n\\begin{align}\n  \\label{eq:<tau>=erf}\n    \\rho_i =  \\frac{\\sqrt{ 1-r } }{2}  \\ \\mathrm{erf} \n   \\bigg[  \\frac{ i -  ( \\langle S \\rangle_{\\mathrm T}  +  \\frac 1  2  )  }{ \\sqrt{2   \\sigma^2  } } \\bigg] \n       +     \\frac{1}{2}   .\n\\end{align} \nWe chose the values of  parameters, such that corresponding  densities and shock positions are given as  \n\\begin{align}\n  \\nonumber \n &    ( r , \\rho   ) = (0.96,  0.5  ) , ( 0.84 , 0.48 ) , ( 0.64,0.44 ) , (0.36,0.38  )   \\Leftrightarrow  \\\\  \n &    ( \\alpha_1 , s  ) =  ( 0.4 , 0.5 ) , (0.3, 0.6   ) ,  ( 0.2 , 0.7  ) ,  ( 0.1, 0.8  ) , \n\\end{align}\nrespectively,  according to eq.~\\eqref{eq:alpha1=,s=}. \n(In the 2nd segment, we expect that the density profile, which is deviated from $  1\/2 $,  can be well written by the exact finite-size effect in the maximal current phase of the open TASEP \\cite{bib:DEHP}.)\n\n\n\\begin{figure}\\begin{center}\n     \\includegraphics[width=0.24\\textwidth]{gauss.pdf}  \\ \n     \\includegraphics[width=0.24\\textwidth]{erf.pdf} \\  \n     \\includegraphics[width=0.24\\textwidth]{sigma-vs-ell.pdf} \\ \n     \\includegraphics[width=0.24\\textwidth]{sigma-vs-r.pdf} \n     \\end{center}  \n\\caption{  \n (a) Distributions of the second-class particle, \n (b) density profiles,\n (c) standard deviation $ \\sigma $  vs segment length   $ \\ell $,  and  \n (d) $\\sigma$ vs hopping rate $r$.   \n For (a) and (b) we  have set  the system length as $ \\ell =200$, and for (d) $ \\ell =10^3 $.   \n  Each  plot marker was obtained by averaging  data of a single simulation run  over \n   $ 10^6  \\le t \\le 10^7$ \n  [exceptionally $ 10^6  \\le t \\le 5\\times 10^7$ for the cases $ \\ell > 10^3 $ in  (c)]. \n   The curves  in  (a) and (b) correspond to  \n    eqs.~\\eqref{eq:P(S)=Gauss}  and  \\eqref{eq:<tau>=erf},  \n    respectively, with $ \\langle S \\rangle  $   and $ \\sigma$ obtained from simulation data. Each line  in (c)  corresponds to  $ ( \\ell  \/10^4 )^{1\/2}    \\times  \\sigma |_{\\ell=10^4} $ with $ \\sigma |_{\\ell=10^4} $ obtained by simulations. \n\\label{fig:2-seg} } \n\\end{figure}\n\n In fig.~\\ref{fig:2-seg}  (c), we observe $ \\sigma \\propto \\sqrt \\ell $  \\cite{bib:BSB}.  Furthermore  $ \\sigma \/ \\sqrt \\ell   $ vs $r $ is shown in fig.~\\ref{fig:2-seg}  (d) with different global densities  $ \\rho $.  So far we have not  found an explicit formula of $ \\sigma \/ \\sqrt \\ell $, but it seems independent of $ \\rho $. \n\n\n\n \n  \n\\section{4-segment TASEP}\\label{sect:4-seg}\nNow we turn to the 4-segment TASEP \\eqref{eq:four_pi=} \\cite{bib:BSB}. By the same argument  as for the 2-segment case,  the density $\\alpha_j  $ of each segment $j$ is flat, when  $ \\rho < \\rho_c  $. (The phase transition line $ \\rho = \\rho_c $ is identical to that of the 2-segment case, fig.~\\ref{fig:diagrams} (a).)\n From the translational invariance,  we have  $ \\alpha_1 = \\alpha_3  $ and  $ \\alpha_2 = \\alpha_4  $.  Furthermore these densities have  the same form  \\eqref{eq:alpha1=,alpha2=}  as for the 2-segment case \\cite{bib:BSB}. \n After the global density exceeds $ \\rho_c  $, the 2nd and 4th segments maintain the density $\\alpha_2 =\\alpha_4 = 1\/2 $, and shocks appear in the 1st and 3rd segments \\cite{bib:BSB}:\n  with   $ \\alpha_1 = \\alpha_3   $,  \n\\begin{align}\\label{eq:rho(x)_SMCSMC}\n   \\rho(x) =   \n   \\begin{cases}  \n     \\alpha_1  \\ (  0  < x < s_1 )   ,    \\    1- \\alpha_1 \\  (  s_1 < x < 1   )   , \\\\ \n        \\frac{1}{2} \\   (  1 < x < 2  )  ,  \\\\ \n     \\alpha_3  \\ (  2  < x < s_3'   )   ,    \\    1- \\alpha_3 \\  (  s_3'  < x < 3    )   , \\\\ \n        \\frac{1}{2} \\   (  3 < x < 4   )    . \n  \\end{cases}\n\\end{align}\nThe shock positions are denoted by  $ s_1 $ and $ s_3'   = s_3 +2$  [$ s_3=0 $ (resp. $ s_3=1  $) corresponds to the boundary between 2nd and 3rd  (resp.  3rd and 4th) segments]. \n The conservation of the number of particles is written as \n\\begin{align}\n 4  \\rho = \n  s_1 \\alpha_1  + (1-s_1) (1-\\alpha_1 )  +   \\alpha_2 \n   +   s_3 \\alpha_3  + (1-s_3)(1- \\alpha_3 ) +   \\alpha_4   .\n\\end{align}\nSolving this together with the current conservation  \\eqref{eq:J=J1=J2}, \nwe find a restriction on the shock positions  \\cite{bib:BSB} \n \\begin{align}  s_1 +  s_3 = 2 s  ,  \\label{eq:s1+s3=s} \\end{align} with $s$ \\eqref{eq:alpha1=,s=}. \nThe form of the density $ \\alpha_1  $ is the same as  for the 2-segment case \\eqref{eq:alpha1=,s=}.\nThe current $J$ is also unchanged \\cite{bib:BSB}, see fig.~\\ref{fig:diagrams}(b). One of the interesting findings in \\cite{bib:BSB} is that  we cannot fix  the shock positions $s_1$ and $s_3$ even in the macroscopic level, but they are synchronized  by the restriction \\eqref{eq:s1+s3=s}, see fig.~\\ref{fig:4-seg} (a).    The right bound of $ s_j $ is 1, and the left  $ \\lambda $ is given by solving $   2 s - \\lambda  = 1  $: \n\\begin{align}   \\label{eq:lambda<sj<1}\n  \\lambda      =   \\frac{ 2 ( 1 - 2\\rho ) }{ \\sqrt{ 1 - r }  }\n  <  s_j <  1 .  \\end{align} \nThe shock positions are uniformly distributed in this range except for the boundaries, see fig.~\\ref{fig:4-seg} (b).  Therefore the density profile   by averaging configurations over a large time interval becomes \n\\begin{align}\\label{eq:rho(x)_SMCSMC_over-time}\n\\langle  \\rho(x) \\rangle_{\\mathrm T}=\n \\begin{cases}  \n     \\alpha_1  \\ (  0  < x < \\lambda )   ,    \\    \\alpha_1    +    (x-\\lambda)  g \\  (  \\lambda  < x < 1   )   , \\\\ \n        \\frac{1}{2} \\   (  1 < x < 2  )  ,  \\\\ \n     \\alpha_1  \\ (  2  < x < \\lambda'   )   ,   \\    \\alpha_1  +   (x-\\lambda')  g \\  (  \\lambda'  < x < 3    )   , \\\\ \n        \\frac{1}{2} \\   (  3 < x < 4   )   ,  \n  \\end{cases}\n\\end{align}\nwhere $ \\lambda' = \\lambda+2 $ and $ g = \\frac{ 1 - 2 \\alpha_1 }{ 1 - \\lambda  } $.  In  fig.~\\ref{fig:4-seg} (c), we compare   simulation results of the density profile  $ \\rho_i $ \\eqref{eq:rho_i:=} with the prediction \\eqref{eq:rho(x)_SMCSMC_over-time} with $ x=i \/\\ell  $. We chose the values of parameters, such that the corresponding densities and left boundaries become \n\\begin{align}\n  \\nonumber \n &    ( r , \\rho   ) = (0.96,  0.5  ) , ( 0.84 , 0.48 ) , ( 0.64,0.44 ) , (0.36,0.38  )   \\Leftrightarrow   \\\\  \n &   ( \\alpha_1 , \\lambda  ) =  ( 0.4 , 0 ) , (0.3, 0.2   ) ,  ( 0.2 , 0.4  ) ,  ( 0.1, 0.6  ) , \n\\end{align}\nrespectively, according to eqs.~\\eqref{eq:alpha1=,s=} and \\eqref{eq:lambda<sj<1}. We expect that the discrepancies between the prediction and simulations are  due to the finite-size effect. In the half-filling case $\\rho = 1\/2$, the profiles of time average in the 1st and 3rd segments are identical to that of the well-studied open TASEP \\eqref{eq:rho_i=open}. Therefore one may naively expect that  the 1st and 3rd segments are effectively equivalent to the co-existence line of the open TASEP; e.g. for the 3rd segment, the 2nd and 4th segments play the role of reservoir of densities $\\alpha_1$ and $1-\\alpha_1$, respectively. Let us observe simulation results of MSD in the 4-segment TASEP, to verify  whether this guess is correct or not. \n\n\n\n\\begin{figure}\\begin{center} \n    \\includegraphics[width=0.24\\textwidth]{kymo.pdf} \\ \n    \\includegraphics[width=0.24\\textwidth]{unif.pdf} \n    \\includegraphics[width=0.48\\textwidth]{density-profiles-4.pdf} \n    \\end{center} \n\\caption{  \n (a)  Kymograph probing  \n   second-class particles and macroscopic density profiles\n    [eq.~\\eqref{eq:macro-density}   with   $\\delta=5$]  \n    in the 1st and 3rd segments \n    for $ (r,\\rho ,\\ell ) = (0.64, 0.5,200 )  $.   \n (b) Distribution  of the second-class particles, and (c) density profiles\n     for $ \\ell =200$, averaged over      $ 10^6 \\le  t \\le 10^8 $.\n The lines in (b)  are the uniform distribution over the range \\eqref{eq:lambda<sj<1}, and in (c) the prediction \\eqref{eq:rho(x)_SMCSMC_over-time} with $ x = i \/\\ell    $. \n\\label{fig:4-seg} } \n\\end{figure}\n\nDenote by $ S_j (t) + (j-1) \\ell  $  the microscopic  position  of the second-class particle  in segment  $ j \\in \\{1,3 \\} $ at time $t$, so   $ 1 \\le    S_j (t)\\le \\ell$.   Figure \\ref{fig:4-MSD}  (a)  shows the simulation results of the MSDs \n\\begin{align} M (t) = \\big\\langle \\big(  S_j (t) - S_j (0) \\big)^2 \\big\\rangle_{\\mathrm E} . \\end{align}\nThanks to the translational invariance, we omit the subscript $j$, and practically  we obtain $  M(t)  $ by averaging simulation data of $j=1$ and $j=3 $. In some time interval, indeed we see that the diffusion coefficient   \\eqref{eq:D=}     with  $\\alpha =  \\alpha_1 $ \\eqref{eq:alpha1=,s=} provides a good estimation of the MSD's behaviour: \n\\begin{align} M(t) \\approx 2 D ( \\alpha_1 ) t . \\end{align} \n Note that the observed  $   M(t) \/ (2t) $ in this \\textit{intermediate diffusive regime} is slightly lower than $ D $, which is expected to be a finite-size effect. \n\n\nFigure \\ref{fig:4-MSD} (a) indicates that the initial and asymptotic behaviours are also diffusive. The current of particles \\eqref{eq:J:SMC} gives a good fitting curve for the diffusion coefficient in the \\textit{initial diffusive regime}, see fig.~\\ref{fig:4-MSD} (b):   \n\\begin{align} \\label{eq:M=2Jt}  M(t) \\simeq 2 J  ( \\alpha_1  ) \\,  t \\quad ( t\\to +0)  , \\end{align} \nwhich implies that the second-class particles obey the symmetric random walk with jump rate $ J ( \\alpha_1  ) $ in a very short time scale.  The same finite-time effect is observed in the open TASEP [fig.~\\ref{fig:openTASEP} (a)], where one can prove that  both probabilities of finding configurations $20$ and $12$ are given by $ J(\\alpha) $ by using matrices \\cite{bib:A1,bib:A2,bib:U,bib:ALS}. \n\n \n\nOn the other hand,  for the  \\textit{asymptotic diffusive regime},\n  \\begin{align}  M(t) \\simeq 2 \\mathcal D ( \\alpha_1  ) \\, t \\quad ( t\\to \\infty) ,    \\end{align} \nwe expect the inequality   $ \\mathcal D ( \\alpha_1  ) < D ( \\alpha_1  )$,  see fig.~\\ref{fig:4-MSD} (a). Note that the conclusion on the asymptotic behaviour is based on our simulations of  up to $t = 10^6$.  Intuitively, no further transition is expected in longer-time simulations. (We considered  the ``asymptotic'' behaviour  in   $ t \\ll \\ell^2 \/ \\mathcal D $, so that the shocks do not reach  boundaries $ S_j(t) =\\lambda \\ell , \\ell  $. When we take the limit $  t\\to \\infty $ with $ \\ell$ fixed,  the MSD converges to a constant.)  The notation $ \\mathcal D (\\alpha_1) $ emphasizes that the diffusion coefficient is a function of $\\alpha_1$, but the dependence on $ \\ell $ and $ \\lambda $ may also be nontrivial. \n \n \nIn the  time interval between the intermediate and asymptotic diffusion regimes, the MSD exhibits sub-diffusion.  The bold lines (slope $-1\/2 $) are drawn in the log-log graph, fig.~\\ref{fig:4-MSD} (a), which agree well with simulations. We also estimate its exponent $\\varepsilon$ $(\\ln  M(t) \\approx  \\varepsilon \\ln t )$ by \n\\begin{align}\\label{eq:eps=}\n\\varepsilon = \\frac{ 1 }{ 3t\/2 + 1 }\n \\sum_{t\/2 \\le  t' \\le 2t } \\frac{ \\ln M ( 10t' ) - \\ln M ( t' ) }{ \\ln 10 } , \n\\end{align}\nin this \\textit{transient regime.}  \nThe exponent by this formula takes the minimum, which is  $\\approx 1\/2$, see  fig.~\\ref{fig:4-MSD} (c).  \n\n \n\n\n\\begin{figure}\\begin{center}  \n    \\includegraphics[width=0.48\\textwidth]{MSD-4.pdf}  \\ \n    \\includegraphics[width=0.24\\textwidth]{initial-diffusion.pdf}\n    \\includegraphics[width=0.24\\textwidth]{epsilon.pdf}\n     \\end{center} \n\\caption{  \n (a) MSD of the shock positions,\n (b) initial diffusion coefficient, and\n (c) exponent $ \\varepsilon$   \\eqref{eq:eps=}.        \n In (a), we divided $M(t)$ by $2t$, so as to see its property more easily.\n In (a,b), the solid, dashed and dotted lines correspond to $J$ [eq.~\\eqref{eq:J:SMC}], $D$ [eq.~\\eqref{eq:D=}] and $\\mathcal D$ [simply $M(10^6)  \/ (2\\times 10^6 )  $ from simulations], respectively. \n To obtain numerical data in (a,c), we took averages over  $ 10^6 $ and $ 10^3 $ simulation runs \n for $ t\\le 2 $ and $ t>2  $, respectively.  For each marker in (b),   we averaged  $  M(t)  \/ (2t)  $   of   $ 10^{-3} \\le t\\le  10^{-2}  $  over $ 10^6 $ simulation runs. \n\\label{fig:4-MSD} }  \n\\end{figure}\n\nLet us investigate the correlation of the shocks.  We expect that, in a very short time,  the two shocks move independently, since they are far from each other. On the other hand,   they are  synchronized in the frame of  a larger time scale, e.g. as  fig.~\\ref{fig:4-seg} (a). In order to  quantify  (in)dependency of the two shocks, we introduce a correlation function \n\\begin{align}\n\\label{eq:C(t)=}\n C(t)  = - \\big\\langle   \\big( S_1 (t) - S_1  (0) \\big) \\big( S_3  (t) - S_3  (0)   \\big)  \\big\\rangle_{\\mathrm E}   . \n\\end{align}\nDue the synchronization $ S_1 (t) + S_3 (t) \\approx \\ell s $, we have  \n\\begin{align} C(t) \\simeq M (t) \\quad ( t \\to + \\infty), \\end{align}  \n   which is observed in fig.~\\ref{fig:4-Correlation} (a).\nOn the other hand, their initial behaviours are completely different, see fig.~\\ref{fig:4-Correlation} (b,c).  In the vicinity of $t= 0$, we conjecture that $C(t)$ increases more slowly than any power function $t^a (a>0)$. \n \n\n\\begin{figure}\\begin{center} \n     \\includegraphics[width=0.24\\textwidth]{long.pdf} \\ \n     \\includegraphics[width=0.24\\textwidth]{medium.pdf} \\ \n     \\includegraphics[width=0.24\\textwidth]{short.pdf} \\ \n     \\includegraphics[width=0.24\\textwidth]{t-vs-ell.pdf}  \\end{center} \n \\caption{  \n (a,b,c) Comparison between the MSD and the correlation function  \n  in different time windows, and\n  (d) crossover time vs segment length. \n   For  (a,b,c),  we have set the values of the parameters as $ (\\ell,r, \\rho) = (1000,0.64,0.44) $,  \n   and averaged over $ 10^3,10^4 $ and $ 10^6 $ simulation runs,  respectively.\n In (d), each marker was plotted by averaging over  $ 10^3 $ simulation runs, \n  and the lines are fitting curves $ c \\, \\ell^z$.  \n\\label{fig:4-Correlation} } \n\\end{figure}\n\nLet us define the time $T$ as \n \\begin{align} T  = \\inf \\big\\{ t > 0 \\big|  2  C(t)   > M(t)  \\big\\} ,  \\end{align}  \ncharacterizing the crossover between the time scales of independency and synchronization.  \nFigure \\ref{fig:4-Correlation} (d) shows  $T$ vs segment length $\\ell$. By assuming the power law  $ T \\simeq c\\, \\ell^z $, we perform fitting from the simulation data that we have. We write the results directly in fig.~\\ref{fig:4-Correlation} (d), and draw corresponding straight lines as well. \n Under the further assumption that the exponent is independent from parameters, \nwe simply average the four obtained exponents:  $z \\approx 1.71$.  \n\n\n\n\\section{Discussions}\\label{sect:disc}\nIn this work,  we investigated synchronization of shocks in the 4-segment  TASEP, by means of  the two second-class particles.  We found that the behaviour of the MSD $ M(t) $ of the shocks is not so simple. In the initial regime (very short time scale), it is diffusive and the coefficient is nothing but the formula for the particle current. In the intermediate diffusive regime, the coefficient $D$ known in the open TASEP provides a good estimation. This fact indicates that,  up to the intermediate diffusion regime,  the shocks' motions are determined by the values of low and high densities, and independent from details of boundary conditions.  Via sub-diffusive regime, the MSD achieves the asymptotic diffusive regime, where the diffusion coefficient is different from $D$.  From the log-log graph and the numerical estimation, fig.~\\ref{fig:4-MSD} (a,c), it seems that $ M(t)\\propto \\sqrt t$ in the sub-diffusive regime.   In some random walks,   similar changes of the diffusivity have been found \\cite{bib:PM,bib:TSJS}.    In our case, the \\textit{regime change} is spontaneously induced by the particle number conservation, without directly imposing any interaction between the two second-class particles in defining the model. \n\n\nWe also investigated the correlation function of the shocks, and the crossover time between independency and synchronization of the shocks. The correlation function $C(t)$ \\eqref{eq:C(t)=} becomes identical to $M(t)$ as $t\\to +\\infty$.  In the vicinity of $  t=0 $, $ C(t) $ increases very slowly, whereas  $ M(t) $ is linear.  We defined the crossover time $T$ as the time when the ratio $ C\/M $ exceeds $ 1\/2 $.   We estimated the dynamical exponent $z$ for $ T $ by using our simulation data, which was found between  the KPZ $ z=3\/2  $ \\cite{bib:GS} and the normal diffusion $ z=2 $.  The  exponent $z$ as well as $ \\mathcal D  $ and $ \\epsilon $ should be  more precisely estimated  in larger systems   with longer simulation time and by more simulation runs. \n\n\n\nThe generalization to the $2n $-segment case is straightforward: the $j$th segment has the rate 1 (resp. $r$) when $j$ is odd (resp. even). When $ \\rho < \\rho_c $, the density profile $ \\rho (x)  \\ ( j - 1 < x < j ) $ of $j$th segment is give as $\\rho (x)  = \\alpha_1$ for odd $j$, and $\\rho (x)  = \\alpha_2$ for even $j$, with the same forms as in the 2-segment case \\eqref{eq:alpha1=,alpha2=}. When the global density $ \\rho_c < \\rho < 1- \\rho_c $, the density profiles are flat with density $ 1\/2 $ in the segments with even numbers.  On the other hand, $n$ shocks appear in segments $j=1,3, \\dots,2n-1 $. Denoting their positions by $ s_j + j - 1 $, we find \n $ s_1 + s_3 + \\cdots + s_{2n-1 } = n s ,  $ from the conservation of the number of particles. As an analogue to the $n=2$ case, we expect that they are not static but synchronized. Since only one equation governs the synchronization of the $ n $ shocks, we naturally expect that the asymptotic diffusion constant $ \\mathcal D_n( \\alpha )$ enjoys \n $   \\mathcal D_1 (  \\alpha ) < \\mathcal  D_2 (  \\alpha ) < \\mathcal  D_3 (  \\alpha ) <  \\cdots $\n with $ \\mathcal D_1 \\equiv 0 $ and $ \\mathcal D_2 ( \\alpha ) = \\mathcal D ( \\alpha ) $. \nA further intuitive conjecture is that the sub-diffusive regime vanishes as $ n \\to + \\infty $, and \n$\\lim_{ n\\to\\infty } \\mathcal  D_n (  \\alpha )   = D(\\alpha)$. \n \n It is known that two shocks are synchronized in the model \\cite{bib:CCB}. A simple generalization of the JL model \\cite{bib:SB}, e.g. $ p_i = r ( i \\in \\{\\ell , 2\\ell\\} ) , 1 ( i \\notin \\{\\ell , 2\\ell\\} ) $ with $L =2 \\ell $, also exhibits the same type of synchronization.  One of important questions is whether the MSDs of shocks in these models behave like fig.~\\ref{fig:4-MSD} (a), and if yes,  whether  the asymptotic  diffusion coefficient is the same as for the 4-segment TASEP.   Note that the positions of the two synchronized shocks are, in general, far from each other in these models as well as the 4-segment TASEP.  (See e.g.  \\cite{bib:MH,bib:JWW,bib:J,bib:DG} for other types of  synchronization.) While the motions of the  second-class particles  are locally defined, the shocks  move as if they \\textit{knew} each other's position.    We believe that this viewpoint gives hints to study self-organization phenomena,  e.g. in biological cells,  as the exclusion process is one of basic models also in biophysics \\cite{bib:CMZ}.  Application to traffic flows would be also an interesting problem. The local inhomogeneities in the JL and generalized JL models are very similar to traffic lights, and combinations of segments that we studied here evoke different limit speeds of cars.\n\n\n\n\n\\section*{Acknowledgements}\n The author thanks Ludger Santen and M Reza Shaebani for useful discussions.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nComposition and behavior of ecological communities are shaped by direct and indirect interactions between the species of these communities, such as the competition for the physical space and the intrinsic and the extrinsic resources. \nThe examples of such  competitive ecosystems are microbial communities in various biomes such as the soil~\\cite{ratzke2020strength}, the ocean~\\cite{tilman1977resource,strom2008microbial} and the human body~\\cite{foster2017evolution} - in particular the human gut which  hosts a diverse microbiome  whose dynamics are important for human health~\\cite{coyte2015ecology,gorter2020understanding}. \nIn the context of cellular populations within organisms, the evolution of neoplasms and tumor cells \\cite{merlo2006cancer,kareva2015cancer,smart2021roles}, interactions within the immune system ~\\cite{tauber2008immune,schmid2003evolutionary}, as well as the appearance of dominant clones during cell reprogramming~\\cite{shakiba2019cell}, exhibit phenomenology akin to ecological competition. \nBeyond biology \\cite{tilman1982resource,morin2009community,tuljapurkar2013population}, competitive interactions shape behaviors in a vast array of systems such as competition economics \\cite{budzinski2007monoculture} and social networks \\cite{koura2017competitive}.\n\nA classical example of the effects of inter-species competition - which inspired important ecological competition paradigms  - is the differentiation in beak forms of finches in the Gal\\'apagos islands \\cite{lewin1983finches,lack1983darwin}. \nOn these islands, dissimilar finch species possess beaks of varying shapes and sizes allowing them to consume different food sources and thus occupy distinct niches; this type of ecosystem structure is commonly referred to as an ecological niche model \\cite{grant1979darwin,pocheville2015ecological}. \nVarious niche models have been used to describe the community structures observed in diverse ecosystems such as plant grassland communities \\cite{zuppinger2014selection,silvertown2004plant}, marine plankton~\\cite{cullen1998behavior} and conservation ecology~\\cite{melo2020ecological,aguirre2015similar}.\nCommonly, niche specialization results in weaker competition for resources between individuals occupying separate niches (inter-species competition) compared to the competition between individuals of the same kind residing in the same niche (intra-species competition). The competition strength (determining the niche overlap \\cite{badali2020effects,capitan2015similar})  defined as the ratio between the inter-specific and intra-specific competition strengths. \n\nAnother paradigmatic class of ecological models comprises neutral models that are often used to describe noisy ecosystems wherein individuals from distinct species are functionally equivalent. \nIn contrast to niche models, interactions between all individuals in neutral models are identical regardless of their species  \\cite{bell2001neutral,hubbell2001unified,chave2004neutral}.\nThus, neutral models have commonly served as the null hypotheses for the exploration of ecological processes in various settings where the differences between inter-specific and intra-specific interaction are functionally negligible \\cite{bell2001neutral,gotelli2006null,blythe2012neutral}.\nNeutral theories can be viewed as a limit of niche theories where inter-specific and intra-specific interactions are equal: in other words, all species reside in completely overlapping niches ~\\cite{grover1997resource,begon2006ecology,pocheville2015ecological}.\n\nIn multi-species communities, the intra- and inter-species interactions as well as interactions with the environment, can lead to complex community composition and population dynamics; some species survive in the long term, while others are driven to extinction.\nHowever, in large communities with high numbers of competing species, it is often impractical or impossible to characterize the entire system composition by the assemblage of abundances for each species.\nHence, coarse-grained paradigmatic descriptions are often used to provide general insights into the common behavior of these ecological communities.\n\nTwo variables commonly used to characterize complex ecological communities are 1) the richness, reflecting the number of co-occurring species \\cite{adams2009species,kery2020applied}, and 2) the species abundance distributions (SAD) - the number of species present at a given abundance. The latter is  closely related to the species rank abundance (SRA) - the species ranked in terms of their abundance \\cite{nias1968clone, rulands2018universality, de2020naive, mcgill2007species, matthews2015species}.\nThese aggregate variables are observable experimentally and serve as the reporters on the underlying community structure, dynamics and the interaction network \\cite{rahbek2001multiscale,hong2006predicting,adler2011productivity,valencia2020synchrony}. \nRichness, for example,  is commonly considered to be an indicator of the competition strength and stability of the ecosystem \\cite{pimm1984complexity, ives2000stability, jousset2011intraspecific,mallon2015microbial,capitan2017stochastic}.\n\n\nThe shape of the SAD is also used as a proxy for the structure of the underlying interactions' network.  For high immigration or weak inter-species competition, the SAD commonly has a peak at high species abundance, away from extinction.\nThis community structure is closely related to the niche models whereby different species co-exist: most species inhabit their own niches with their species abundance fluctuating around the peak of the SAD. Conversely, other ecosystems, such as many microbial communities and T-cell repertoires, exhibit few high-abundance species  alongside highly diverse populations of low-abundance species \\cite{lynch2015ecology, de2020naive}.\nThis `rare biosphere' or `hollow-curved distribution' is described by a unimodal, monotonically decreasing SAD. Interestingly, this unimodal behaviour is empirically observed in many different ecosystems and is often considered universal (see \\cite{leidinger2017biodiversity} and references therein). Neutral models have been championed to describe the emergence of this universality, although other theoretical explanations for the `rare biosphere' SAD in competitive ecosystems have been suggested \\cite{mcgill2007species,magurran2013measuring}.\n\n\n\n\n\n\n\n\n\n\nMany theoretical studies that have aimed to quantify the competitive dynamics, the richness and the abundance distributions in ecological populations applied to  various systems, commonly employ a small number of paragidmatic models.\nOne common model of ecological competition is the deterministic, competitive Lotka-Voltera (LV) model, which has been especially useful in characterizing the niche regime by describing stable species coexistence as stable fixed points of the model. \nDepending on the ratios of inter- and intra-species competition strengths, deterministic LV models provide examples of both the `niche-like' regimes of multiple species coexistence, and the competitive exclusion where species with weaker intra-species interactions drive others to extinction \\cite{hardin1960competitive,macarthur1967limiting,MacArthur1969species, gause2019struggle}.\nIn complex scenarios, such as when the strengths of inter-specific interactions are randomly distributed among different species pairs, multi-species deterministic LV models can exhibit not only deterministic fixed point coexistence but also chaotic behavior reflected in the SAD shapes and richness \\cite{scheffer2006self,vergnon2012emergent,kessler2015generalized,bunin2016interaction,roy2020complex}. \nBeyond disorder in the interaction network, dynamical noise from various sources - both extrinsic and intrinsic - can have important effects on the system composition and dynamics, especially in the neutral regime.\nIn order to capture experimentally observed stochastic fluctuations of population abundances,  environmental noise is often introduced into the mathematical models \\cite{fisher2014transition,lynch2015ecology,verberk2011explaining,fowler2013colonization,barabas2016effect}.\nIn particular, by tuning the strength of environmental noise the shape of the SAD can change from unimodal to bimodal \\cite{fisher2014transition}, indicating a transition between `niche-like' and `neutral-like' regimes.\n\nRegardless of the presence of the external environmental noise or randomness in the interaction network, the demographic noise - the inherent randomness of birth and death events - is ever-present and has fundamental impact on the community structure and stochastic population dynamics \\cite{hubbell2001unified,alonso2006merits,haegeman2011mathematical}. \n\nIn particular, demographic noise has been suggested to be responsible for the SAD shape in neutral systems; these are characterized by the power law decay with an exponential cutoff that may account for `rare biosphere' abundance and distributions observed in many experimental systems \\cite{hubbell2001unified,baxter2007exact,mckane2004analytic}. On the other hand, birth-death-immigration processes with demographic noise have also been shown to exhibit bimodal SADs at very low immigration rates \\cite{xu2018immigration} breaking from the paradigm wherein neutrality is synonymous to an SAD of `rare biosphere' type.\nAlthough demographic noise models have been shown to reproduce observed features of a number of ecological systems \\cite{haegeman2011mathematical,capitan2015similar,capitan2017stochastic,capitan2020competitive}, a complete picture of the different regimes of community structures, is still missing. \nIn particular, it remains to be fully understood how the interplay of the competition strength, the immigration rate, demographic noise and the resulting dynamics of species turnover shape transitions between these different community structure regimes.\n\n\n\n\\begin{figure}[t!]\n   \\begin{flushleft}\n        A\n   \\end{flushleft}\n    \\includegraphics[width=\\columnwidth]{figures\/neutral-vs-niche.pdf}\n    \\begin{flushleft}\n        B\n   \\end{flushleft}\n    \\includegraphics[width=\\columnwidth]{figures\/island-2.pdf}\n    \\caption{Island model.  Panel A: Conventionally, weak competition is associated with `niche-like' bimodal SAD, while strong competition is linked to `neutral-like' monotonously decreasing SAD. However, this paradigm is not complete, since the dependence on other parameters, such as immigration rate $\\mu$ or diversity $S$, is not fully investigated. Thus, the entire phase space, e.g. $(\\mu, \\rho)$ or $(S, \\rho)$,  remains unexplored. Panel B: The model illustration. An island with $J$ individuals from $S^*$ species.  Each individual may proliferate and die with some rate corresponding to inter- and intraspecific interactions within the island. Here we consider deterministic, symmetric, fully-connected interspecific interactions network, governed by single parameter; the competitive overlap $\\rho$. Additionally, individuals may migrate from a cloud\/mainland, contains $S$ species, into the island with a constant rate $\\mu$.       }\n    \\label{fig:fig1}\n\\end{figure}\n\nIn this paper, we systematically study the full parameter space of the community composition and structure using a competitive LV model with the demographic noise and an interaction network of minimal complexity; more complex scenarios may be examined by building on this paradigmatic null model. We show that, beyond the perception of dichotomous neutral-niche regimes, many different regimes of richness and shape of the abundance distribution emerge from the interplay between the competition strength and immigration in the presence of stochasticity as illustrated in Fig.~\\ref{fig:fig1}).\nThese regimes exhibit contrasting dynamics that underpin the differences in the community structures in different regimes, and the transitions between them. \n\nThe paper is structured as follows. In Sec.\\ref{sec:model} we introduce the minimal model. In Sec.~\\ref{sec:results} we present our main results, including the regimes boundaries, their richness and the abundance distributions, as well as their associated underlying dynamics, and the species correlation structure. Lastly, in Sec.~\\ref{sec:discussion}, we discuss our results in the context of experimental observations. \n\n\n\n\n\\section{Mathematical models and methods}\n\\label{sec:model}\n\n\n\nThe minimal model studied in this paper incorporates three essential features of the ecological processes: competitive interactions, immigration and intrinsic demographic noise~\\cite{black2012stochastic,haegeman2011mathematical}.\nIn the model, illustrated  in Fig.~\\ref{fig:fig1}B, the community composition is characterized by the  species abundances, $\\vec{n}=(n_1,\\dots n_i\\dots n_S)$ where the discrete random variable $n_i$ represents the  number of individuals of the $i$-th species, and $S$ is the total number of species.\nThe dynamics of the system are described by a birth-death process with interactions, whereby the abundance (number of individuals) of any species can increase by one with the birth rate $q^+$ or decrease by one with the death rate $q^-$ defined as \n\\begin{align}\nq_i^+(\\vec{n})&=r^+ n_i +\\mu,  \\\\\nq_i^-(\\vec{n})&=r^- n_i + \\frac{r}{K} n_i \\left(n_i +\\sum_{j\\neq i} \\rho _{j,i} n_j\\right) \\nonumber\n\\end{align}\nfor each species $i\\in \\{1,2,\\dots,S\\}$.\n\nThe birth rate incorporates two factors: the per-capita birth rate $r^+$ corresponding to procreation, and the constant and positive immigration rate $\\mu$ from an external basin which ensures that the system possesses no global absorbing extinction state \\cite{capitan2015similar}. \nThe death rates include the `bare' per-capita death rate of the organisms $r^-$ and the competitive interactions effects that increase the mortality at high population numbers, incorporated through a quadratic term in the death rates; Parameter $\\rho_{j,i}$ quantifies the competition strength  between species $i$ and $j$.\nThe carrying capacity for each species is represented by $K$.\nThe per-capita turnover rate is $r=r^+-r^-$.\n\nThese aggregate coarse-grained parameters are determined by a variety of system factors such as the efficiency of resource consumption, interactions with the environment and external forces. Although it is possible to derive these rates from explicit resource competition models in several special cases, the expressions are highly model-dependent and are not explicitly modeled here\n~\\cite{macArthur1970species,chesson1990macarthur,o2018whence}.\nFor biological reasons,  $K$, $r^+$, $r^- > 0$ are all positive, which results in strictly positive transition rates for all $ n_i\\geq 0$. In this paper, we focus on the homogeneous case where the parameters ($\\mu$, $K$, $\\rho$, $r^+$, and $r^-$) are identical for all species and the competitive interactions  $\\forall i,j :\\rho_{j,i}=\\rho$ for all species pairs. \nThis symmetric and homogeneous interaction network \nhas been used in \\cite{badali2020effects,capitan2017stochastic,capitan2020competitive,haegeman2011mathematical} in contrast to the models   wherein the competition strengths are inhomogeneous and drawn from a distribution \\cite{fisher2014transition,allesina2012stability}. \nThis  minimal complexity model allows us to investigate the full phase space of the system to examine the underlying principle without impractical multi-parameter sweeps.\n\nThe stochastic evolution of the system is described by the master equation\n\\iffalse\n\\begin{multline}\n\\label{master-eq}\n\\partial_t  {\\rm \\mathcal{P}}(\\vec{n};t)= \\sum_{i}\\left\\{ \\vphantom{\\left[ q^+_i\\right]} q^+_i (\\vec{n}-\\vec{e}_i){\\rm \\mathcal{P}}(\\vec{n}-\\vec{e}_i;t) \\right.\\\\\n+q^-_i (\\vec{n}+\\vec{e}_i){\\rm \\mathcal{P}}(\\vec{n}+\\vec{e}_i;t)\\\\ \n-\\left. \\left[q^+_i(\\vec{n})+q^-_i(\\vec{n})\\right]{\\rm \\mathcal{P}}(\\vec{n};t)\n\\right\\}\n\\end{multline}\n\\fi\n\\begin{multline}\n\\label{master-eq}\n\\partial_t  {\\rm \\mathcal{P}}(\\vec{n};t)= \\sum_{i}\\left\\{ -\\left[q^+_i(\\vec{n})+q^-_i(\\vec{n})\\right]{\\rm \\mathcal{P}}(\\vec{n};t) \\vphantom{\\left[ \\sum q^+_i\\right]} \\right.\\\\\n\\left. \\vphantom{\\left[ \\sum q^+_i\\right]} +q^+_i (\\vec{n}-\\vec{e}_i){\\rm \\mathcal{P}}(\\vec{n}-\\vec{e}_i;t)+q^-_i (\\vec{n}+\\vec{e}_i){\\rm \\mathcal{P}}(\\vec{n}+\\vec{e}_i;t)\\right\\},\n\\end{multline}\nwhere $\\vec{e}_i$ is the standard basis vector and ${\\rm \\mathcal{P}}(\\vec{n},t)$ is the joint probability density function for the system to exhibit the species composition $\\vec{n}$ at time $t$ \\cite{gardiner1985handbook}. In the long time limit, the system reaches a stationary state where $\\partial_t \\mathcal{P}=0$.\n\nThe species abundance distribution (SAD) describing the mean fractions of species with $n$ individuals, can be related to the marginal single species probability distribution $P(n)$:\n\\begin{align}\n    {\\rm SAD}&(n) = \\frac{1}{S}\\left\\langle \\sum_{i=1}^{S}\\delta (n_i -n) \\right\\rangle\\\\\n   \\nonumber &=\\frac{1}{S}\\sum_{i=1}^S\\left[ \\sum_{n_1=0}^{\\infty}\\cdots \\sum_{n_{i-1}=0}^{\\infty}\\sum_{n_{i+1}=0}^{\\infty}\\cdots \\sum_{n_S=0}^{\\infty}{\\rm \\mathcal{P}}(\\vec{n})|_{n_i=n}\\right] \\\\  &=\\nonumber  P_i(n) \\equiv  P(n),\n\\end{align}\nwhere $\\delta$ is the Kronecker delta function, and using the fact that in this homogeneous system the marginal distributions $P_i(n)=P(n)$ of population abundance are identical for all species. \n\n\\iffalse\nThe full master of the Equation...  can be reduced to the one dimensional master equation for the marginal distribution $P(n)$ with effective birth-death rates (see SM for derivation)[AZ: is this an exact reduction, or there is an approximation involved?]\n\n\\begin{eqnarray}\nq^+(n)&=&r^+ n +\\mu,  \\\\\nq^-(n)&=&r^- n + \\frac{r}{K} n \\left((1-\\rho)n + \\rho \\langle J | n_i = n \\rangle \\right). \\nonumber\n\\end{eqnarray}\n\\fi\n\nIn the Fokker-Planck approximation, the continuous deterministic limit of the master equation (\\ref{master-eq}) recovers the well-known competitive Lotka-Volterra (LV) equations\n\\begin{align}\n    \\frac{\\partial x_i}{\\partial t}&= q_i^+(\\vec{x}) - q_i^-(\\vec{x})\\nonumber\\\\\n    &=r x_i \\left( 1 - \\frac{x_i}{K} + \\sum_{j\\neq i} \\rho \\frac{x_j}{K} \\right) + \\mu \n    \\label{eq:LV}\n\\end{align}\nfor the variable $x_i$, which corresponds to the continuous deterministic limit\nof the discrete variable $n_i$  \\cite{gardiner1985handbook}; see Supplementary Materials (SM) for further details. \n\nThe deterministic steady state is given by\n\\begin{equation}\n    \\tilde{x}(S) = \\frac{K}{2[1+\\rho(S-1)]}\\left\\{1 +  \\sqrt{1+\\frac{4\\mu[1+\\rho(S -1)]}{r K}}\\right\\}.\n    \\label{eq:fixed-LV}\n\\end{equation}\nNote that in the deterministic LV process all species survive with abundance $\\tilde{x}$ as long as $\\rho\\leq 1$ with $\\mu>0$ \\cite{capitan2015similar}.\nConversely, in the stochastic competitive environment the numbers of individuals of each species fluctuate, occasionally reaching extinction. \nThus, the number of (co-)existing species $S^*$ is a stochastic variable as well, and may be smaller than the overall number species $S$ in the immigration flux from the larger basin, with $S^*\\leq S$.\nThe richness, denoted as $\\langle S^* \\rangle $, is defined as the average number of the (co-)existing species, and is related to the SAD via\n\\begin{equation}\n\\label{eq:richness}\n\\langle S^* \\rangle = S(1-P(0))\n\\end{equation}\nwhich, intuitively, is determined by the probability that a species is present in the system,  $1-P(0)$.\n\nNo exact analytical solution for the high-dimensional master equation  \\eqref{master-eq} is known for a general competition strength $\\rho$. To understand the principles of the community organization and the impact of competition, immigration and demographic noise, we developed approximate analytical solutions to the master equation verified by Gillespie simulations (see SM for details).\n\n\\section{Results}\n\\label{sec:results}\n\n\\subsection{Mean-Field Approximation}\nThe full master equation \\eqref{master-eq} can be reduced to a one dimensional approximation for the marginal distribution $P(n)$ with effective birth-death rates (see SM). The SAD, $P(n)$, is obtained as a self-consistent stationary solution of this equation as\n\\begin{multline}\n   P(n) \\equiv P_i(n_i=n)\\\\\n   =P(0)\\frac{(r^+)^{n}(\\mu\/r^+)_{n}}{n!\\prod_{n_i=1}^{n}\\left(r^-+r n_i\/K+r\\rho \\sum_{j\\neq i}^S\\langle n_j |n_i \\rangle \/K\\right)}.\\\\\n   \\label{eq:mean-field}\n\\end{multline}\nTo obtain an analytical approximation to $P(n)$ we use a mean field closure for the unknown conditional averages $\\langle n_j |n_i\\rangle$ as\n$\\left\\langle \\sum_{j\\neq i} n_j |n_i\\right\\rangle\\approx  (S-1) \\langle n \\rangle$.\nThus, \\eqref{eq:mean-field} becomes a closed-form implicit equation for the probability distribution $P(n)$ which can be solved numerically.\nWe have found a good agreement between exact stochastic simulation results and this mean-field approximation for most of the parameter space examined (see SM). \n\nFollowing \\eqref{eq:richness}, the average richness in the mean-field approximation is (See SM)\n\\begin{eqnarray}\n\\langle S^*\\rangle = S \\left( 1 - \\frac{1}{{_1}F_1[a,b+1;c]}\\right ) , \n\\label{eq:mf-richness}\n\\end{eqnarray}\nwhere $P(0)=1\/{_1F_1}[a,b+1+1;c]$ is the normalization constant of $P(n)$ where ${_1F_1}[a,b;c] $ is the hypergeometric Kummer confluent function, with $a=\\mu\/r^+$, ${b}=\n[r^-K+r\\rho (S-1)\\langle n\\rangle  ]\/r$, and ${c}\n{r^+ K}\/{r}$.\n\n\n\n\n \n  \n\n\n\n\\subsection{The system exhibits rich behavior with distinct regimes of population structures controlled by competition strength, immigration rate and the species number} \n\\label{sec:Phases}\n\nDepending on the values of the competitive strength and the immigration rate, the number of species and the system size, the population can exhibit a number of different regimes of behavior, which can be categorized by their richness and the shape of their SAD, as visualized in Fig.~\\ref{fig:phases_sim} and described below.\n.\n\n\\subsubsection{Richness regimes}\nIn the classical deterministic LV model, the systems exhibits either an interior fixed-point with full coexistence of all species at abundances given by Eq.~\\ref{eq:fixed-LV}, or mass extinction with a single surviving species, in agreement with the well-known Gause's law of deterministic competitive exclusion \\cite{capitan2015similar}. By contrast, the stochastic model may also exhibit partial coexistence due to the\nabundance fluctuations arising from the demographic noise whereby a subset of the species are driven to temporary extinction. Overall, the number of co-existing species and their abundances are determined by the balance between the immigration and stochastic competitive exclusion events.\nThree distinct richness regimes can be discerned as shown in Fig.~\\ref{fig:phases_sim}, based on the variations of the richness of the system {$\\langle S^* \\rangle$} in different regimes in the ($\\rho$,$\\mu$,$S$) parameter space.\n\nAt low competition strength - region (a) in Fig.~\\ref{fig:phases_sim}A - all species co-exist so that the richness of the system is equal to the number of species, same as in the deterministic regime $\\langle S^*\\rangle \\approx S$. \nIn this regime, each species effectively inhabits its own niche because the inter-species competition is not sufficiently strong  to drive any of the species to extinction even in the presence of abundance fluctuations arising from the demographic noise.\nThe probability for a species to be present is determined by the balance of its immigration rate and the death rate.\nAt higher immigration rates this regime extends into regions with higher competition strength $\\rho$: high immigration rates stabilize full richness populations even with a relatively high competition strength.\n\n\n\nIn the second regime - region (b) in Fig.~\\ref{fig:phases_sim}A - only a fraction of the species are simultaneously present on average, which we denote as the partial coexistence regime.\nIn this regime, the immigration influx is not high enough to prevent temporary stochastic extinctions of some species resulting from the competition.\n\nAt the very high competition strengths a complete exclusion regime - region (c) in Fig.~\\ref{fig:phases_sim}A - is found. \nHigh competition along with the very low immigration rates act in unison such that the richness is less than two species.\nAlthough regime (c) may appear similar to regime (b) since both present partial coexistence, they are distinguished by key behavioral features as explained below. \n\nNote that the stochasticity is central to the  effect of the competition on the observed richness.\nStochastic fluctuations increase the risk of extinction with increasing competitive overlap, unlike in the deterministic case where the richness is independent of the interaction strength for $\\rho < 1$ \\cite{capitan2015similar}.\n\n\n\n\n\n\n\n\n\\subsubsection{SAD shape and modality regimes}\n\\begin{figure*}[ht!]\n    \\begin{minipage}{.30\\linewidth}\n   \\begin{flushleft}\n        A\n   \\end{flushleft}\n   \\includegraphics[width=\\textwidth]{figures\/Fig3-A.pdf}\n   \\end{minipage}\n    \\begin{minipage}{.65\\linewidth}\n   \\begin{flushleft}\n        B\n   \\end{flushleft}\n   \\includegraphics[width=\\textwidth]{figures\/Fig3-B.pdf}\n   \\end{minipage}\n    \\begin{minipage}[t]{.48\\textwidth}\n   \\begin{flushleft}\n    C\n    \\end{flushleft}\n    \\vspace{0pt}\n    \\includegraphics[width=\\textwidth]{figures\/Fig3-C.pdf}\n  \\end{minipage}\n  \\hfill\n  \\begin{minipage}[t]{.48\\textwidth}\n    \\begin{flushright}\n        \\begin{flushleft}\n        D\n        \\end{flushleft}\n    \\vspace{8pt}\n    \\includegraphics[width=\\textwidth]{figures\/Fig3-D.pdf}\n    \\end{flushright} \n  \\end{minipage}\n    \\caption{Phenomenology of the population structures. Panel \\textbf{A}: The system possesses three distinct richness phases. (a): full coexistence of all the species $\\langle S^* \\rangle \\approx S$; (b): partial coexistence with $\\langle S^* \\rangle < S$; (c): a single species exists on average. Panel \\textbf{B}: Different population regimes are distinguished by different SAD modalities. (I): immigration dominated regime with unimodal SAD at a typical abundance given by the positive root of $\\tilde{n}$; (II): bimodal regime with species at non-zero abundance $\\tilde{n}$ and a rapid species turnover peak a zero abundance; (III): `rare biosphere' regime of a unimodal SAD with peak at zero abundance resulting from the rapid turnover of the temporarily extinct species; (IV) multimodal regime.\n    Panels \\textbf{C} and \\textbf{D}: Intersection of the modality and richness regimes in the $(\\mu,\\rho)$ plane (\\textbf{ C}) and $(S,\\rho)$ plane (\\textbf{D}); see text for discussion. In panels \\textbf{ A}, \\textbf{B} and \\textbf{C} the number of  species $S=30$.  In panel \\textbf{D} the immigration rate is $\\mu=10^{-1}$. For all panels; Colored regions represent data from simulation (see Methods), whereas boundaries from the mean-field approximation are represented by solid black lines.\n   \n    The solution for the master equation \\eqref{master-eq} is simulated using the Gillespie algorithm with $6 \\cdot 10^8$ time steps, $r^+=2$, $r^-=1$, and $K=100$.\n   \n    }\n    \\label{fig:phases_sim}\n\\end{figure*}\nBesides the richness, the balance between immigration and stochastic competitive extinctions  also dictates the mean abundances of individual species and the species abundance distribution (SAD).\nWhen the immigration influx of individuals into the system is higher than the average out-flux due to the transient extinctions, shown in Fig.~\\ref{fig:phases_sim}B as region (I), most species are forced away from extinction.\nIn this regime, the SAD is unimodal with a peak at relatively high species abundances $\\tilde{n}$  which is approximately located at \n\\begin{equation}\n    \\label{eq:dom-level}\n    \\tilde{n} =\n    \\frac{K-\\rho(S-1)\\langle n\\rangle}{2}\\left\\{1 \\pm \\sqrt{1+4\\frac{(\\mu-r^+) K}{r(K-\\rho(S-1)\\langle n\\rangle)^2}}\\right\\},\n\\end{equation}\nwhich agrees with the simulation results, see Fig.~\\ref{fig:Fig3}; see  SM.\n\n\n\nAt lower immigration rates - regime (II) in Fig.~\\ref{fig:phases_sim}B - the immigration rate is insufficiently strong to overcome the competition driven temporary extinctions of some of the species, and the SAD develops an additional peak around $n=0$ corresponding to the temporarily extinct species. \nThe subset of the `quasi-stably' co-existing species whose abundances fluctuate around $\\tilde{n}$ within the high `niche-like' abundance peak dominate the population number, punctuated by rare  fluctuation-driven extinctions and the occasional invasion of a temporarily extinct species into the dominant population.\nBy contrast, the dynamics of species in the $n=0$ zero peak is characterized by the rapid turnover of the remaining species close to extinction.\n\n\n\n\nAt low immigration rates, the peak at \\eqref{eq:dom-level} coincides with the deterministic stable solution in \\eqref{eq:fixed-LV}  (see SM)\n\\begin{equation}\n\\lim_{\\mu\\rightarrow 0}\\tilde{n}=\\lim_{\\mu\\rightarrow 0}\\tilde{x}\\left(\\langle S^* \\rangle \\right) = \\frac{K}{1+\\rho(\\langle S^*\\rangle -1)}.\n\\label{eq:Lim_ss}\n\\end{equation}\nNamely, in the bimodal regime the coexisting dominant species are fluctuating around $\\tilde{n}$ which, at low immigration, is the deterministic fixed point with $\\langle S^* \\rangle$ species. \nThus, the dynamics of abundant species around the $\\tilde{n}$ can be heuristically understood as spatially dependent diffusion in an effective potential well of the Fokker-Plank Equation (See Sec.\\ref{sec:model} and SM). \n\nSomewhat unexpectedly, at low immigration rate $\\mu \\lesssim .05$, the bimodal regime extends onto the neutral line at $\\rho = 1$  where the SAD has been commonly believed to have the monotonically decreasing `rare biosphere' shape \\cite{hubbell2001unified,baxter2007exact}.\nHowever, in this regime the competition is so strong that most of the time either no species are present at high abundance, or only one species survives in a kinetically `frozen' long lived quasi-stable state with the abundance $\\tilde{n}\\simeq K$, as observed previously \\cite{xu2018immigration}. \n\nFurthermore, at the intermediate immigration rates and relatively high competition strengths we observe a unimodal behaviour with a peak at zero rather than at finite $\\tilde{n}$ - region (III) in Fig.~\\ref{fig:phases_sim}B.\nIn this regime, the competition is strong enough so that the fluctuations competitively drive populations\nto temporary extinction before any species is able to establish a `quasi-stable' state at a high abundance.\nAll species undergo rapid turnover around zero resulting from the balance between random immigration and extinction events.\nThis regime corresponds to what was previously described as the `rare biosphere': fewer number of species are found at higher abundances resulting in a monotonically decreasing SAD. \nThis SAD shape is classically recognized as a hallmark of a `neutral-like' regime. However, as shown in Figure~\\ref{fig:phases_sim} the unimodal regime (III) unexpectedly extends substantially beyond the neutral manifold $\\rho=1$, into the non-neutral regions with $\\rho < 1$, and the monotonic-decreasing SAD persists even for competitions strengths as low as $\\rho \\approx 0.1$ - an order of magnitude weaker than the classical neutral regime.\nThis challenges the common perception that the SAD of the `rare biosphere' is necessarily closely related to neutrality.\n\nFinally, we have found an entirely novel multimodal regime with more than two peaks - regime (IV) in Fig.~\\ref{fig:phases_sim} -  which possesses one rapid turnover peak around extinction and multiple peaks at non-zero abundances. \nSimilar to region (IIc), the peak at $n=0$ comprises species which rapidly turnover around extinction. However, in addition to the peak at positive abundances $K$ formed by one surviving species ($S^*=1$) in a meta-stable frozen state, this regime possesses a second peak  at $\\sim K\/(1+\\rho)$ with two simultaneously surviving quasi-stable species ($S^*=2$), following \\eqref{eq:Lim_ss}. The slow fluctuations between the states with $S^*=1$ and $S^*=2$ result in the appearance of the SAD with two non-zero modes at quasi-stable dominance abundance, $\\tilde{n} \\sim K$ and $\\tilde{n}\\sim K\/(1+\\rho)$ observed in the region (IV); these two peaks are only visibly separated when the richness is low and carrying capacity is high, as follows from \\eqref{eq:Lim_ss}.\n  \n\n\n\n\n\nThe transitions between the different regimes and the corresponding changes in the SAD shapes are illustrated in Fig. ~\\ref{fig:Fig3}. Generally, at low competition strength $\\rho$, where the species are practically independent of each other and reside in largely non-overlapping niches, their typical abundance $\\tilde{n}$ is close to the carrying capacity $K$. Increasing competition strength $\\rho$ makes it harder to sustain co-existing species at high abundances, and accordingly $\\tilde{n}$ decreases, as illustrated in the top panels of Fig.~\\ref{fig:Fig3}A) and  Fig.~\\ref{fig:Fig3}B). With further increase in $\\rho$ the system behavior bifurcates depending on the immigration rats $\\mu$. \nAt high immigration rates, $\\mu \\gtrsim 0.05$, the competition-driven decrease in $\\tilde{n}$ continues up to the critical competition strength (calculated in the next section) where the peak around $\\tilde{n}$ disappears (top right panel of Fig.~\\ref{fig:Fig3}A) and  Fig.~\\ref{fig:Fig3}B), as the system is not able to sustain `quasi-stable' niche-like species co-existence. This corresponds to the transition from bimodal region (II) to the `rare biosphere' neutral-like region (III) in Fig.~\\ref{fig:phases_sim}).\nAt lower immigration rates (top left panel of Fig.~\\ref{fig:Fig3}A) and  Fig.~\\ref{fig:Fig3}B)), further increases in the competition strength eventually cause mass species extinctions which allows the remaining few dominant species to maintain higher abundances (region (III) Fig.~\\ref{fig:phases_sim}). As $\\rho \\rightarrow 1$, the system transitions to the region (IIc) of the Fig.~\\ref{fig:phases_sim}: only one dominant species remains, as described in \\cite{xu2018immigration}, with abundance $K$. \n\n\n\\subsubsection{Global Phase Diagram and Regime Boundaries}\n\\label{sec:regime-bound}\n\n\\begin{figure}[t!]\n   \\begin{flushleft}\n        A\n   \\end{flushleft}\n  \n   \n    \\includegraphics{figures\/Fig2-A.pdf}\n    \\begin{flushleft}\n        B\n   \\end{flushleft}\n   \n    \\includegraphics[width=\\columnwidth, trim= 0 0 0 15,clip]{figures\/Fig2-B.pdf}\n    \\caption{SAD changes between different regimes. Panel \\textbf{A}: (upper left) Simulation results for species abundance distributions (SADs) for fixed $\\mu=10^{-3}$ as a function of $\\rho$. (upper right) same for $\\mu=1$. Different values of the interaction strength  $\\rho$ are emphasized with different colors indicated in the color-bar. (lower left) Simulation results for SADs as a function of $\\mu$ for fixed $\\rho=0.5$  (lower right) same for $\\rho=1$. Different immigration rates $\\mu$ are emphasised with different color shown in the color-bar. \n    \\textbf{B}: The non-zero mode of the SAD  given by the positive solution of  $\\tilde{n}$ representing the dominant species abundance as a function of $\\rho$ for different values of $\\mu$. Markers and dotted lines represent simulation results, while solid lines are given from analytic analysis, \\eqref{eq:dom-level}.}\n \\label{fig:Fig3}\n\\end{figure}\n\nIn this section we describe the complete phase diagram of the system defined by the intersection of the different richness and the SAD shape\/modality regimes, derive the regime boundaries and discuss the transitions between them, as shown in the ($\\mu,\\rho$) space in Fig.~\\ref{fig:phases_sim}C, and in ($S,\\rho$) space in Fig.~\\ref{fig:phases_sim}D.\nWe show that the boundaries between different regimes observed in simulations can be understood within simple mean field theories, and discuss the underlying physical factors responsible for the transitions between different regimes.\n\n\\iffalse\n\\begin{center}\n\\begin{tabular}{lccc}\n\\hline\n\\multirow{2}{4em}{\\textcolor{black}{modality:}} & &\n      \\textcolor{black}{$R_n(\\tilde{x} \\rightarrow \\tilde{x} +1)=R_n(\\tilde{x}+1 \\rightarrow \\tilde{x})$} \\\\ &\n     & \\textcolor{black}{$R_n(0 \\rightarrow 1)=R_n(1 \\rightarrow 0)$}   \\\\\n    \\hline \n    \\multirow{2}{4em}{\\textcolor{black}{richness:}}  & & \\textcolor{black}{$R_{S^*}(1 \\rightarrow 2)=R_{S^*}(2 \\rightarrow 1)$} \\\\ &\n     & \\textcolor{black}{$R_{S^*}(S \\rightarrow S-1)=R_{S^*}(S-1 \\rightarrow S)$}\n\\\\\n\\hline\n\\end{tabular}    \n\\end{center}\n\n\\fi\n\nWe define the boundary between the full coexistence (a) and partial coexistence (b) regimes to be at $\\langle S^* \\rangle=S-1\/2$: the midpoint between full richness $S^*=S$ and the loss of 1 species on average.\nSimilarly, the boundary between the partial coexistence (b) and exclusion (c) regimes is located at $\\langle S^* \\rangle=3\/2$, that is to say where the richness is between one and two species such that on average only 1 species is present in regime (c).\n\nTo derive the boundaries corresponding to the transitions of the SAD modality regimes, we use discrete derivatives of the approximated SAD to determine the existence of peaks and their location (See SM).\nThe immigration dominated regime (I) is characterized by a unimodal SAD with a peak at the positive root of $\\tilde{n}$ given in \\eqref{eq:dom-level}.\nCompared to this immigration dominated regime, the neighboring bimodal and monotonically-decreasing unimodal regimes - regions (II) and (III) respectively - differ by the emergence of a new mode at zero abundance. \n\nThus, the boundary that defines transitions to either regime (II) or (III) from the immigration dominated regime (I) is described by a flattening of SAD at $n=0$: $\\partial P(n)\/ \\partial n|_{n=0} = 0$ which, in the discrete case, heuristically corresponds to $P(0)=P(1)$. Combining this condition for the boundary with the global-balance of the master equation \\eqref{master-eq}  results in the rate balance equation,\n$\\langle q_i^+(\\vec{n})|n_i=0 \\rangle=\\langle q_i^-(\\vec{n})|n_i=1 \\rangle$.\n\nIn the mean-field approximation, this boundary is found at\n\\begin{equation}\n    \\label{eq:boundary-I}\n    \\mu = r^- +\\frac{r}{K}[1+\\rho(S-1)\\langle n \\rangle ].\n\\end{equation}\nThis equation recovers the similar transition for $\\rho=1$ derived independently in \\cite{xu2018immigration}. \n\nThe boundary between the bimodal regime (II) and the `rare biosphere' regime (III) is characterized by the disappearance of the peak at high abundance $\\tilde{n}$ in \\eqref{eq:dom-level}.\nIn the bimodal regime at least one solution to $\\tilde{n}$ is real and positive, then a maximal, real peak-location exists. \nConversely, in the `rare biosphere' regime, both solutions of $\\tilde{n}$ are negative or imaginary.  \nWe find that the boundary between the real and imaginary $\\tilde{n}$ is\n\\begin{equation}\n\\label{eq:boundary-IIIa}\nr(K-\\rho (S-1) \\langle n \\rangle )^2=4(r^+-\\mu){K}\n\\end{equation}\nand the transition line between positive and negative solutions, $\\tilde{n}=0$, is\n\\begin{equation}\n    \\frac{(K- \\rho (S-1) \\langle n \\rangle )^4}{16}=1+ \\frac{K(\\mu-r^+)}{r}.\n    \\label{eq:boundary-IIIb}\n\\end{equation}\nThe intersection of these two conditions defines the `rare biosphere' regime and is shown as the blue line in Fig.~\\ref{fig:phases_sim}B,C.\n\nThe modality and the richness of the system are also affected by the number of species $S$ as shown in  Fig.~\\ref{fig:phases_sim}D.\nIn brief, the frequency of the immigration events rises as more species are present in the immigration flux.\nIncreased immigration causes the total population to rise without providing more room for each species in the system; this  increases the stochastic competition, driving more species to extinction.\nHence, as $S$ increases, the transition from the bimodal regime (II) to the unimodal regime (III) occurs at lower values of competition strength $\\rho$, and the fraction of the concurrently surviving species decreases. This effect has been qualitatively observed experimentally \\cite{gore2021}, and we return to it in the Discussion.\n\nThese analytical expressions for the regime boundaries - confirmed by stochastic simulations - provide insights into the effects of different control parameters on the regime boundaries. In particular, using the low $\\mu$ deterministic approximation for $\\langle n \\rangle \\approx K \/ \\left[ 1+\\rho (S-1) \\right]$, shows that the location of the boundary of the `rare biosphere' regime grows proportionally to the carrying capacity and is a decreasing function of the number of species $S$. Thus, the size of the `rare biopshere' neutral-like regime increases with the number of species $S$ as shown in Fig.~\\ref{fig:Fig3}D,\nwhereas increasing the carrying capacity shrinks this regime (See SM).\n\n\n\n\n\n\n\n\n\n\\subsection{Kinetics of the species turnover, extinction and recovery underlie the transitions between different regimes}\n\\label{sec:Dynamics}\n\nSo far we have focused on the steady-state properties of the system, such as the dominant species abundance, SAD modality, and richness to categorize the different regimes. \nHowever, the transitions between different regimes are closely related to the underlying kinetics of species turnover and fluctuations, which are investigated in this section.\n\nThe kinetics of an individual species change drastically between the unimodal `rare biosphere' regime (III) and the regimes that exhibit a peak in SAD at non-zero abundance as shown in Fig.~\\ref{fig:turnover}A  that contrasts the kinetics in these cases. \nIn regime (III), all species undergoes rapid turnover in the relatively broad range of abundances around extinction.\nIn contrast, in the `niche-like' regimes (I, II, and IV) the `quasi-stable' dominant species undergo fast fluctuations around the co-existence peak at $\\tilde{n}$ in addition to fast turnover of the remaining species near extinction. The `niche-like' regimes also possess slow timescales corresponding to individual species leaving the high abundance peak as they are temporarily driven towards extinction, and the reverse invasions of temporarily extinct species into the dominant `niche-like' peak.\n\nTo characterize the differences in the kinetics in different regimes, we calculate the mean first-passage times (MFPT) of the  transitions between different abundance levels characterizing different regimes, denoting  the MFPT for a species transition from an abundance $a$ to another abundance $b$ as  $T(a \\rightarrow b)$.  Similarly, $T(a\\rightarrow a)$ refers to the mean time of return to an abundance $a$ having left that same abundance.\nThe MFPT is calculated from the one-dimensional backward Master equation (see SM).\n\nAlthough these times are important indicators of the system dynamics, the MFPT ratios of two processes\/events is more informative than the MFPT of each event separately; this ratio measures the discrepancy between the timescales at which these events occur. \n\nTo understand the intrinsic kinetics that give rise to the different regimes in Fig.~\\ref{fig:phases_sim}, we first focus on the ratio of the MFPTs of the transitions from dominance to exclusion over the mean time of return to the dominant abundance level (starting from the dominant abundance level), $T(\\tilde{x}(\\langle S^* \\rangle )\\rightarrow 0) \/ T(\\tilde{x}(\\langle S^* \\rangle )\\rightarrow \\tilde{x}(\\langle S^* \\rangle ))$, shown in (Fig.~\\ref{fig:turnover}B).\nRecall that $\\tilde{x}$ is the solution given in ~\\eqref{eq:fixed-LV}; this deterministic solution is a natural extension of the peak $\\tilde{n}$ in regimes where positive modes are non-existent.\nAs explained below, this ratio underlies the changes in the modality of the SAD as a function of the immigration rates and the competition strength.\n\nLarge values of the ratio $T(\\tilde{x}(\\langle S^* \\rangle )\\rightarrow 0)\/ T(\\tilde{x}(\\langle S^* \\rangle )\\rightarrow \\tilde{x}(\\langle S^* \\rangle ))$\nimply that the extinction rate from $\\tilde{x}(\\langle S^* \\rangle )$ is much slower than the rate of local fluctuations in the effective potential well around $\\tilde{x}(\\langle S^* \\rangle )$. Accordingly, Fig.~\\ref{fig:turnover}B shows that this ratio is high in the bimodal and immigration-dominated regimes. Conversely, this ratio is lower within the `rare biosphere' regime that does not possess a high abundance peak with `quasi-stable' co-existing species.\nThis ratio approximately delineates the `rare biosphere' neutral-like  regime from the `niche-like' regimes, as shown in Fig.~\\ref{fig:turnover}B: the contour lines qualitatively recover the boundaries of region IIIb in Fig.~\\ref{fig:phases_sim}C.\n\nThe second ratio, which underlies the richness transitions in the system, \n$T(0\\rightarrow \\tilde{x}(\\langle S^* \\rangle ))\/T(0\\rightarrow0)$ (Figure.~\\ref{fig:turnover} panel C) relates the mean return time to extinction to the invasion time from extinction at zero abundance  to dominance at $\\tilde{x}$. \nHigh values of this ratio indicate that, for an extinct species, the mean invasion times are longer than return times back to extinction.\nThis MFPT ratio approximates the ratio of the average number of temporarily extinct species, $S - \\langle S^* \\rangle$ to the average number of existing species, $\\langle S^* \\rangle$, see Fig.~\\ref{fig:turnover}C. \nConsequently, the ratio quantitatively recovers the boundaries of richness regimes in Fig.~\\ref{fig:phases_sim} in most regions of the parameter space.\nThese MFPT ratios can be understood as the reciprocal of the rates ratios which describe how much more frequently an event occurs than the other.\nFurther discussion on the dynamical features are presented in the SM.\n\n\\begin{figure*}[t!]\n    \\centering\n    \\begin{minipage}{11cm}\n    \\begin{flushleft}\n        A\n   \\end{flushleft}\\includegraphics{figures\/Fig4-A.pdf}\n   \\begin{flushleft}\n        B\n   \\end{flushleft}\n   \\includegraphics{figures\/Fig4-B.pdf}\n    \\begin{flushleft}\n        C\n   \\end{flushleft}\n        \\includegraphics{figures\/Fig4-C.pdf}\n    \\end{minipage}\n    \\caption{Kinetics of species extinction, invasion and turnover. Panel \\textbf{A}:  Sample of the trajectories of the species abundances (grey lines).  We highlight five species' trajectories for visibility.\n    (Upper panel): stable `niche-like' dynamics, where the abundances of the dominant species mostly fluctuate in the vicinity of $\\tilde{n}$, with occasional transitions from  the dominance to nearly-extinct states. The red curve represents the corresponding bimodal SAD. Lower panel shows the erratic dynamics obtained in the `rare biosphere' regime, where all species are rapidly fluctuating close to extinction. The SAD in this case is unimodal monotonically decreasing function.  Panel \\textbf{B}: The MFPT ratio $ T(\\tilde{x}\\rightarrow 0) \/ T(\\tilde{x}\\rightarrow \\tilde{x})$ as a function of $\\rho$ (left). For very weak immigration rates $\\mu\\approx 10^{-3}$ the ratio is non-monotonic in competition strength, revealing regime (c) in Fig.~\\ref{fig:phases_sim}. The MFPT ratios as a function of both $\\mu$ and $\\rho$ is represented with a color-map (right). \\textbf{C}:  The MFPT ratio $ T(0\\rightarrow \\tilde{x}) \/ T(0\\rightarrow 0)$ as a function of the competition overlap (left) and as a function of both $\\rho$ and $\\mu$ (right). This ratio qualitatively captures the richness behaviour in Fig.~\\ref{fig:phases_sim}. Both panels (B) and (C): the values are represented with the logarithmic color scale.}\n    \\label{fig:turnover}\n   \n\\end{figure*}\n\nOverall, the `niche-like' Regimes I, II and IV  are characterized by a relatively stable behavior; generally species stay longer about the\ndominant species abundances\npunctuated by the occasional crossings between dominance to nearly-extinct states and the reverse invasions from extinction into the dominance.\nThe transition between partial coexistence (b) to regime of competitive exclusion (c) is captured by the non-monotonic behaviour of the ratio as shown in Fig.~\\ref{fig:turnover}C.\nSurprisingly, unlike the other regimes, increasing the competition strength in the competitive exclusion regime (c) increases the stability of the dominant species abundance: return times to the dominant abundance are much shorter than the time to extinction for the single species in the frozen `quasi-stable' state.\nConversely, the `rare biosphere' regime (III) features rapid dynamics where species cycle rapidly between extinction and a broad range of abundances without establishing `quasi-stable' states with slow turnover. \nThese different dynamic types are illustrated by illustrative trajectory plots in Fig.~\\ref{fig:turnover}A. \n \n\n\n\n\n\n\n\n\n\n\\subsection{The abundances of different species are weakly anti-correlated}\n\\label{sec:Correlation}\n\nSo far, we have investigated the single species marginal abundance distribution $P(n)$ and the average richness $\\langle S^*\\rangle $. However,\nthe species are not independent of each other due to the inter-species competitive interactions. It has been suggested that inter-species correlations reflect on the underlying community structure and the phase space \\cite{carr2019use,chance2019native}\nTo investigate the connection between the population structure and the cross-species correlations, we calculated the cross-species abundances correlations, quantified via the Pearson correlation coefficient, as shown in Fig.~\\ref{fig:correlation}A. The population  exhibits weak cross-species anti-correlation that increases with the competition strength $\\rho$. This is expected given that the death rate of each species increases with the abundance of the other species and, consequently, these cross-species influences are more pronounced at high competition strengths.\nConversely, higher immigration rates ensure that the abundance fluctuations of different species are less likely to be correlated with those of other species.\nThus, the anti-correlation is most pronounced in the high competition and low immigration regime\n\n\nFurthermore, the impact of individual species on the total population size varies between community structures, which can be quantified by the correlation between the total population size $J$ and individual abundances.\nWe found that the individual species abundances are positively correlated with the total abundance $J=\\sum_i n_i$, which also fluctuates as the individuals of all species undergo birth and death events.\nInterestingly, as shown in Fig.~\\ref{fig:correlation}B, the magnitude of this correlation $\\text{cov}(J,n_i)\/\\sigma_n \\sigma_J$ exhibits inverse \ntrends compared to the inter-species anti-correlation: the correlation  $\\text{cov}(J,n_i)\/\\sigma_n \\sigma_J$ is weaker when the cross-species anti-correlation is stronger.\nThe magnitude of the correlation between the total population size $J$ and a species abundance $n$ exhibits similar behavior to the average richness: $\\text{cov}(J,n_i)\/\\sigma_n \\sigma_J$ is high in the high immigration, low competitive overlap regime and is low otherwise.\nThis behaviour may be understood heuristically:\nwhereas each species in a system with $S^*$ dominant species only contributes $\\sim J\/S^*$ to the total population size. \n\nSomewhat unexpectedly, neither the inter-species correlations nor the correlations between the species abundance and the total abundance distinguish between the different modality regimes but rather both increase with the richness. As expected, our mean-field approximation works best at very low ${\\rm cov}(n_i,n_j)\/\\sigma_{n_i}\\sigma_{n_j}$, whereas our mean-field deviates from the solution at anti-correlation gets stronger (see SM). \n\n\\begin{figure}[t!]\n    \\centering\n    \\begin{flushleft}\n        A\n    \\end{flushleft}\n    \\includegraphics[width=0.75\\columnwidth,trim= 10 10 10 10, clip]{figures\/Fig5-A.pdf}\n     \\begin{flushleft}\n        B\n    \\end{flushleft}\n    \\includegraphics[width=0.75\\columnwidth,trim= 10 10 10 10, clip]{figures\/Fig5-B.pdf}\n    \\caption{Abundance correlations. Panel \\textbf{A}: Pearson correlation coefficient between the abundances of any two different species. Panel \\textbf{B}: Pearson correlation coefficient between  the total population size $J=\\sum n_j$ and an abundance of any species. Correlations were calculated from Gillespie simulation time course data with $6 \\cdot 10^8$ time steps\n    }\n    \\label{fig:correlation}\n\\end{figure}\n\n\\section{Summary and Discussion}\n\\label{sec:discussion}\n\nEcological systems display a wide variety of different behavior regimes that have been commonly analysed through a limited number of paradigmatic models such as the `niche' and `neutral' theories. However, it remains incompletely understood what features of ecological population structure and dynamics are universal and which are system specific, how different models relate to  each other, and what behavior is expected in the full range of the parameter space. Using a minimal model of the competitive population dynamics with demographic noise, we have investigated\nthe different regimes of the population structures and dynamics as a function of the immigration rate $\\mu$, competitive overlap $\\rho$, and the number of species $S$.\nAlthough this minimal model may not fully capture the more complex interaction structures of many ecological communities, it already exhibiting rich and unexpected behaviours paralleling many experimentally observed ones (see Table \\ref{tab:my_label} below and Table S1 in SM), and illuminates the underlying mechanisms that shape population structures in different ecosystems.\n\nWe have focused on the system richness reflecting the number of the co-existing species, and the SAD shape as the characteristics of the different population regimes, using a combination of simulations and analytical mean-field approaches. Our analysis shows that the ecosystem behaviors can be partitioned into different regimes of richness and SAD shape\/modality, parameterized by the immigration rate and the competitive overlap\n\nOur model recovers the expected limits of  the well known `neutral-like' and the `niche-like' regimes. In particular, at $\\rho = 1$ and intermediate values of $\\mu$, the SAD has the `rare-biosphere' monotonously decreasing shape characteristic of the classical neutral regime. On the other hand, at low competition strength, the system SAD exhibits a peak at high species abundance where all species co-exist, effectively occupying distinct ecological niches. \n\nNote that even independent species with no inter-species competition with $\\rho=0$ may present either a unimodal or bimodal SAD depending on the immigration rate, see Fig.~\\ref{fig:phases_sim}B\\&C. Unlike the immigration dominated high abundance peak at high immigration rates, at the very low immigration rates the SAD is peaked around zero due to high extinction probability solely from the intra-species competition\n\nChanges in the SAD between different regimes, reflected in the changes in the locations and the heights of the SAD peaks as a function of the immigration and the competition strength, occur through different routes.\nFor instance, starting at the bimodal regime (II) at $\\rho=1$, as the immigration rate increases,  the SAD peak height gradually decreases without changing significantly its location, until it finally disappears at the boundary of the `rare biosphere' regime (III). \nBy contrast, at lower competition strengths $\\rho<1$, the transition from the bi-modality to the `rare biosphere' regime occurs via   simultaneous changes in the peak's height and location. This is discussed in Sec.~\\ref{sec:results}B.\n\nWe found that for the intermediate immigration rates the system maintains the monotonically decaying `neutral-like' SAD in regime (III) even at rather low competition strength ($\\rho \\approx 0.1$) contrary to the common expectation that different species inhabit separate niches away from neutrality. Conversely, at low immigration rates, the system SAD unexpectedly maintains the peak at non-zero abundance characteristic of `niche-like' regimes even for the high values of the competition strength $\\rho$ usually considered to be in the `neutral-like' domain (regime (IIc)); see Sec.~\\ref{sec:results}B and Figure \\ref{fig:phases_sim}.\n\nWe have also uncovered an unusual - and to the best of our knowledge hitherto not described - regime characterized by the multi-modal SAD with more than one positive, `quasi-stable' abundance peak (Regime (IV) in Fig~\\ref{fig:phases_sim}). This multi-modality arises from the richness fluctuations in this regime: the number of co-existing species is switching randomly between two relatively long-lasting states with  $S^*=1$ and $S^*=2$. Thus, one peak of the SAD is found around $\\sim K$ and the other one in the vicinity of $\\sim K\/2$, as explained in Sec.~\\ref{sec:results}B. We observe that for low $K$, the multimodal regime is non-existent and appears as $K$ increases; see the corresponding phase diagrams in SM.\n\nWe show that the population structures in different regimes stem from  the underlying dynamics of species fluctuations, extinctions and invasions. In the `rare biosphere' neutral-like regimes, all species undergo relatively fast turnover around extinction. This is reflected in the low ratio of the turnover to the extinction mean first-passage times. Conversely, in the `niche-like' regimes the system develops two additional time scales: relatively fast fluctuations about the high abundance peak, and the long waiting times for the transitions from the `quasi-stable' co-existence at high abundance to extinction.  Accordingly, the ratio of the mean extinction time to the mean time of return to dominance is higher in the `niche-like' regime, as discussed in Sec.~\\ref{sec:results}C. \n\nInterestingly, ecological regimes  akin to those predicted by our demographic noise model (except for the multimodal SAD regime) have been also found using deterministic, noiseless LV models with a random  matrix of inter-species competitive interaction strengths  \\cite{may1972will,allesina2008network,allesina2012stability,kessler2015generalized,gore2021}.\nHowever, the underlying mechanisms that give rise to the apparently similar regimes in the two model types are very different. In the demographic noise model, the partial richness `niche-like' regime is formed by the coexistence of some species at a positive abundance and temporary extinctions of other species induced by the stochastic abundance fluctuations. By contrast, in the deterministic LV models with random asymmetric interactions, this regime is formed from a large number of fixed points where different sets of species are deterministically excluded. At higher interaction strengths, the system transitions to the deterministically chaotic behavior that resembles the `neutral-like' regimes, however the nature of the species turnover and the SAD shape are very different to the behaviour we described above \\cite{bunin2017ecological,kessler2015generalized,gore2021}\n\nThe existence of the predicted regimes and the transitions between them can be tested experimentally by measuring the SAD and the dynamics of the species abundances in ecosystems with varying immigration and competition strengths.\nLong-term observations may provide measurements of the stationary species abundance distributions \\cite{weigelt2010jena} although the steady-state SAD may be difficult to estimate due to the limited amount of data.\nIt may be  difficult to experimentally determine and control the immigration rate, the competition strength, and carrying capacity, but practically useful proxies for these parameters exist. For example, the flow rate carrying bacteria into a chamber of a microfluidic device is a well controlled quantity that approximates well the immigration rate for populations encased in the chamber~\\cite{duran2021slipstreamin}.\nFurthermore,   measurements of the SADs and the community compositions have become more attainable due to the advances in single cell gene sequencing techniques \\cite{ratzke2020strength, gore2021,shakiba2019cell}. Another commonly used and robustly  estimated experimental observable is the species rank abundance (SRA), which can be used to infer the SAD to which it is closely mathematically related, although in practice the conversion might be constrained by limitations of noise and quantity of the experimental data. \n\nDespite these difficulties, the asymptotic behaviour of the SADs may provide indication of qualitative dissimilarities between the various regimes to discern different regimes of behavior among the experimental observations. In the mean-field approximation in our model the asymptotic behaviour of the SAD on the neutral line $\\rho=1$ approximates a power law with an exponential cutoff (see SM) - similar to the commonly used neutral birth-death models with the fixed total population size \\cite{baxter2007exact,mckane2004analytic}.\nNotably, the Yule process that is often used to model neutral processes also results in the SAD of a similar form. However, the Yule process is substantially different from the model of this paper because it does not include inter-species interactions and reaches the steady state SAD only if $r^+ < r^-$\n\nIn Table 1, we qualitatively compare the family of the regimes predicted by our model to the various behaviors inferred from experimental findings based on the SAD measurements and population abundance time series.\nThe notable abundance of the neutral ecosystems observed experimentally may pertain to our finding (Section~\\ref{sec:results}B) that the `neutral-like' `rare biosphere' regime extends substantially beyond the neutral line $\\rho =1$: non-neutral communities appear neutral as they exhibit SAD's characteristic of neutral communities, such as gastrointestinal microbiomes \n\\cite{jeraldo2012quantification}. Furthermore, multimodal SAD's predicted by our model that are related to the richness fluctuations may provide an explanation for the multimodal SADs observed in some ecological data,  complementary to the current explanations such as spatial heterogeneity or emergent neutrality \\cite{dornelas2008multiple,vergnon2012emergent}\n\n\n\\begin{table}[ht]\n    \\centering\n    \\begin{tabular}{|l|l|\n   \n       System (Ref.) & Regimes  \\\\ \\hline \nmicrobial competition\\cite{gore2021}  & stable full coexistence (IIa) \\\\   &  stable partial coexistence (IIb)\\\\ & persistent fluctuation (IIIb) \\\\ \nglobal birds species \\cite{callaghan2021global} &\tunimodal - log skew (I) \\\\ \nplankton \\cite{ser2018ubiquitous}  &\tpower-law decay (III)  \\\\ \ncoral \\cite{dornelas2008multiple}\t& multimodal (IV)\t\n\\\\ \narthropods \\cite{matthews2014multimodal} &\tmultimodal (IV)\n\\\\\nT-cell receptors \\cite{oakes2017quantitative} & bimodal (II)\\\\\n& unimodal (III)\n\\\\\nmicrobial competition  \\cite{descheemaeker2020stochastic} & `neutral-like' (III) \\\\ & `niche-like' (I \\& II)\n\\\\ \ngastrointestinal  &\t`neutral-like' (III)\n\\\\ \nmicrobiomes \\cite{jeraldo2012quantification} & \\\\\n    \\end{tabular}\n    \\caption{Qualitative classification of observed population regimes in various ecological systems.  }\n    \\label{tab:my_label}\n\\end{table}\n\n\n\n\nOne quantity that is experimentally relatively easy to control is the total number of species $S$. The regimes predicted by the model and the transitions between them shown in Fig.~\\ref{fig:phases_sim}D show similarities with the experimentally observed ones - which were previously explained within the deterministic LV models with random interaction matrix~\\cite{gore2021}. As shown in Fig.~\\ref{fig:phases_sim}D; our model yields `neutral-like' regimes for high $S$ and $\\rho$, which are characterized by erratic dynamics, and `niche-like' behavior with more stable behavior for either low $S$ or $\\rho$. This qualitatively agrees with observed phase-space the in \\cite{gore2021}, which for  strong competition and large pool of species the system presents `persistent fluctuation' regime, while for small species pool or weak competition exhibit `stable full\/partial coexistence'. \nThe fact that both the deterministic LV model with random interaction matrix and the homogeneous LV model with demographic noise are in qualitative agreement with the experimental data raises interesting and important questions concerning the role of stochastic and deterministic dynamics on community composition.\n\nAnother quantity that may enable qualitative and quantitative testing of different models is the carrying capacity $K$ which may be controllable experimentally in some systems.\nAs shown in SM, `rare biosphere' regime shrinks in size with increasing $K$ because a higher carrying capacity sustains higher average abundance,  and larger (less likely) fluctuations are needed for the extinction events to occur.\nHigher average abundance together with insufficiently\nstrong fluctuations,\nresult in longer MFPTs from\ndominance to extinction abundances and vice-versa.\nThis might be captured by the dependence of $K$ in \\eqref{eq:boundary-I}, but further work on the mean-field approximations is needed.\nUnfortunately, rarer turnover events imply longer times to reach the steady state, thus comparing our analytical prediction of the dependence on very large $K$ becomes unfeasible using simulations.\n\nWe expect that the minimal model of this paper can be used for more complicated scenarios, including incorporating speciation  to probe the interaction of the natural selection, inter-species interactions and population diversity and structure.\nFurthermore, the deterministic models inspire further extension\nof our framework to more complex  distributions of the interaction network $\\rho_{i,j}$.\nFinally, our examination of the local ecosystem within an island in the mainland-island model (see Fig.~\\ref{fig:fig1}),\ncan be expanded to a many-island model which allows studying differences in dynamics between the local community and metacommunity, a prominent topic for conservation ecologists and the study of the human microbiome, amongst others.\n \n\n\n\n \n \\iffalse\n\\section{Deterministic Resilience}\n\nIn this section we examine how fast the deterministic system approaches its fixed point. The time for an ecosystem to return to its steady-state, also known as the system resilience, is one of the properties associated with the system stability.   As was mentioned, when $0\\leq \\rho\\leq 1$ the deterministic fixed point, given in ~\\eqref{eq:solstat}, is stable. The direction and pace\/rate\\textcolor{red}{\/another term instead of pace\/rate? } of the flow toward the fixed point are determined by the eigenvalues and eigenvectors of the considered system.  The eigenvalues are obtained as \n\\begin{equation}\n\\lambda_i =\n\\begin{cases}\n\\frac{r}{K}\\left( K - 2 \\tilde{x}(1+\\rho(S-1)) \\right) & \\text{, if } i=1 \\\\\n\\frac{r}{K}\\left( K - \\tilde{x}(2+\\rho(S-2)) \\right) & \\text{, otherwise}.\n\\end{cases}\n\\end{equation}\nand the eigenvectors are \n\\begin{equation}\n\\vec{v}_i =\n\\begin{cases}\n(1,1,\\cdots,1,1)^T & \\text{, if } i=1 \\\\\n(-1,\\delta_{2,i},\\delta_{3,i},\\cdots,\\delta_{S-1,i},\\delta_{S,i})^T & \\text{, otherwise}.\n\\end{cases}\n\\end{equation}\nFor analyzing $\\{\\lambda_i,\\vec{v}_i\\}|_{\\tilde{x}}$ we can deduce the following behaviour of the flow toward the fixed point.  \n\nFirst, we found that $\\lambda_1|_{\\tilde{x}} \\leq \\lambda_{i\\neq 1}|_{\\tilde{x}}$ (the equality is for $\\rho=0$), which means that the system approach faster to constant $J$. Then, in the vicinity of the circle $||\\vec{n}||_1=J$ in $\\ell_1$, it approaches slower toward its fixed point. \n\nSecond, $|\\lambda_1|$ increases with $S$. It means that for highly diverse system, the process reaches faster to the vicinity of its stable community size. In addition, increasing the immigration rate $\\mu$ increase the pace to approaching constant $J$.   \n\nThird, the other (degenerated) eigenvalues describe the behaviour of the flow in the vicinity of constant $J$ (note that it is not necessarily on the {\\em exact} constant J surface). Similarly to $|\\lambda_1|$, the flow in the neighborhood of $||\\vec{n}||_1=J$ toward the fixed point is faster for higher immigration rate. \n\nForth, we have found that $\\lambda_{i\\neq 1}|_{\\tilde{x}}(\\rho,S)$ is not a necessarily a monotone function. For example, when $\\mu=0.01, K=100$ and $r=1$, we find that for $\\rho=0.1$ higher $S$ gives slower approach on the direction of $\\vec{v}_{i\\neq 1}$. However, for $\\rho=1$ an opposite effect is found; high diversity gives faster flow to the fixed point.   \n\n\n\n\\section{Extinction and Invasion Rates}\n\n\\subsection{Stochastic Stability}\n\nHow to define stability? In stochastic models it is not well defined (depends on the paper you read). Note that the boundaries (or any other point) are not absorbing ($\\mu>0$). Stability might mean: recurrence (with probability 1) for any point, $\\rho$-stability, ergodicity, absorbing region, Lyapanov stability for the mean.   \n\n\\textcolor{red}{Q:can we state something about, the bimodality of the abundance distribution,  the level of dominate species, and the mean time of extinction?  }\n\n\\subsubsection{Mean First Passage Time}\nOne of the properties associated with stability is the first passage time. One may ask how long would it take for a species with low abundance to become dominant (or vice versa). \n\nAssume that a species almost surely reaches $x=a$. In addition, we assume that the species level is described by the one-dimensional probability. Then, the mean first passage time from point $x$ to $a$ (where $x>a$) is given by \n\\begin{equation}\n   \\langle T_a(x) \\rangle =   \\sum_{y=a}^{x-1}\\frac{1}{F_{\\rm right}(y)}{\\rm Prob}\\left[z\\geq y+1\\right] \\label{eq:MFPT}\n\\end{equation}\nwhere $F_{\\rm right}(y)\\equiv q^+(y)P(y)$ is the flux right from point $y$ and ${\\rm Prob}[z\\geq y+1]\\equiv \\sum_{z=y+1}^{\\infty}P(z)$ is the probability to be found on larger (or equal) level than $y+1$.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth,trim= 130 300 120 300,clip]{figures\/MFPT_differentMu.pdf}\n    \\caption{Mean first passage time from level $x_0$ to extinction. Here the total number of species is $S=20$. We choose $r^+=2$, $r^-=1$ and $\\rho=0.5$. The immigration rate is $\\mu=1$ (blue circles), $\\mu=0.1$ (green rectangles) and $\\mu=0.01$ (purple diamonds). The results are generated from $10^5$ statistically similar systems. The initial position is uniformly distributed in $[1,60]$, i.e. $x_0\\in U[1,60]$.      }\n    \\label{fig:MFPT_differentMu}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth,trim= 130 300 120 300,clip]{figures\/MFPT_rho.pdf}\n    \\caption{Mean first passage time from level $x_0$ to extinction. Here the total number of species is $S=20$. We choose $r^+=2$, $r^-=1$,  $\\mu=1$, and $\\rho=1$ (blue circles), $\\rho=0.5$ (green rectangles) and $\\rho=0.1$ (purple diamonds). The results are generated from $10^5$ statistically similar systems. The initial position is uniformly distributed in $[1,60]$, i.e. $x_0\\in U[1,60]$.  \\textcolor{red}{no agreement with $\\rho=1$}    }\n    \\label{fig:MFPT_rho}\n\\end{figure}\n\n\\subsubsection{Mean Extinction Time from  Level $n_0$}\n\nIn fig.~\\ref{fig:MFPT_differentMu} we present the mean time to extinction (i.e. $a=0$) where the species has started at the level $x_0$. We find that higher immigration rate $\\mu$ give lower time to extinction. This is due to the fact that the influx is proportional to $\\mu$, and the process is stationary (i.e. inbound flux, from zero to positive numbers, is equal to outbound flux, from positive number to extinction). Moreover, In Fig.~\\ref{fig:MFPT_rho} we fixed the immigration rate, and we have found that for small $\\rho$ (weak mutual competition) the MFPT is higher than high $\\rho$. \n\nImportantly, is some cases, the abundance (one  dimension) distribution is not sufficient to evaluate the mean first passage time (see SM). \n\n\\subsubsection{Mean Extinction Time for the Core Species}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth,trim= 130 280 130 300,clip]{figures\/DominantSpecies_differentS1.pdf}\n    \\caption{The mean extinction time of the core species \\textcolor{red}{(approximated analytic solution, no simulation yet)} versus competition strength $\\rho$.  The number of species varies between $S=10$ (blue circles), $S=50$ (green rectangles), $S=100$ (pink crosses) and $S=200$ (yellow starts).    }\n    \\label{fig:DeterminsticVsStochastic}\n\\end{figure}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth,trim= 130 270 110 300]{figures\/BimodalUnimodalRegionAnalytic_3regions_sim_K50_S30}\n    \\includegraphics[trim= 150 270 130 300,width=\\columnwidth]{figures\/BimodalUnimodalRegionAnalytic_3regions_conv_K50_S30.pdf}\n\\includegraphics[trim= 150 270 130 300,width=\\columnwidth]{figures\/BimodalUnimodalRegionAnalytic_3regions_MeanField_K50_S30.pdf}\n    \\caption{Unimodality and bimidality of SAD depending in immigration rate $\\mu$ and competition $\\rho$. The upper panel is obtain from simulation, and the middle and button panels are given from the approximations. This results are obtained from the approximated abundance distribution given using \\textcolor{red}{ 1st approximation (upper panel) and mean-field approximation (lower panel)}. Here we choose the following parameters $S=30$, $r^+=2$, $r^-=1$, $K=50$. The yellow, turquoise and blue regions represent the values of $(\\mu, \\rho)$ where $P(n)$ is a unimoal distribution with maximum at zero, a bimodal distribution with two maxima, or unimodal distribution with maximum at an existence level, respectively. \\textcolor{red}{More or less all have similar map, accept $\\rho=1$ where 1st approx cannot find bimodality there.  }   }\n    \\label{fig:BimodalUnimodal}\n\\end{figure}\n\n \\begin{figure}\n    \\centering\n    \\includegraphics[trim=150 270 150 270,width=\\columnwidth]{figures\/14Oct2.pdf}\n    \\caption{Simulation results for the location and height of the `most-right' peak (2nd peak in bimodality and the only peak at the unimodal phases) .  Upper:Location of the peak. Lower:Height of the peak. }\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n        \\includegraphics[width=\\columnwidth,trim= 130 270 120 300,clip]{figures\/Richness.pdf} \n    \\caption{Richness vs competition.\n    The total number of species change between $S=50$ (red stars), $S=30$ (green squares) and $S=10$ (blue circles). The lines correspond to the approximated analytic solution: 1st and 2nd methods are represented with black and pink curves (respectively). The immigrating rate is $\\mu=1$,\n    $r^+=50$, $r^-=0.1$, and $K=50$.\n    \\textcolor{red}{To run the same figures for $r^+=2$ and $r^-=1$. $K=50$\n    }\n    }\n    \\label{fig:Ricness}\n\\end{figure}\n\n   \\begin{figure}\n        \\centering\n        \\includegraphics[width=\\columnwidth,trim= 130 270 120 280,clip]{figures\/Richness_10to7_mu.pdf}\n        \\includegraphics[width=\\columnwidth,trim= 130 270 120 280,clip]{figures\/Richness_10to7_rho.pdf}\n        \\caption{Upper: The richness versus $\\rho$ for different $\\mu$. Lower: The richness vs $\\mu$ for different $\\rho$.  Here $K=50$, $S=30$, $r^+=2$ and $r^-=1$. The simulation results are given from $10^7$ reactions.}\n        \\label{fig:Richness_mu}\n    \\end{figure}\n    \n    \\begin{figure}\n        \\centering\n        \\includegraphics[width=\\columnwidth,trim= 140 280 140 300,clip]{figures\/Richness_10to7_sim.pdf}\n        \\caption{Simulation results for the richness for different $\\mu$ and $\\rho$. The values of the richness are represented with colors corresponding the colorbar; from high richness (yellow) to low richness (dark blue).  Here $K=50$, $S=30$, $r^+=2$ and $r^-=1$. The simulation results are given from $10^7$ reactions.   }\n        \\label{fig:Richness_heatMap_sim}\n    \\end{figure}\n    \n    \\fi\n    \n\\section*{Methods}\nThe solution for the master equation \\eqref{master-eq} is simulated using the Gillespie algorithm with $10^8$ time steps. We use $r^+=2$, $r^-=1$, $K=100$. \nModalities' classification   is numerically executed after smoothing the simulated SAD. The MFPT is evaluated via the simulated SAD ($\\tilde{x}(S^*)$ is rounded), where a uni-dimensional approximation of the process is considered, see details in SM.\n\n\n\\begin{acknowledgments}\nThe authors acknowledge helpful discussions and comments from all the members of the Goyal and Zilman Groups. AZ acknowledges the support from the National Science and Engineering Research Council of Canada (NSERC) through the Discovery Grant Program. SG acknowledges the support from the National Science and Engineering Research Council of Canada (NSERC) through the Discovery Grant Program and from the Medicine by DEsign Program at the University of Toronto.\n\\end{acknowledgments}\n\n\n\\section*{Approximations of Species Abundance Distribution }\n\n\\subsection*{Derivation of zero flux in $n_i$ - Global balance equation - derivation for the exact SAD}\n\\label{App:ZeroFlux}\n\n\nConsider the multi-dimensional  master equation \n\\begin{eqnarray}\n    \\partial_tP(n_1,n_2\\dots,n_S) &=& \\sum_{i}\\left\\{ q_{i}^+(\\vec{n}-\\vec{e_i})P(\\vec{n}-\\vec{e_i})+q^-_{i}(\\vec{n_i}+\\vec{e_i}) P(\\vec{n}+\\vec{e_i})-\\left[q_{i}^+(\\vec{n})+q_{i}^-(\\vec{n}) \\right]P(\\vec{n}) \\right\\}\n\\end{eqnarray}\n where $q^+_{n_i}(\\vec{n})$ and $q^-_{n_i}(\\vec{n)}$ represents the birth and death rate of species $i$ (respectively), which are generally depends in $\\vec{n}=(n_1,\\dots, n_s)$. Here, $e_i=\\{0, \\dots, 1, \\dots , 0\\}$ (the one is located in the $i$-th component). \n To find Master equation for $n_1$ we sum over all other components; i.e. \n\\begin{eqnarray}\n  &&\\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty}  \\partial_tP(n_1,n_2\\dots,n_S) = \\\\ \\nonumber &&=  \\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty} \\left\\{\\sum_{i}\\left\\{ q_{i}^+(\\vec{n}-\\vec{e_i})P(\\vec{n}-\\vec{e_i})+q^-_{i}(\\vec{n_i}+\\vec{e_i}) P(\\vec{n}+\\vec{e_i})-\\left[q_{i}^+(\\vec{n})+q_{i}(\\vec{n}) \\right]P(\\vec{n}) \\right\\}\\right\\} \n  \\end{eqnarray}\n  thus\n  \\begin{equation}\n     \\partial_t P_1(n_1) = \\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty} \\left\\{\\sum_{i}\\left\\{ q_{i}^+(\\vec{n}-\\vec{e_i})P(\\vec{n}-\\vec{e_i})+q^-_{i}(\\vec{n_i}+\\vec{e_i}) P(\\vec{n}+\\vec{e_i})-\\left[q_{i}^+(\\vec{n})+q_{i}(\\vec{n}) \\right]P(\\vec{n}) \\right\\}\\right\\} .\n  \\end{equation}\n  We can now use the fact that for every $n_i$:\n \\begin{eqnarray}\n      \\sum_{n_i=0}^{\\infty}  q_{i}^+(n_1, \\dots n_i-1, \\dots, n_S)P(n_1, \\dots n_i-1, \\dots, n_S) &=& \\sum_{n_i=0}^{\\infty}  q_{i}^+(n_1, \\dots n_i, \\dots, n_S)P(n_1, \\dots n_i, \\dots, n_S)\n,  {\\rm \\ \\ and} \\\\ \\nonumber\n \\sum_{n_i=0}^{\\infty}  q_{n_i}^-(n_1, \\dots n_n+1, \\dots, n_S)P(n_1, \\dots n_i+1, \\dots, n_S)&=& \\sum_{n_i=0}^{\\infty}  q_{n_i}^-(n_1, \\dots n_i, \\dots, n_S)P(n_1, \\dots n_i, \\dots, n_S)\n \\end{eqnarray}\n  [note that   $q_{n_i}^+(n_1, \\dots -1, \\dots, n_S)P(n_1, \\dots, -1, \\dots, n_S)=q_{n_i}^-(n_1, \\dots 0, \\dots, n_S)P(n_1, \\dots 0, \\dots, n_S)=0$]. Thus, the above equation is given by\n\\begin{eqnarray}\n  \\partial_t P_1(n_1) &=& \\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty} \\left\\{ q_{1}^+(\\vec{n}-\\vec{e_1})P(\\vec{n}-\\vec{e_1})+q^-_{1}(\\vec{n}+\\vec{e_1}) P(\\vec{n}+\\vec{e_1})-\\left[q_{1}^+(\\vec{n})+q^-_{1}(\\vec{n}) \\right]P(\\vec{n})\\right\\}.\n\\end{eqnarray}\nFor simplicity, we define $F^{+}(\\vec{n})\\equiv q_1^+(\\vec{n})P(\\vec{n})$ and $F^{-}(\\vec{n})\\equiv q_1^-(\\vec{n})P(\\vec{n})$, thus Eq. \ncan be written as \n\\begin{eqnarray}\n    \\partial_t P(n_1) =&& \\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty} \\left\\{ F^{+}(n_1-1,n_2,\\dots)-F^{+}(n_1,n_2,\\dots)\n     +F^{-}(n_1+1,n_2,\\dots)-F^{-}(n_1,n_2,\\dots) \\right\\}.\n\\end{eqnarray}\nBy using z-transform ($n_1\\rightarrow z$), which is defined for a function $k(n_1)$ as $K(z)=\\sum_{n_1=0}^{\\infty} k(n_1) z^{-n_1} $, we obtain\n\\begin{eqnarray}\n    \\partial_t P(z) = \\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty} F_{\\rm right}(z,n_2,\\dots)(1-z^{-1})+F_{\\rm left}(z,n_2,n_3\\dots)(1-z) \n\\end{eqnarray}\n\\{used ${\\cal Z}[g(n)-g(n-1)]=[1-z^{-1}]\\hat{G}(z)$, and ${\\cal Z}[g(n+1)-g(n)]]=[1-z]\\hat{G}(z)-zg(0)$ \\}. Stationary solution; $\\partial_t P(z)=0 $ and re-organize the equation yields \n\\begin{eqnarray}\n     \\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty}F_{\\rm right}(z,n_2,n_3,\\dots)=\\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty} F_{\\rm left}(z,n_2,n_3,\\dots)\\frac{1-z}{z^{-1}-1}=\\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty}F_{\\rm left}(z,n_2,n_3\\dots)z .\n\\end{eqnarray}\nThen, we use the inverse z-transform ($z\\rightarrow n_1$), and find\n\\begin{eqnarray}\n      \\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty}q^+_{1}(\\vec{n})P(\\vec{n})= \\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_s=0}^{\\infty}q^-_{1}(\\vec{n}+\\vec{e_1})P(\\vec{n}+\\vec{e_1}). \n\\end{eqnarray}\nWe use Bayes formula; $P(n_1,n_2,n_3,\\dots,n_S)=P(n_2,n_3,\\dots, n_S|n_1)P(n_1)$ and obtain\n\\begin{eqnarray}\n     \\langle q_{n_1}^+(\\vec{n})|n_1\\rangle_{n_2,n_3,\\dots,n_S}P_1(n_1) = \\langle q_{n_1}^-(\\vec{n}+\\vec{e_1}) |n_1+1\\rangle_{n_2,\\dots n_S} P_1(n_1+1), \n\\end{eqnarray}\nwhere $ \\langle *|n_1\\rangle_{n_2, \\dots, n_S}\\equiv\\langle *|n_1\\rangle \\equiv\\sum_{n_2=0}^{\\infty}\\dots \\sum_{n_S}^{\\infty} (*)P(n_2, \\dots, n_S|n_1) $. \nUp to now there are no assumption in the derivation, and the above is general. For our case, as specified in the main text,  $\\langle q_{1}^+(\\vec{n})|n_1\\rangle=\\mu + r^+ n_1$ (note that $q^+_{i}$ depends solely on $n_i$) and $\\langle q_{1}^-(\\vec{n})|n_1\\rangle=n_1\\left(r^-+r n_1\/K + r \\rho \\sum_{j\\neq 1 }\\langle n_j |n_1\\rangle \/K \\right)$.  Additionally, from symmetry,  $P_i(n_i)=P_j(n_j) = P(n)$ for every $i,j$.\nThus, solving the recursive equation and obtain\n\\begin{eqnarray}\n    P(n)&=&P(0)\\prod_{n'=1}^{n}\\frac{q^{+}(n'-1)}{\\langle q^-(\\vec{n})|n'\\rangle}= \\label{eq:exact_appendix}\n    \\\\ \\nonumber\n    &=&P(0)\\prod_{n'=1}^{n}\\frac{r^+(n'+a)}{n\\left(r^-+r n'\/K + r\\rho \\sum_{j\\neq 1 }\\langle n_j |n'\\rangle \/K \\right)}= P(0)\\frac{(r^+)^{n}(a)_{n}}{n!\\prod_{n'=1}^{n}\\left(r^-+r n'\/K + r\\rho \\sum_{j\\neq 1 }\\langle n_j |n'\\rangle \/K \\right)}.\n\\end{eqnarray}\nwhere\n $a=\\mu\/r^+$ and $(a)_{n} \\equiv a(a+1)\\dots (a+n-1)$, in the Pochhammer symbol.  Here, $P(0)$ is given from normalization.\nWe emphasize that the above abundance distribution $P(n)$ in \\eqref{eq:exact_appendix} is exact, means no approximations have been taken so far.\n\nNote, that the denominator in the exact solution depends on the effect of interactions from all other species over $n_1$,  through the term $\\sum_{j\\neq 1}\\langle n_j |n_1 \\rangle $. Therefore, in order to provide an explicit expression to $P_1(n_1)$, we need to use some approximations. Three approximation approaches, and a discussion about their limitations are given in the following subsections. We note that all the presented approximations below provide decent results. However, we find that  none of them manage to provide adequate fit for every set of parameters, see Figures.  \n\n\\subsection*{Approximation Approach I: Estimating $\\sum_{j\\neq i}\\langle n_j|n_i\\rangle $ Using Mean-Field Approximation }\nWe assume $\\langle n_j |n_i \\rangle = \\langle n_j \\rangle $. Thus, \n\\begin{equation}\n    P(n)\\approx P(0)\\frac{(r^+)^{n}(a)_{n}}{n!\\prod_{n'=1}^{n}\\left(r^-+r n'\/K + r\\rho \\sum_{j\\neq 1 }\\langle n_j\\rangle \/K \\right)} = P(0)\\frac{(r^+)^{n}(a)_{n}}{n!\\prod_{n'=1}^{n}\\left(r^-+r n'\/K + r\\rho (S-1)\\langle n\\rangle \/K \\right)},\n    \\label{eq:MF1}\n\\end{equation}\nwhere the last equality is given from symmetry;  $\\langle n_j \\rangle = \\langle n_i \\rangle = \\langle n\\rangle $ for every species $i,j$. In addition, by definition, $\\langle n \\rangle = \\sum_{n=0}^{\\infty }n P(n)$, hence\n\\begin{equation}\n    \\langle n \\rangle \\approx P(0)\\sum_{n=0}^{\\infty}n\\frac{(r^+)^{n}(a)_{n}}{n!\\prod_{n'=1}^{n}\\left(r^-+r n'\/K + r\\rho (S-1)\\langle n\\rangle \/K \\right)}\n    \\label{eq:MF1_closure}\n\\end{equation}\nwhere $P(0)=1\/{_1}F_1[a,b;c]$ is the normalization coefficient,  with ${_1}F_1[a,b;c]$ is the Kummer confluent hypergeometric function, $a=\\frac{\\mu}{r^+}$, $b=\\frac{r^- K + r \\rho (S-1) \\langle n \\rangle }{r}+1$ and $c=\\frac{r^+ K}{r}$. Solving numerically the above implicit equation and evaluate $\\langle n\\rangle$. The last step is to substitute the numerical solution of $\\langle n\\rangle$, obtained from \\eqref{eq:MF1_closure}, into \\eqref{eq:MF1}.    \n\n\\subsection*{Approximation Approach II: Estimating $\\langle J|n\\rangle $ using Convolution }\nThe exact solution can be written as \n\\begin{eqnarray}\n    P_1(n_1)=P(n)= P(0)\\frac{(r^+)^{n}(a)_{n}}{n!\\prod_{n'=1}^{n}\\left(r^-+r (1-\\rho)n'\/K + r\\rho \\langle J |n_1\\rangle \/K \\right)},\n\\end{eqnarray}\nwhere $J=\\sum_{i=1}^{\\infty} n_i $ is the total population size. Here we assume that the total number of individuals in the system, $J$, is weekly depends on $n_1$. Thus $J$ is an independent random variable. Hence, \n\\begin{equation} \n    P_1(n_1)=P(n_1|\\langle J |n_1 \\rangle ) \\approx P(n_1|J)=P(0)\n    \\frac{(a)_{n_1} \\Tilde{c}^{n_1}}{n_1 ! (\\Tilde{b}+1)_{n_1} } \n\\end{equation}\nwith $a=\\frac{\\mu}{r^+}$, $\\tilde{b}= \\frac{r^-K+r\\rho J}{r(1-\\rho)}$, and $\\tilde{c}=\\frac{r^+ K}{r(1-\\rho)}$ [note that both $\\tilde{b}$ and $\\tilde{c}$ differ from $b$ and $c$ defined in previous subsection].\nMoreover, we assume that the species levels are mutually independent, means ${\\cal P}(n_1,\\dots n_S) \\approx \\prod_i P_i(n_i)$. Thus, the PDF of $\\sum_i n_i$ reads \n\\begin{equation}\n    P\\left(\\left.\\sum_i n_i\\right|J\\right)=\\underbrace{P_1(n_1|J)*P_2(n_2|J)* \\dots * P_S(n_S|J)}_{S {\\rm \\ times}}\n\\end{equation}\nwhere $A*B$ means the convolution of $A$ with $B$. $P\\left(\\sum_i n_i|J\\right)$ is the `analytical' PDF to have $\\sum_i n_i$ individuals where we assume that a single species PDF is $P_1(n_1|J)$ with a given $J$. \nTo capture the fact that $J$ has a meaning of number of individuals as well, we consider\n\\begin{equation}\n    P(J)\\approx \\frac{{\\rm Prob}\\left(\\left.\\sum_i n_1=J\\right|J\\right)}{\\sum_J {\\rm Prob}\\left(\\left.\\sum_i n_1=J\\right|J\\right)},\n\\end{equation}\nwhere $P(J)$ is the approximated distribution of $J$. \nThen\n\\begin{equation}\n    P_1(n_1) = \\sum_{J}P_1(n_1|J) P(J)\n\\end{equation}\nis the approximated PDF. \n\nNote that when $S$ is large, we find\n\\begin{equation}\n    P\\left(\\left.\\sum_i n_i\\right|J\\right) \\sim  {\\cal N}\\left(S\\langle n_1 |J \\rangle, S \\cdot Var(n_i) \\right),\n\\end{equation}\nthus $P(J)\\approx {\\rm Prob}(\\sum_i n_i =J|J)$ reaches its maximum in the vicinity of $J$ which satisfies $J\\approx S \\langle n_i |J \\rangle = \\left\\langle \\sum_i n_i |J \\right\\rangle $. Furthermore, for the approximation $P(J)\\approx {\\rm Prob}(\\sum_i n_i =J|J)$, the values of $J$ where $J\\ll S\\langle n_i |J \\rangle  $ or $J\\gg S\\langle n_i |J \\rangle $ are highly improbable, due to the Gaussian nature of $P(\\sum_i n_i|J)$ for large $S$.\n \n\\subsection*{Approximation Approach III: Estimating $\\langle J|n\\rangle $ using Mean-Field Approximation } In a similar fashion to previous approximation approaches, we assume $\\langle J|n\\rangle \\approx \\langle J\\rangle  $, thus\n\\begin{eqnarray}\n    P(n) \\approx P(0)\\frac{(r^+)^{n}(a)_{n}}{n!\\prod_{n'=1}^{n}\\left(r^-+r (1-\\rho)n'\/K + r\\rho \\langle J \\rangle \/K \\right)}=  P(0)\\frac{(r^+)^{n}(a)_{n}}{n!\\prod_{n'=1}^{n}\\left(r^-+r (1-\\rho)n'\/K + r\\rho S\\langle n\\rangle  \/K \\right)}.\n\\end{eqnarray}\nThen, $\\langle n \\rangle$ is given by the numerical solution of\n\\begin{eqnarray}\n    \\langle n \\rangle = \\sum_{n=0}^{\\infty} n P(0)\\frac{(r^+)^{n}(a)_{n}}{n!\\prod_{n'=1}^{n}\\left(r^-+r (1-\\rho)n'\/K + r\\rho S\\langle n\\rangle  \/K \\right)}.\n\\end{eqnarray}\nwhere here the normalization factor is $P(0)=1\/{_1}F_1[a,b;c]$ with $a=\\frac{\\mu}{r^+}$, $\\tilde{\\tilde{b}}= \\frac{r^-K+r\\rho \\langle J\\rangle }{r(1-\\rho)}$, and $\\tilde{c}=\\frac{r^+ K}{r(1-\\rho)}$.\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=0.45\\columnwidth,trim=100 100 100 100]{figures\/Quality.pdf}\n    \\includegraphics[width=0.45\\columnwidth,trim=100 100 100 100]{figures\/examples_approx.pdf}\n    \\caption{Quality of Approximation}\n    \\label{fig:quality}\n\\end{figure}\n\n\\subsection*{Interspecies Correlations and Limitations of the Approximations Approaches}\nFor approximation approaches described above, we assume the species are mutually independent, meaning $P(\\vec{n})=\\prod_{i=1}^S P(n_i)$.\nOf course, this mutual independence cannot exactly be obtained, except for $\\rho=0$, since for every $\\rho>0$ the dynamics of a species, i.e its birth and death rates, depends on other species levels.  Therefore, we expect deviations of our approximation from the true simulated SAD. In particular,  we expect a relation between the interspecies correlation and the quality of approximation. \n\nWe quantify the quality of approximation, using two metrics. The first one, Kolmogorov-Smirnov (KS), is defined with $KS(P,Q)\\equiv\\max \\left|{\\rm CDF}(P)-{\\rm CDF}(Q)\\right|$ for two PDFs, $P$ and $Q$. The second metric we use is the Kullback\u2013Leibler divergence, which is defined by $KL(P,Q)\\equiv \\sum_x (P-Q)\\ln\\left(P\/Q\\right)$. \nIntuitively, the KS metric captures the difference between the approximation and the simulation results, and the KL divergence capture the ratio between the distributions. \nFig.~\\ref{fig:quality} shows the KS and KL metrics comparing the three approximations presented above and the simulation results.\n\nIn the main text, we choose to present the results for the regime boundaries obtained from the Approximation I above. Even so, all approximations present reasonable agreement with the simulation. \nHowever, in some regimes, some approximations work better than the others. \nAdditionally, one approximation may better capture  some features of the system, while other approximation show better agreement with other features. \nFor example, only approximation (I) captures the bi-modality at very low $\\mu$ and $\\rho=1$, where the other approximations seem to align with the simulated SAD slightly better.\n\n\\subsection*{Tail-end of distributions}\n\nThe above model does not describe an ecological zero-sum game as the total population size $J=\\sum_i n_i$ may fluctuate in this proposed formulation.\nConversely, classical Moran models describe stochastic processes in which the finite population is fixed. \nThe species abundance distribution can be solved for exactly in certain cases where this assumption of a fixed total population size holds.\n\nIn this limit,\n\\begin{equation}\n    P(n) = P(0)\\frac{(a)_{n}}{n!}\\approx P(0)\\frac{n^{a-1}}{\\Gamma[a]}\n\\end{equation}\nIn the case where $\\langle J|n_1\\rangle \\approx \\langle J \\rangle $:  \\begin{equation}\n    P(n)= P(0) \\left(\\frac{r^+}{r^- + r \\langle J \\rangle \/K}\\right)^{n}\\frac{ (a)_{n}}{n!} \\approx  P(0) \\left(\\frac{r^+}{r^- + r \\langle J \\rangle \/K}\\right)^{n_1}\\frac{ n_1^{a-1}}{\\Gamma[a]}.\n\\end{equation}\nNote that this is valid only when $n \\gg a$, that is to say for the tail-end of the distributions.\n\n\nMoran models are generally neutral $\\rho = 1$), however using our demographic noise model allows for observing non-neutral systems. \nWe can approximate the tail of the distribution for non-neutral models ($0 < \\rho < 1$) using similar arguments as above, that is to say approximating $\\langle J |n \\rangle \\approx J$.\nUsing our previous solution and this approximation, we find that\n\\begin{equation}\n    P(n) = P(0)\\frac{(r^+)^{n}(a)_{n}}{n!\\prod_{n'=1}^{n}(r^-+r(1-\\rho)n'\/K+\\rho \\langle J |n' \\rangle \/K)}\\approx P(0)\\frac{(a)_n \\tilde{c}^n}{n!(\\tilde{b}+1)_n}\n\\end{equation}\nwhere $a=\\frac{\\mu}{r^+}$, $\\tilde{b}= \\frac{r^-K+r\\rho J}{r(1-\\rho)}$, and $\\tilde{c}=\\frac{r^+ K}{r(1-\\rho)}$.\nLooking at the tail of the distribution ($n \\gg a, \\tilde{b}, \\tilde{c}$) we find that \n\\begin{equation}\n    P(n) \\xrightarrow{n \\rightarrow \\infty} P(0)\\frac{\\Gamma[\\tilde{b}+1]}{\\sqrt{2 \\pi} \\Gamma[a]} n^{a-\\tilde{b}-\\frac{3}{2}}(\\tilde{c}\/n)^{n}e^n\n\\end{equation} \nwhere we have used Stirling's approximation that $n! \\approx \\sqrt{2\\pi n}n^n e^{-n}$.\nIn the high $n$ limit, we can further approximate $J\\sim n$ such that $P(n)\\sim n^{-n\/(1-\\rho)}e^n$\n\nIn the case where $\\rho \\rightarrow 0$, there is no need for approximating $\\langle J |n\\rangle$, as that term disappears in the exact solution.\nWe are left with the solution\n\\begin{equation}\n    P(n) \\xrightarrow{\\rho\\rightarrow 0}P(0) \n    \\frac{(\\mu\/r^+)_{n} (r^+ K\/r)^{n}}{n ! (r^-K\/r+1)_{n} }\n\\end{equation}\nwhich is exact.\nThe tail of this distribution goes as\n\\begin{equation}\n    P(n) \\xrightarrow{\\rho\\rightarrow 0,n\\rightarrow \\infty} P(0)\\frac{\\Gamma[\\tilde{r^-K\/r}+1]}{\\sqrt{2 \\pi} \\Gamma[\\mu\/r^+]} n^{\\frac{\\mu}{r^+}-\\frac{r^-K}{r}-\\frac{3}{2}}(r^-K\/rn)^{n}e^n.\n\\end{equation}\n\nNote that the denominator of $\\tilde{b}$ and $\\tilde{c}$ goes to 0 as $\\rho \\rightarrow 1$, as such the limit must be taken carefully: $\\tilde{b} \\xrightarrow{\\rho \\rightarrow 1} \\infty $.\nWe use the fact that $(x+1)_n \\xrightarrow{ x \\rightarrow\\infty} x^{n}$ to write\n\\begin{equation}\n     P(n) \\xrightarrow{\\rho\\rightarrow 1}P(0) \n    \\frac{(a)_{n} \\{r^+ K\/[r(1-\\rho)]\\}^{n}}{n ! \\{(r^-K+rJ)\/[r(1-\\rho)]\\}^{n} }= P(0) \n    \\frac{(a)_{n} (r^+ K)^{n}}{n ! (r^-K+rJ)^{n} }  \\xrightarrow{n \\rightarrow \\infty } P(0) \\left( \\frac{r^+K}{r^-K + rJ} \\right)^n \\frac{n^{a-1}}{\\Gamma[a]}\n\\end{equation}\nwhich agrees with what we found earlier for $\\rho=1$ and constant $J$. For $\\rho = 1$, we know that if any one species abundance gets large it should dominate the system. Therefore, we can approximate $J\\approx n$ for large $n$ in the neutral regime.\n\nThis asymptotic behaviour may be compared to analytical solutions for which $J$ is held constant.\nThese Moran type models are often solvable exactly, we choose to show their results wherein $J=S n_{det}$ where $n_{det}$ is the solution to the mean deterministic equation Lotka-Voltera equation. \nIn~\\cite{mckane2004analytic}, an analytical solution to the Hubbel model with immigration is found such that\n\\begin{equation}\n    P(n)={J \\choose n}\\frac{\\beta (n+p,n^*-n)}{\\beta (p^*,n^*-J)} \n\\end{equation}\nwhere $p=1\/S$, $n^*=(J-m)\/(1-m) - p$, and $p^*=m p (J-1)\/(1-m)$ .\nIn this model, $m$ is defined as the probability of immigration at any step.\nThis is different from our immigration rate, however we find a suitable transformation to be $m \\approx \\mu \/ \\langle r^+_n + r^-_n \\rangle$: the probability of immigration is the rate of immigration divided by the mean rate of a reaction.\nNote that the function $\\beta(a,b)=\\Gamma (a) \\Gamma (b)\/ \\Gamma (a+b)$.\n\nIn~\\cite{baxter2007exact}, a continuum Fokker-Planck equation is solved to evaluate a similar multi-allelic diffusion model abundance. \nHowever, in this formalism, immigration is replaced by mutations wherein $u_i$ is the rate of mutation of cell allele $i$.\nAssuming all the mutation rates are equivalent, $u_i=u$\nThe steady state joint probability distribution is \n\\begin{equation}\n    P(\\vec{x})=\\Gamma (2 S u ) \\delta (1-\\sum_i x_i)\\prod^S_{i=0} \\frac{x_i^{2u-1}}{\\Gamma (2 u )}\n\\end{equation}\nwhich may be integrated to find the SAD\n\\begin{equation}\n    SAD(n) = \\langle \\sum_j \\delta (x_j - n\/J) \\rangle_{P(\\vec{x})} \\approx \\left( \\frac{n}{J} \\right)^{2 u - 1 } e^{- (2 u (S-1) -1 ) n \/ J }  \n\\end{equation}\nAlthough mutations and immigration are not completely equivalent, mutations may take on a heuristic role similar to immigration that allows for no species to be truly extinct.\nAs such, we assume $u=\\mu\/r$.\nComparisons of these different asymptotic behaviours are found in Fig. ~\\ref{fig:asymptotic}.\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=0.7\\columnwidth]{figures\/asymptotic_neutral.pdf}\n   \\caption{Asymptotic behaviour of various neutral models compared to simulations. (top panels) Using $\\langle J |n \\rangle \\approx n $, our approximation does not recover a bimodality, however the analytical approximation clearly follows the simulation's power law with exponential cutoff. (bottom panels) Moran-like models in the literature of power-law SADs with exponential cutoff. Here, the total population size used is the total population size of the steady-state Lotka-Voltera equation, $J=S n_{LV}$. The continuous Fokker-Planck diffusion model of Baxter, Blythe \\& McKane\\cite{baxter2007exact} shows the immigration dominated peak. However, the Hubbel community model solved by Alonso, McKane \\& Sole\\cite{mckane2004analytic} shows a bimodality in the low immigation regime. Both have power laws with exponential cutoff in different regimes.}\n    \\label{fig:asymptotic}\n\\end{figure}\n\n\\section*{Derivations of Boundary Equations}\n\n\\subsection*{Boundaries for Richness Regimes} For the boundaries defined from richness, we use\n$    \\langle S^* \\rangle = S \\left(1-P(0)\\right) \n$, where $P(0)$ obtained numerically from the approximated SAD. Note that in the mean-filed approximations $P(0)$ is explicitly given as Kummer confluent hypergeometric function. Then, the transition between full richness to partial coexistence is given with $S P(0) = 1\/2$ (as the arithmetic mean between the two boundaries). Similarly, the transition boundary between partial coexistence and excluded regime is drawn where $SP(0)=S-3\/2$.  \n\n\\subsection*{Derivation of $\\tilde{n}$ } The boundaries defined by the modalities can be given directly from the above approximations, see Figures. However, we have found that using the mean-field approximation allows us to derived a closed expression for the boundaries.     \n\nThe transition between neutral-like to bimodal regimes, is defined by the presence  or absence of a local maximum in a real positive level. In another words, in the neutral-like regime $P(n)>P(n+1)$, since the SAD is monotony decreasing, while in the bimodal regime, there is $n>0$ where $P(n)<P(n+1)$. Thus the boundary between the regimes occurs where\n  \\begin{eqnarray}\n      P(n)=P(n+1).\n    \\end{eqnarray}\nUsing the $P(n)$ from the mean-field approximation, we find\n    \\begin{eqnarray}\n      P(0)\\frac{(a)_n c^n}{n! (b+1)_n }&=&P(0) \\frac{(a)_{n+1} c^{n+1}}{(n+1)! (b+1)_{n+1} } \n      \\\\\n      \\frac{(b+1)_{n+1}(n+1)!}{n! (b+1)_n }&=& \\frac{(a)_{n+1} c^{n+1}}{(a)_n c^n } \n      \\\\\n      n(b+n)&=& c (a+n-1)\n      \\\\\n      n&=& \\frac{(c-b)\\pm \\sqrt{(c-b)^2+4(a-1)c}}{2}\n  \\end{eqnarray}\nSubstitute $a=\\frac{\\mu}{r^+}$, $b=\\frac{r^- K + r \\rho (S-1) \\langle n \\rangle }{r}+1$ and $c=\\frac{r^+ K}{r}$ yields\n\\begin{eqnarray}\n   \\tilde{n}\\approx\n    \\frac{K-\\rho(S-1)\\langle n\\rangle}{2}\\left\\{1 \\pm \\sqrt{1+4\\frac{(\\mu-r^+) K}{r(K-\\rho(S-1)\\langle n\\rangle)^2}}\\right\\}.\n    \\end{eqnarray}\nIf we use $S \\langle n \\rangle = \\langle S^* \\rangle \\tilde{n}$, we find\n\\begin{equation}\n    \\tilde{n}=\\frac{K}{2(1-\\rho+\\rho \\langle S^* \\rangle)}\\left\\{1+ \\sqrt{1+\\frac{4(\\mu-r^+)(1-\\rho+\\rho \\langle S^* \\rangle)}{rK}}\\right\\} \\approx \\frac{K}{1-\\rho+\\rho \\langle S^* \\rangle}- \\frac{\\mu-r^+}{r}.\n\\end{equation}\n   \n\\subsection*{Boundaries for Modalities Regimes}\nIn the neutral-like regime the SAD is monotonously decreasing. Therefore, the neutral-like regime is sufficiently define by either imaginary $\\tilde{n}$, or real but negative $\\tilde{n}$. The transition line between real and imaginary $\\tilde{n}$, i.e. where $\\Im(\\tilde{n})=0$, is given by  \n\\begin{eqnarray}\n    {r\\left[K-\\rho  (S-1)\\langle n\\rangle\\right]^2}=4{(r^+-\\mu) K}.\n\\end{eqnarray}\nThe transition line between the negative to positive $\\tilde{n}$, where $\\tilde{n}=0$, is drawn where\n\\begin{equation}\n    \\frac{(K-\\rho(S-1)  \\langle n\\rangle)^2}{4}=1+ \\frac{4K(\\mu-r^+)}{r(K-\\rho(S-1)  \\langle n\\rangle)^2}.\n\\end{equation}\nTherefore, the neutral-like regime is defines as the union of the regions defined by both two equations above.\n\nThe second boundary we derive is the border of uni-modality region with positive probable abundance. At zero abundance, the abundance distribution poses either local maximum ($P(0)>P(1)$; correspond to bimodality or neutral-like regimes) or local minimum ($P(0)<P(1)$; unimodality). Consequently, the boundary between these two possibilities is given by $P(0)=P(1)$.  Thus, following the zero-flux equation, we obtain\n\\begin{equation}\n    q^+(0) = \\langle q^-(\\vec{n})|1\\rangle  \\Longrightarrow  \\mu = r^- +\\frac{r}{K} \\left[1-\\rho + \\rho \\langle J\\rangle\\right]. \n\\end{equation}\n\n\n\n\\section*{Dependence on Carrying Capacity $K$}\n\nThe regime boundaries depend on the carrying capacity, as can be seen in Fig.~\\ref{fig:carry_capacity}; as the carrying capacity increases, the `neutral-like' regime shrinks in size.\nWe also find that the full richness regime extends to larger competitive overlap, intuitively suggesting that interspecies competition is decreased in the larger carrying capacity regimes.\nThe larger carrying capacity allows for more individual species to immigrate into the system and coexist.\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=0.45\\columnwidth]{figures\/K50.pdf}\n    \\includegraphics[width=0.45\\columnwidth]{figures\/K200.pdf}\n    \\caption{Varying carry capacity. Left panel: $K=50$. Right panel: $K=200$. Note that the the erroneous classification in certain regimes is due to numerical difficulty in determining optima of the SAD.}\n    \\label{fig:carry_capacity}\n\\end{figure}\n\n\\section*{Kinetics}\n\nAs was mentioned in the main text, we concentrate on time scales associated with mean-first-passage-time (MFPT) from some initial abundance $i$, to a final one $f$. This MFPT, denoted as $\\langle T(i\\rightarrow f)\\rangle$, is inversely proportional to the transition rate from $i$ to $j$, where the first-passage times are exponentially distributed (see Figures). \nIn a unidimensional process in an interval $[0,\\infty)$, the MFPTs we consider are  \n\\begin{eqnarray}\n      \\langle T (\\tilde{x} \\rightarrow 0) \\rangle =   \\sum_{y=0}^{\\tilde{x}-1}\\frac{1}{q^+(y)P(y)}\\sum_{z=y+1}^{\\infty}{P(z)}, & &\n  \\langle T (0 \\rightarrow 0) \\rangle =  \\frac{1}{q^+(0)P(0)}=  \\frac{1}{\\mu P(0)}, \\\\ \\nonumber \n  \\langle T (0 \\rightarrow \\tilde{x}) \\rangle =  \\sum_{y=0}^{\\tilde{x}}\\frac{1}{q^+(y)P(y)}\\sum_{z=0}^{y}{P(z)}, & &\n  \\langle T (\\tilde{x} \\rightarrow \\tilde{x}) \\rangle =    \\frac{1}{P(\\tilde{x})[q^+(\\tilde{x})+q^-(\\tilde{x})]} .\n\\end{eqnarray}\nThese expressions are exact for processes in one dimension \\cite{gardiner1985handbook}, i.e. for a single species were no other species affect its dynamics.  Here we have found, using simulation, that these MFPTs agree with the multi-dimensional scenario, where many species evolve (see Figures). Hence, we substitute the SAD obtained from simulation,  as $P(n)$ in the expressions above and estimated the rates with $ R(a \\rightarrow b)=1\/\\langle T(a \\rightarrow b)\\rangle $. A similar approach is used to estimate the other rate ratio. \n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.7\\columnwidth]{figures\/supp-MFPT.pdf}\n\\end{center}\n\\includegraphics[trim=140 260 140 230, width=0.24\\columnwidth]{figures\/MFPT_rho1_mu03162new.pdf} \n\\includegraphics[trim=140 260 140 230, width=0.24\\columnwidth]{figures\/MFPT_rho1_mu1_new.pdf} \n\\includegraphics[trim=140 260 140 230, width=0.24\\columnwidth]{figures\/MFPT_rho01_mu03162.pdf} \n\\includegraphics[trim=140 260 140 230, width=0.24\\columnwidth]{figures\/MFPT_rho03162_mu3162.pdf}\n\\\\\n\\includegraphics[trim=140 230 140 230, width=0.24\\columnwidth]{figures\/MFPT_rho1_mu03162_new.pdf}  \n\\includegraphics[trim=140 230 140 230, width=0.24\\columnwidth]{figures\/MFPT_rho1_mu1_new_part2.pdf}\n\\includegraphics[trim=140 230 140 230, width=0.24\\columnwidth]{figures\/MFPT_rho01_mu03162_part2.pdf}\n\\includegraphics[trim=140 230 140 230, width=0.24\\columnwidth]{figures\/MFPT_rho03162_mu3162_part2.pdf}\n\\caption{Turnover time statistics}\n\\end{figure}\n\n\\section*{Fokker-Planck formulation}\n\nIn the analysis so far we have used $n_i$ to represent a discrete variable, however we may approximate the master equation in a continuum should some large characteristic system size exist.\nAssuming here that $K >> 1$ is the system size, we define $x_i$ to be the corresponding continuous limit of $n_i$.\nThe variable may be rescaled by the characteristic system size, $y_i = x_i \/ K$\nThis continuous approximation has as probability density $ P(\\vec{n},t) = p(\\vec{y},t)\/K$ and we may write the the scaled rates as $q_i^{+\/-}(\\vec{n})=K Q_i^{+\/-}(\\vec{y})$ where\n\\begin{align*}\n    Q_i^{+}(\\vec{y}) &= r^+ y_i + \\mu \/ K \\\\\n    Q_i^{+}(\\vec{y}) &= r^- y_i + r y_i \\left( y_i + \\sum_{j\\neq i}\\rho_{j,i} y_j \\right)\n\\end{align*}\nis defined on the continuum.\nThen we can write the corresponding master equation as\n\\begin{equation}\n    \\label{eq:master-eq-cont}\n    \\partial_t  p(\\vec{x};t) =\n    K \\sum_{i}\\left\\{Q^+_i(\\vec{y}-\\vec{e}_i)p(\\vec{y}-\\vec{e}_i,t)+ \n    Q^-_i (\\vec{y}+\\vec{e}_i)p(\\vec{y}+\\vec{e}_i,t) - \\left[Q^+_i(\\vec{y})+Q^-_i(\\vec{y})\\right]p(\\vec{y},t)\\vphantom{\\vec{e}_i} \\right\\}\n\\end{equation}\nwherein $\\vec{e}_i$ is the change in abundance $\\vec{y}$ from the respective event, which in our single birth-death process will be $\\vec{e}_i=(\\delta_{1i}\/K, \\delta_{2i}\/K, ..., \\delta_{Si}\/K)$, in other words a vector of zeros except for $1\/K$ located at species $i$.\nTo go from the master equation to the Fokker-Planck equation, we Taylor expand each of the expressions from the right-hand side of \\ref{eq:master-eq-cont}.\nAs such,\n\\begin{multline*}\n    Q^+_i(\\vec{y}-\\vec{e}_i)p(\\vec{y}-\\vec{e}_i,t)= Q^+_i(\\vec{y})p(\\vec{y},t)+\\sum_j (-(\\vec{e}_i)_j) \\frac{\\partial}{\\partial y_j}\\left( Q^+_i(\\vec{y})p(\\vec{y},t)\\vphantom{\\frac{1}{1}}\\right)\n    + \\frac{1}{2!}\\sum_j \\sum_k (\\vec{e}_i)_j(\\vec{e}_i)_k \\frac{\\partial^2}{\\partial y_j \\partial y_k}\\left( Q^+_i(\\vec{y})p(\\vec{y},t)\\vphantom{\\frac{1}{1}}\\right)+...\n\\end{multline*}\n\\begin{multline*}\n    Q^-_i(\\vec{y}+\\vec{e}_i)p(\\vec{y}+\\vec{e}_i,t)= Q^-_i(\\vec{y})p(\\vec{y},t)+\\sum_j (\\vec{e}_i)_j \\frac{\\partial}{\\partial x_j}\\left( Q^-_i(\\vec{y})p(\\vec{y},t)\\vphantom{\\frac{1}{1}}\\right)\n    + \\frac{1}{2!}\\sum_j \\sum_k (\\vec{e}_i)_j (\\vec{e}_i)_k \\frac{\\partial^2}{\\partial y_j \\partial y_k}\\left( Q^-_i(\\vec{y})P(\\vec{y},t)\\vphantom{\\frac{1}{1}}\\right)+...\n\\end{multline*}\nNote that $(\\vec{e}_i)_j=\\delta_{ij}\/K$ in our single birth-death event process, which simplifies these equations considerably. \nNow, replacing these expressions in our master equation \\ref{eq:master-eq-cont}, we note that we can write our equation in orders of $1\/K$.\nThus, we obtain\n\\begin{equation}\n\\label{eq:fokker-planck}\n    \\partial_t  p(\\vec{y};t) =\n    -\\sum_j  \\frac{\\partial}{\\partial y_j}\\left[\\left( Q^+_i(\\vec{y})-Q^-_i(\\vec{y})\\right)p(\\vec{y},t)\\vphantom{\\frac{1}{1}}\\right]\n    + \\frac{1}{2 K}\\sum_j  \\frac{\\partial^2}{\\partial y_j \\partial y_k}\\left[\\left( Q^+_i(\\vec{y})+Q^-_i(\\vec{y})\\right)p(\\vec{y},t)\\vphantom{\\frac{1}{1}}\\right] + \\mathcal{O}(1\/K^2).\n\\end{equation}\nUp to order $1\/K$, \\ref{eq:fokker-planck} is the Fokker-Planck Equation (FPE) of the process.\nUsing Ito's prescription for SDEs, this corresponds to the Langevin equation\n\\begin{equation}\n    \\label{eq:langevin}\n    d y_i = \\left( Q^+_i(\\vec{y})-Q^-_i(\\vec{y}) \\right)dt + \\sqrt{ \\frac{Q^+_i(\\vec{y})+Q^-_i(\\vec{y})}{K} } dW_i\n\\end{equation}\nwhere $W_i$ is a standard Wiener process. By multiplying both sides of this equation by the characteristic size $K dy_i = dx_i$,\n\\begin{equation}\n    \\label{eq:langevin-LV}\n    d x_i = \\left( q^+_i(\\vec{x})-q^-_i(\\vec{x}) \\right)dt + \\sqrt{ K \\left( q^+_i(\\vec{x})+q^-_i(\\vec{x})\\right) } dW_i\n\\end{equation}\nwhere the force term in the Langevin equation recovers the Lotka-Voltera equation.\n\nIn particular, a common choice is for the diffusion term to be proportional to the square root of the abundance, such that the noise is independent of other species abundances, as in  \n\\begin{equation}\n    \\label{eq:langevin-LV-sqrt-noise}\n    d x_i = \\left( q^+_i(\\vec{x})-q^-_i(\\vec{x}) \\right)dt + \\sqrt{ K r x_i } dW_i.\n\\end{equation}\nHowever not all choice of birth and death rate appropriately recovers this form.\n\nUsing an Euler integration method, simulations of the Langevin equation assess how well the Fokker-Planck approximates the SAD. \nWe find that the Fokker-Planck approximation does not reproduce the complete phase space of the modality regimes for either noise specified in \\eqref{eq:langevin-LV} and \\eqref{eq:langevin-LV-sqrt-noise}, see Fig.~\\ref{fig:langevin}.\nIn both cases, the `neutral-like' regime at high competitive overlap ($\\rho > 2\\cdot10^-1$) is present even at low immigration  such that no bimodality is observed on the neutral manifold ($\\rho = 1$).\nThe boundary between the immigration dominated unimodal regime and other regimes is recovered; in this regime, few fluctuations arise that are sufficiently strong enough to bring any species close to the excluded state, $x_i=0$.\nNote that the force term is always positive for some $x_i>0$, which implies that species will assuredly be deterministically pushed back from exclusion.\nAdditionally, the multimodal regime is absent in all Langevin results.\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=0.7\\columnwidth]{figures\/supp-langevin.pdf}\n    \\caption{Langevin numerical simulations. Left: Modality regimes with noise \\eqref{eq:langevin-LV} from the rates defined in the main text. Right: Modality regimes with $\\sqrt{n}$ noise as in \\eqref{eq:langevin-LV-sqrt-noise}. Colours correspond to the modality regime presented in the main text: yellow is the `neutral-like' regime, teal is the bimodal regime and purple is the immigration dominated unimodal regime.}\n    \\label{fig:langevin}\n\\end{figure}\n\n\\section*{Species Rank Abundances vs Species Abundance Distribution }\nIn the vast majority of the manuscript we use species abundance distributions (SADs), together with some dynamical properties, in order to examine and classify processes into deferment regimes.  However, in many experimental studies, the species rank abundances (SRAs) are frequently reported instead, e.g see \\cite{ser2018ubiquitous,mora2018quantifying,hubbell1979tree,descheemaeker2020stochastic,jeraldo2012quantification}. The SRA is closely related to the cumulative distribution correspond to SAD, as is described in the following. First, the cumulative abundance distribution is computed with ${\\rm CAD}(n)\\equiv\\sum_0^n P(n')$. Then, we note that the most abundance species, namely species with rank 1, has abundance between ${\\rm CAD}^{-1}(1-1\/S)$ to  ${\\rm CAD}^{-1}(1)$. The second most abundance species, i.e. the ranked 2,  has abundance between ${\\rm CAD}^{-1}(1-2\/S)$ to  ${\\rm CAD}^{-1}(1-1\/S)$, and so on. Therefore, the x-axis in Fig.~\\ref{fig:SRAvsSAD_rho1} is computed with $1+S(1-{\\rm CAD}(n))$ and the y-axis are the abundances $n$. Using this approach, we generated the SRAs correspond to SAD, see   Fig.~\\ref{fig:SRAvsSAD_rho1} for the results for $\\rho=1$.  However, as is shown in Fig.~\\ref{fig:SRAvsSAD_rho1}, classification through the SRAs is less significant.\n\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[trim=140 220 100 230, width=0.49\\columnwidth]{figures\/SAD_rho1.pdf}\n    \\includegraphics[trim=140 220 100 230, width=0.49\\columnwidth]{figures\/SRA_rho1.pdf}\n    \\caption{A comparison between the species abundance distribution (left panel) and its corresponding species rank abundances (right panel). Here $\\rho=1$ is fixed, and the immigration rate $\\mu$ varies. Solid lines represent $\\mu\\in [10^{-3}, 10^{-1}, 10]$ following the legend. The dashed, dotted and dashed-dotted lines are in-between $\\mu$-s. The color scheme corresponds the modality classification of SAD; teal (bimodal, very low $\\mu$), yellow (rare-biosphere, intermediate $mu$) and blue (unimodal, high $\\mu$).       }\n    \\label{fig:SRAvsSAD_rho1}\n\\end{figure}\n\n\\section*{Experimental Studies}\n\n\\begin{table}[h!]\n    \\centering\n    \\begin{tabular}{|p{3cm}|p{4.5cm}|p{5cm}|p{3cm}|}\n       System (Ref.) & Regimes &\tConclusions & Observations  \\\\ \\hline \nMicrobial competition \\cite{gore2021} & Stable full coexsitence (IIa), stable partial coexistnce (IIb), persisnt fluctuation (IIIb) &\tPositive correlation between species and instability (rapid fluctuations)& \tCommunity composition\/ richness\/  fluctuating communities   \n\\\\\nGlobal birds species \\cite{callaghan2021global} &\tUnimodal - log skew &\t&\tSAD  \\\\\nPlankton \\cite{ser2018ubiquitous}  &\tAbundance is power-law decay , neutral-like &\t& SAD and SRA \\\\\nCoral \\cite{dornelas2008multiple}\t& Multimodal distribution &\tSAD Not from habitat preferences, most likely due to spatial effects.  Common of large samples  &\tSAD\t\n\\\\\nLymphocyte repertoire \\cite{mora2018quantifying} &\tPower law distributions&\tFunctional repertoire is more relevant than actual repertoire, overlap in antigen coverage reduces size for repertoire &\tSRA \n\\\\\nArthropods \\cite{matthews2014multimodal} &\tMulti-modal distribution\t& Propose that multiple modes come from ecologically distinct communities &\tSAD \n\\\\\nTrees \\cite{hubbell1979tree} &\tPower law distributions &\tDifferent forest types show curves with different slopes, explained by random walk model &\tSRA \n\\\\\nBacteria \\cite{bell2005larger}\t& Increasing K, related to Species Area Relation (SAR) &\tLarger islands have more bacteria taxa on them (increased diversity)\t& Diversity per islands size\t\n\\\\\nT-cell receptors \\cite{oakes2017quantitative}\t& Bimodal and unimodal (exponential) &\tAlthough total population (CD8 naive cells) has bimodal distirbution, subpopulations have exponential\tclonotype & SAD\t\n\\\\\nMicrobial competition  \\cite{descheemaeker2020stochastic} &\tNeutral-like and niche-like & Heavy tailed rank abundance (microbiome tongue seems to have 2 slopes). Model with linear (extrinsic noise) reproduces this & SRA and time series \n\\\\ \nCompetition in gastrointestinal microbiomes \\cite{jeraldo2012quantification} &\tNon-neutral (niche) &\tThe species abundance patterns are seemingly well fit by the neutral theory, however the operational taxonomic units (OTUs)  classify it as niche   &\tSRA and OTUs\n\\\\ \n    \\end{tabular}\n    \\caption{Experimental studies consider competitive species and their reported results.  }\n    \\label{tab:my_label}\n\\end{table}\n\n\n\\iffalse\n\\begin{table}\\centering\n\\caption{This is a table}\n\n\\begin{tabular}{lrrr}\nSpecies & CBS & CV & G3 \\\\\n\\midrule\n1. Acetaldehyde & 0.0 & 0.0 & 0.0 \\\\\n2. Vinyl alcohol & 9.1 & 9.6 & 13.5 \\\\\n3. Hydroxyethylidene & 50.8 & 51.2 & 54.0\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\\fi\n\n\\iffalse\n\\movie{Type legend for the movie here.}\n\n\\movie{Type legend for the other movie here. Adding longer text to show what happens, to decide on alignment and\/or indentations.}\n\n\\movie{A third movie, just for kicks.}\n\n\\dataset{dataset_one.txt}{Type or paste legend here.}\n\n\\dataset{dataset_two.txt}{Type or paste legend here. Adding longer text to show what happens, to decide on alignment and\/or indentations for multi-line or paragraph captions.}\n\\fi\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}