diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzgodl" "b/data_all_eng_slimpj/shuffled/split2/finalzzgodl" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzgodl" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nAs usual in algebraic dynamics, given a self-map $\\Phi\\colon X\\longrightarrow X$ of a quasi-projective variety $X$, we denote by $\\Phi^n$ the $n$-th iterate of $\\Phi$. Given a point $x\\in X$, we let $\\mathcal{O}_\\Phi(x)=\\{\\Phi^n(x)\\colon n\\in\\mathbb N\\}$ be the orbit of $x$. Recall that a point $x$ is periodic if there exists some $n\\in\\mathbb{N}$ such that $\\Phi^n(x)=x$; a point $y$ is preperiodic if there exists $m\\in\\mathbb{N}$ such that $\\Phi^m(y)$ is periodic. Our first result is the following.\n\n\n\\begin{theorem}\n\\label{thm:uniform-bound-fibers}\nLet $X$ and $Y$ be quasi-projective varieties defined over a field $K$ of characteristic $0$, let $f\\colon X\\longrightarrow Y$ be a morphism defined over $K$, let $\\Phi\\colon X\\longrightarrow X$ be an \\'etale endomorphism, and let $x\\in X(K)$. \nIf $|\\mathcal{O}_\\Phi(x)\\cap f^{-1}(y)|<\\infty$ for each $y\\in Y(K)$, then there is a constant $N$ such that $$|\\mathcal{O}_\\Phi(x)\\cap f^{-1}(y)|0,\n\\]\nwhere $h(\\cdot)$ is the logarithmic Weil height for algebraic numbers.\n\\end{theorem}\n\nNote that if $X=\\mathbb{A}^1$, the map $\\Phi\\colon X\\longrightarrow X$ is given by $\\Phi(x)=x+1$, and $f\\colon X\\hookrightarrow \\mathbb{P}^1$ is the usual embedding, then $h(f(\\Phi^n(0)))=\\log(n)$ for $n\\in\n\\mathbb{N}$. This example shows that Theorem~\\ref{thm:gaps} is, in some sense, the best possible. However, we believe that this gap result should hold more generally for rational self-maps. Specifically, we make the following conjecture.\n\n\\begin{conjecture} (Height Gap Conjecture)\n\\label{conj:gaps}\nLet $X$ be a quasi-projective variety defined over $\\overline{\\mathbb{Q}}$, let $\\Phi\\colon X\\dashrightarrow X$ be a rational self-map, and let $f\\colon X\\dashrightarrow \\mathbb{P}^1$ be a rational function. Then for $x\\in X(\\overline{\\mathbb{Q}})$ with the property that $\\Phi^n(x)$ avoids the indeterminacy locus of $\\Phi$ for every $n\\ge 0$, either $f(\\mathcal{O}_\\Phi(x))$ is finite or $$\\limsup_{n\\to\\infty} \\frac{h(f(\\Phi^n(x)))}{\\log(n)}>0.$$\n\\end{conjecture} \n\nTheorem \\ref{thm:gaps} proves this conjecture in the case of endomorphisms. Many interesting number theoretic questions fall under the umbrella of the gap conjecture stated above. As an example, we recall that a power series $F(x)\\in \\overline{\\mathbb{Q}}[[x]]$ is called $D$-\\emph{finite} if it is the solution to a non-trivial homogeneous linear differential equation with rational function coefficients. It is known that if $\\sum_{n\\geq0} a(n) x^n$ is a $D$-finite power series over a field of characteristic zero, then there is some $d\\ge 2$, a rational endomorphism $\\Phi\\colon\\mathbb{P}^d\\dashrightarrow \\mathbb{P}^d$, a point $c\\in \\mathbb{P}^d$ and a rational map $f\\colon \\mathbb{P}^d\\dashrightarrow \\mathbb{P}^1$ such that $a(n)=f\\circ \\Phi^n(c)$ for $n\\ge 0$, see \\cite[Section 3.2.1]{DML-book}. Heights of coefficients of $D$-finite power series have been studied independently, notably by van der Poorten and Shparlinski \\cite{vdPS}, who showed a gap result holds in this context that is somewhat weaker than what is predicted by our height gap conjecture above; specifically, they showed that if $\\sum_{n\\geq0} a(n)x^n\\in \\overline{\\mathbb{Q}}[[x]]$ is $D$-finite and $$\\limsup_{n\\to\\infty} \\frac{a(n)}{\\log\\log(n)}=0,$$ then the sequence $\\{a(n)\\}$ is eventually periodic. This was improved recently \\cite{BNZ}, where it is shown that if $\\limsup_{n\\to\\infty} \\frac{a(n)}{\\log(n)}=0$, then the sequence $\\{a(n)\\}$ is eventually periodic. We see this then gives additional underpinning to Conjecture \\ref{conj:gaps}. Furthermore, with the notation as in Theorem~\\ref{thm:gaps}, assume now that \n\\begin{equation}\n\\label{limsup is zero}\n\\limsup_{n\\to\\infty} \\frac{h\\!\\left(f(\\Phi^n(x))\\right)}{\\log(n)}=0.\n\\end{equation}\nThen Theorem~\\ref{thm:gaps} asserts that Equation~\\eqref{limsup is zero} yields that $f(\\mathcal{O}_\\Phi(x))$ is finite. We claim that actually this means that the set $\\{f(\\Phi^n(x))\\}_{n\\in\\mathbb{N}}$ is eventually periodic. Indeed, for each $m\\in\\mathbb{N}$, we let $Z_m$ be the Zariski closure of $\\{\\Phi^n(x)\\}_{n\\ge m}$. Then $Z_{m+1}\\subseteq Z_m$ for each $m$ and thus, by the Noetherian property, we get that there exists some $M\\in\\mathbb{N}$ such that $Z_{m}=Z_M$ for each $m\\ge M$. So, there exists a suitable positive integer $\\ell$ such that $\\Phi^\\ell$ induces an endomorphism of each irreducible component of $Z_M$; moreover, each irreducible component of $Z_M$ contains a Zariski dense set of points from the orbit of $x$. Furthermore, because $f(\\mathcal{O}_\\Phi(x))$ is a finite set, we get that $f$ must be constant on each irreducible component of $Z_M$ and thus, in particular, $f$ is constant on each orbit $\\mathcal{O}_{\\Phi^\\ell}(\\Phi^r(x))$ for $r$ sufficiently large. Hence, Theorem~\\ref{thm:gaps} actually yields that once Equation~\\eqref{limsup is zero} holds, then $\\{f(\\Phi^n(x))\\}_{n\\in\\mathbb{N}}$ is eventually periodic. \n \nIt is important to note that one cannot replace $\\limsup$ with $\\liminf$ in Conjecture \\ref{conj:gaps}, even in the case of endomorphisms. To see this, consider the map $\\Phi\\colon\\mathbb{A}^3\\to\\mathbb{A}^3$ given by $(x,y,z)\\mapsto (yz, xz, z+1)$.\nThen, letting $c=(0,1,1)$, it is easily shown by induction that for $n\\ge 0$, we have\n\\[\n\\Phi^{2n}(c)=(0, (2n)!, 2n+1)\\quad\\quad\\textrm{and}\\quad\\quad\\Phi^{2n+1}(c)=((2n+1)!,0,2n+2).\n\\]\nConsequently, if $f\\colon\\mathbb{A}^3\\to \\mathbb{A}^1$ is given by $f(x,y,z)=x+1$, then we see that $f(\\Phi^{2n}(c))=1$ and $f(\\Phi^{2n+1}(c))=(2n+1)!+1$ for every $n\\ge 0$, and so\n\\[\n\\liminf_{n\\to\\infty} \\frac{h(f(\\Phi^{n}(c)))}{\\log(n)}=0, \\quad\\quad\\textrm{while}\\quad\\quad \\limsup_{n\\to\\infty} \\frac{h(f(\\Phi^n(c)))}{\\log(n)}=\\infty.\n\\]\nDespite the fact that the conjecture does not hold when one replaces $\\limsup$ with $\\liminf$, we believe the following variant of Conjecture \\ref{conj:gaps} holds:\n\n\n\\begin{conjecture}\n\\label{conj:gaps-dense}\nLet $X$ be an irreducible quasi-projective variety defined over $\\overline{\\mathbb{Q}}$, let $\\Phi\\colon X\\dashrightarrow X$ be a rational self-map, and let $f\\colon X\\dashrightarrow \\mathbb{P}^1$ be a non-constant rational function. Let $x\\in X(\\overline{\\mathbb{Q}})$ with the property that $\\Phi^n(x)$ avoids the indeterminacy locus of $\\Phi$ for every $n\\ge 0$, and further suppose that $\\mathcal{O}_\\Phi(x)$ is Zariski dense in $X$. Then $$\\liminf_{n\\to\\infty} \\frac{h(f(\\Phi^n(x)))}{\\log(n)}>0.$$\n\\end{conjecture} \n\n\n\nWe point out that, if true, this would be a powerful result and would imply the Dynamical Mordell--Lang conjecture for rational self-maps when we work over a number field. To see this, let $Z$ be a quasi-projective variety defined over $\\overline{\\mathbb{Q}}$, let $\\Phi\\colon Z\\dashrightarrow Z$ be a rational self-map, $Y$ be a subvariety of $Z$, and suppose that the orbit of $x\\in Z(\\overline{\\mathbb Q})$ avoids the indeterminacy locus of $\\Phi$. As before, denote by $Z_n$ the Zariski closure of $\\{\\Phi^j(x) \\colon j\\ge n\\}$. Since $Z$ is a Noetherian topological space, there is some $m$ such that $Z_n=Z_m$ for every $n\\ge m$. Letting $X=Z_m$, and replacing $Y$ with $Y\\cap X$, it suffices to show that the conclusion to the Dynamical Mordell--Lang conjecture holds for the data $(X,\\Phi, x, Y)$. We let $X_1,\\ldots,X_d$ denote the irreducible components of $X$ and let $Y_i=Y\\cap X_i$. Since $\\Phi|_X$ is a dominant self-map, it permutes the components $X_i$, so there is some $b$ such that $\\Phi^b(X_i)\\subset X_i$ for each $i$. Then if we let $x_1,\\ldots ,x_d$ be elements in the orbit of $x$ with the property that $x_i\\in X_i$, then it suffices to show that the conclusion to the statement of the Dynamical Mordell--Lang conjecture holds for the data $(X_i, \\Phi^b, x_i, Y_i)$ for $i=1,\\ldots,d$. Then by construction, the orbit of $x_i$ under $\\Phi^b$ is Zariski dense. We prove that either $\\mathcal{O}_{\\Phi^b}(x_i)\\subset Y_i$ or that $\\O_{\\Phi^b}(x_i)$ intersects $Y_i$ finitely many times. If $Y_i=X_i$ or $Y_i=\\emptyset$ then the result is immediate; thus we may assume without loss of generality that $Y_i$ is a non-empty proper subvariety of $X_i$. We pick a non-constant morphism $f_i\\colon X_i\\longrightarrow \\mathbb{P}^1$ such that $f_i(Y_i)=1$. If $\\Phi^{bn}(x_i)\\in Y_i$, then $h(f(\\Phi^{bn}(x_i)))=0$. Conjecture \\ref{conj:gaps-dense} implies that this can only happen finitely many times, and so $\\{n\\colon \\Phi^{bn}(x_i)\\in Y_i\\}$ is finite.\n\n\n\\section{Proof of our main results}\nWe recall the following definitions. The ring of strictly convergent power series $\\mathbb{Q}_p\\langle z\\rangle$ is the collection of elements $P(z):=a_0+a_1 z+a_2 z^2 + \\cdots \\in \\mathbb{Q}_p[[z]]$ such that $|a_n|_p\\to 0$ as $n\\to \\infty$ and which thus consequently converge uniformly on $\\mathbb{Z}_p$. The \\emph{Gauss norm} is given by\n$\n|P(z)|_{\\rm Gauss}:=\\max_{n\\geq0} |a_n|_p.\n$ \nThe ring $\\mathbb{Z}_p\\\\subset\\mathbb Q_p\\$ is the set of $P(z)$ with $|P(z)|_{\\rm Gauss}\\leq1$, i.e.~the set of $P$ with $a_i\\in\\mathbb{Z}_p$.\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:uniform-bound-fibers}.]\nClearly, we may reduce immediately (at the expense of replacing $\\Phi$ by an iterate of it) to the case $X$ and $Y$ are irreducible.\n\nA standard spreading out argument (similar to the one employed in the proof of \\cite[Theorem~4.1]{DML-etale}) allows us to choose a model of $X$, $Y$, $f$, $\\Phi$, and $x$ over an open subset $U\\subseteq\\spec R$, where $R$ is an integral domain which is a finitely generated $\\mathbb{Z}$-algebra. In other words, $K$ is a field extension of the fraction field of $R$, we can find a map $\\mathcal{X}\\longrightarrow\\mathcal{Y}$ over $U$, a section $U\\longrightarrow\\mathcal{X}$, and an \\'etale endomorphism $\\mathcal{X}\\longrightarrow\\mathcal{X}$ over $U$ which base change over $K$ to be $f\\colon X\\longrightarrow Y$, $x\\colon\\spec K\\longrightarrow X$, and $\\Phi\\colon X\\longrightarrow X$, respectively. After replacing $U$ by a possibly smaller open subset, we can assume $U=\\spec R[g^{-1}]$ for some $g\\in R$. Since $R[g^{-1}]$ is a finitely generated $\\mathbb{Z}$-algebra, it is of the form $\\mathbb{Z}[u_1,\\dots,u_r]$. Applying \\cite[Lemma 3.1]{generalized-SML}, we can find a prime $p\\geq 5$ and an embedding $R[g^{-1}]$ into $\\mathbb Q_p$ which maps the $u_i$ into $\\mathbb{Z}_p$. Base changing by the resulting map $\\spec\\mathbb{Z}_p\\longrightarrow U$, we can assume $U=\\spec\\mathbb{Z}_p$. We will abusively continue to denote the map $\\mathcal{X}\\longrightarrow\\mathcal{Y}$ by $f$, the \\'etale endomorphism $\\mathcal{X}\\longrightarrow\\mathcal{X}$ by $\\Phi$, and the section $\\spec\\mathbb{Z}_p=U\\longrightarrow\\mathcal{X}$ by $x$. We let $\\overline{\\mathcal{X}}=\\mathcal{X}\\times_{\\mathbb{Z}_p}{\\mathbb F}_p$, let $\\overline{\\Phi}\\colon\\overline{\\mathcal{X}}\\longrightarrow\\overline{\\mathcal{X}}$ be the reduction of $\\Phi$, and let $\\overline{x}\\in\\mathcal{X}({\\mathbb F}_p)$ be the reduction of $x\\in\\mathcal{X}(\\mathbb{Z}_p)$.\n\nNotice that if $f(\\Phi^n(x))=y$, then since $x$ extends to a $\\mathbb{Z}_p$-point of $\\mathcal{X}$, necessarily $y\\in Y(K)$ extends to a $\\mathbb{Z}_p$-point of $\\mathcal{Y}$ as well. In particular, it suffices to give a uniform bound on the sets $\\{n:f(\\Phi^n(x))=y\\}$ as $y$ varies through the elements $\\mathcal{Y}(\\mathbb{Z}_p)$.\n\nTo prove Theorem \\ref{thm:uniform-bound-fibers}, we may replace $x$ by $\\Phi^\\ell(x)$ for some $\\ell\\in\\mathbb N$. Since $|\\mathcal{X}({\\mathbb F}_p)|<\\infty$, we can therefore assume $\\overline{x}$ is $\\overline{\\Phi}$-periodic, say of period $D$. It suffices to show that for each $1\\leq i$ such that in a $p$-adic analytic neighborhood, we have $$\\Phi^n(x)=(\\phi_1(n),\\dots,\\phi_d(n))\\in\\mathbb{Z}_p^d;$$ more precisely, letting $\\phi(z):=(\\phi_1(z),\\dots,\\phi_d(z))$, if $B\\subset\\mathcal{X}(\\mathbb{Z}_p)$ is the $p$-adic ball of points whose reduction mod $p$ is $\\overline{x}$, then there is an analytic bijection $\\iota\\colon B\\longrightarrow\\mathbb{Z}_p^d$, such that $\\iota\\!\\left(\\Phi^n(x)\\right)=\\phi(n)$.\n\nNext, fix an embedding $\\mathcal{Y}\\subset\\mathbb{P}^r_{\\mathbb{Z}_p}$, let $\\{V_i\\}_i$ be an open affine cover of $\\mathcal{Y}$, and for each $i$, let $\\{U_{ij}\\}_j$ be an open affine cover of $f^{-1}(V_i)$. We can further assume that each $V_i$ is contained in one of the coordinate spaces $\\AA^r_{\\mathbb{Z}_p} \\subset \\mathbb{P}^r_{\\mathbb{Z}_p}$. Since $\\mathcal{X}$ and $\\mathcal{Y}$ are quasi-compact, we can assume the $\\{U_{ij}\\}_{i,j}$ and $\\{V_i\\}_i$ are finite covers. Then we can view $f|_{U_{ij}}\\colon U_{ij} \\longrightarrow V_i \\subseteq \\AA^r_{\\mathbb{Z}_p}$ as a tuple of polynomials $(p_{ij0}, \\dots, p_{ijr})$. Letting $P_{ijk}(z)=p_{ijk}\\iota^{-1}\\phi(z)$, we see $f|_{\\O_{\\Phi}(x)}$ is given by the following piecewise analytic function: $$f(\\Phi^n(x))=(P_{ij0}(n), \\dots, P_{ijr}(n))$$ whenever $\\Phi^n(x)\\in U_{ij}$.\n\n\nIt therefore suffices to prove that for each $i,j$, there exists $N_{ij}$ such that for all $(y_1,\\dots,y_r)\\in V_i(\\mathbb{Z}_p)\\subseteq\\AA^r(\\mathbb{Z}_p)$, the number of simultaneous roots of $P_{ijk}(z)-y_k$ (for $k=1,\\dots,r$) is bounded by $N_{ij}$. In other words, we have reduced to proving the lemma below, where $S=\\{n:\\Phi^n(x)\\in U_{ij}\\}$ and $V=V_i(\\mathbb{Z}_p)$. \n\n\n\\begin{lemma}\n\\label{l:uniform-bound-piecewise-analytic}\nLet $r$ be a positive integer, let $V\\subset\\mathbb{Z}_p^r$, and let $S\\subset\\mathbb N$ be an infinite subset. For each $1\\leq k\\leq r$, let $P_k\\in\\mathbb{Z}_p\\$\nand consider the function $P\\colon S\\longrightarrow\\mathbb{Z}_p^r$ given by\n\\[\nP(n):=(P_1(n),\\dots,P_r(n)).\n\\]\nSuppose the set $\\{n\\in S:P(n)=y\\}$ is empty if $y\\in \\mathbb{Z}_p^r\\setminus V$ and is finite if $y\\in V$. Then there exists $N\\geq0$ such that\n\\[\n|\\{n\\in S:P(n)=y\\}|\\leq N\n\\]\nfor all $y \\in V$.\n\\end{lemma}\n\\begin{proof}\nWe may assume $S$ is infinite since otherwise we can take $N=|S|$. We claim that $P_k(z)$ is not a constant power series for some $k$. Suppose to the contrary that $P_k(z)=c_k\\in\\mathbb{Z}_p$ for each $k$. If $y:=(c_1,\\dots,c_r)\\in\\mathbb{Z}_p^r\\setminus V$, then we can take $N=0$. If $y\\in V$, then $\\{n\\in S:P(n)=y\\}=S$ which is infinite, contradicting the hypotheses of the lemma.\n\nWe have therefore shown that some $P_k(z)$ is non-constant. Let $\\mathcal{K}$ be the set of $k$ for which $P_k(z):=\\sum_{m\\geq0} c_{k,m}z^m$ is non-constant. Given any non-constant element $Q(z):=\\sum_{m\\geq0} c_mz^m$ of $\\mathbb{Z}_p\\$, let\n\\begin{equation}\n\\label{eqn:max-coeff-Gauss}\nD(Q):=\\max\\{m:|c_m|=|Q|_{\\rm Gauss}\\}.\n\\end{equation}\nRecall from Strassman's Theorem (see \\cite{strassman} or \\cite[Theorem 4.1, p.~62]{cassels}) that the number of zeros of $Q(z)$ is bounded by $D(Q)$. As a result, if $\\alpha\\in\\mathbb{Z}_p$, then the number of zeros of $Q(z)-\\alpha$ is bounded by $1+D(Q)$. Letting\n\\[\nN:=1+\\max_{k\\in\\mathcal{K}} D(P_k),\n\\]\nwe see then that for all $(y_1,\\dots,y_r)\\in\\mathbb{Z}_p^r$, the number of simultaneous zeros of $P_1(z)-y_1$, $\\dots$, $P_r(z)-y_r$ is bounded by $N$. In particular, $|\\{n\\in S:P(n)=y\\}|\\leq N$ for all $y\\in V$.\n\\end{proof}\nThis concludes the proof of Theorem~\\ref{thm:uniform-bound-fibers}.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:gaps}.] \nAs before, at the expense of replacing $\\Phi$ by an iterate, we may assume $X$ is irreducible. Furthermore, arguing as in the last paragraph of the introduction, we may assume $\\O_\\Phi(x)$ is Zariski dense.\n\nLet $K$ be a number field such that $X$, $\\Phi$, and $f$ are defined over $K$ and moreover, $x\\in X(K)$. As proven in \\cite{Schanuel}, there exists a constant $c_0>0$ such that for each real number $N\\ge 1$, there exist less than $c_0N^2$ algebraic points in $K$ of logarithmic height bounded above by $\\log(N)$. So, there exists a constant $c_1>1$ such that for each real number $N\\ge 1$, there are less than $c_1^N$ points in $K$ of logarithmic height bounded above by $N$.\n\nArguing as in the proof of Theorem~\\ref{thm:uniform-bound-fibers}, we can find a suitable prime number $p$, a model $\\mathcal{X}$ of $X$ over some finitely generated $\\mathbb{Z}$-algebra $R$ which embeds into $\\mathbb{Z}_p$ such that the endomorphism $\\Phi$ extends to an endomorphism of $\\mathcal{X}$, and a section ${\\rm Spec}(\\mathbb{Z}_p)\\longrightarrow \\mathcal{X}$ extending $x$; we continue to denote by $\\Phi$ and $x$ the endomorphism of $\\mathcal{X}$ and the section ${\\rm Spec}(\\mathbb{Z}_p)\\longrightarrow \\mathcal{X}$, respectively. At the expense of replacing both $\\Phi$ and $x$ by suitable iterates, we may assume the reduction of $x$ modulo $p$ (called $\\overline{x}$) is fixed under the induced action of $\\overline{\\Phi}$ on the special fiber of $\\mathcal{X}$. Consider the $p$-adic neighborhood in $B\\subset\\mathcal{X}(\\mathbb{Z}_p)$ consisting of all points whose reduction modulo $p$ is $\\overline{x}$. Then there is an analytic isomorphism $\\iota\\colon B\\to\\mathbb{Z}_p^m$ so that in these coordinates\n $$\\overline{x}=(0,\\ldots, 0)\\in \\mathbb{F}_p^m$$ \n and \n$\\Phi$ is given by $(x_1,\\cdots, x_m)\\mapsto (\\phi_1(x_1,\\dots, x_m),\\cdots, \\phi_m(x_1,\\dots, x_m))$, where\n$$\\phi_i(x_1,\\dots, x_m)\\equiv \\sum_{j=1}^m a_{i,j}x_j\\pmod{p}$$\n for each $i=1,\\dots, m$, for some suitable constants $a_{i,j}\\in\\mathbb{Z}_p$ (for more details, see \\cite[Section~11.11]{DML-book}). Applying \\cite[Theorem~11.11.1.1]{DML-book} (see also the proof of \\cite[Theorem~11.11.3.1]{DML-book}), there exists a $p$-adic analytic function $G\\colon\\mathbb{Z}_p\\longrightarrow \\mathbb{Z}_p^m$ such that for each $n\\ge 1$, we have\n\\begin{equation}\n\\label{eq:p-adic approximation}\n\\|\\Phi^n(x)-G(n)\\|\\le p^{-n},\n\\end{equation}\nwhere for any point $(x_1,\\dots, x_m)\\in\\mathbb{Z}_p^m$, we let\n$$\\|(x_1,\\dots, x_m)\\|:=\\max_{1\\leq i\\leq m} |x_i|_p.$$\n\n\n\nAs in the proof of Theorem~\\ref{thm:uniform-bound-fibers}, let $V_1\\simeq\\AA^1$ and $V_2\\simeq\\AA^1$ be the standard affine cover of $\\mathbb{P}^1$, and let $\\{U_{ij}\\}$ be a finite open affine cover of $\\mathcal{X}$ minus the indeterminacy locus of $f$ such that $f(U_{ij})\\subset V_i\\simeq\\AA^1$. Let\n\\[\nS_{ij}:=\\{n:\\Phi^n(x)\\in U_{ij}\\}.\n\\]\nSince $f|_{U_{ij}}$ is given by a polynomial with $p$-adic integral coefficients, there exist $H_{ij}(z)\\in\\mathbb{Z}_p\\$ such that\n\\[\nf(G(n))=H_{ij}(n)\n\\]\nwhenever $n\\in S_{ij}$. Notice that if $f(\\Phi^n(x))=y$, then since $x$ extends to a $\\mathbb{Z}_p$-point of $\\mathcal{X}$, necessarily $y\\in \\mathbb{P}^1(K)$ extends to a $\\mathbb{Z}_p$-point of $\\mathbb{P}^1$ as well. Thus, we need only concern ourselves with roots of $H_{ij}(z)-t$ for $t\\in\\mathbb{Z}_p$.\n\n\\begin{lemma}\n\\label{l:constant-rate-of-return}\nThere is some choice of $i$ and $j$ with the following properties:\n\\begin{enumerate}[label=$(\\arabic*)$]\n\\item\\label{constant-rate-of-return::infinite} $\\{f(\\Phi^n(x)):n\\in S_{ij}\\}$ is an infinite set,\n\\item\\label{constant-rate-of-return::Banach-density} $\\mathbb N\\smallsetminus S_{ij}$ has upper Banach density zero,\n\\item\\label{constant-rate-of-return::return} there exists a constant $\\kappa$ and a sequence $M_1$, there exists some $L\\ge 1$ such that $|a_L|_p>|a_j|_p$ for all $j>L$. As proven in Lemma~\\ref{l:uniform-bound-piecewise-analytic}, since $H(z)$ is not constant, there exists a uniform bound $C$ such that for each $t\\in \\mathbb{Z}_p$, the number of solutions to $H(z)=t$ is at most $C$. Furthermore, if $n$ is an element of $S$ such that $f(\\Phi^n(x))=t$, then equation \\eqref{eq:p-adic approximation}\nyields $$|H(n)-t|_p\\le p^{-n}.$$ \nAs mentioned above, by the Weierstrass Preparation Theorem, we can write\n\\[\nH(z)-t= q_t(z)u_t(z)\n\\]\nwith $q_t(z)$ a polynomial of degree $D(H-t)\\leq L$ and $u_t(z)$ a unit of Gauss norm $1$; moreover, the leading coefficient of $q_t(z)$ has $p$-adic norm equal to the Gauss norm of $H-t$. Hence, we can write\n\\[\nq_t(z)=b_t(z-\\beta_{1,t})\\cdots (z-\\beta_{D(H-t),t})\n\\]\nwith $b_t\\in \\mathbb{Q}_p$, the $\\beta_{j,t}\\in \\overline{\\mathbb Q}_p$, and\n\\[\n|b_t|_p=|H-t|_{\\rm Gauss}\\geq |a_L|_p.\n\\]\nWe have therefore bounded $|b_t|_p$ below independent of $t\\in\\mathbb{Z}_p$. As noted before the proof of the lemma, we know $|u_t(n)|_p=1$ for all $t\\in\\mathbb{Z}_p$ and $n\\in\\mathbb N$. Hence, there is a constant $c_2>0$ (independent of $t$) such that for all $t\\in\\mathbb{Z}_p$, if $|H(n)-t|_p\\le p^{-n}$ then there exists $1\\leq j\\leq D(H-t)$ such that\n\\[\n|n-\\beta_{j,t}|_pc_2 p^{\\min(n_{k_1},n_{k_2})\/L};\n\\]\ntherefore there exists a positive constant $c_3$ (independent of $t$, since both $L$ and $c_2$ are independent of $t$) such that for all $M\\ge 1$ and all $t\\in\\mathbb{P}^1(K)$,\n\\begin{equation}\n\\label{eq:c3-bound}\n\\#\\{n\\le M:n\\in S \\textrm{\\ and\\ } f(\\Phi^n(x))=t\\}\\leq c_3\\log(M).\\footnote{In fact, we have a substantially better bound. Let $\\exp^k$ denote the $k$-th iterate of the exponential function and let $L_p(M)$ be the smallest integer $k$ such that $\\exp^k(p)>M$. Then $\\#\\{n\\le M: n\\in S \\textrm{\\ and\\ } f(\\Phi^n(x))=t\\}\\leq c_3 L_p(M)$, however we will not need this stronger bound.}\n\\end{equation}\nAs an aside, we note that this type of gap is similar to the one obtained for the Dynamical Mordell--Lang problem in \\cite{gap-Compo}.\n\nNow, let $\\kappa$ be as in Lemma \\ref{l:constant-rate-of-return}, and choose a constant $c_4>1$ such that\n\\begin{equation}\n\\label{eq:c43}\nc_3\\cdot \\log(\\kappa c_4^r)\\cdot c_1^rc_4^{N_\\ell-1}.\n\\end{equation}\n\nTo conclude the proof, we show that for all $\\ell$ sufficiently large, there exists some $n_\\ell\\le \\kappa c_4^{N_\\ell}$ with the property that $n_\\ell\\in S$ and $h(f(\\Phi^{n_\\ell}(x)))\\ge N_\\ell$. If this were not the case, then since there are less than $c_1^{N_\\ell}$ algebraic numbers $t\\in\\mathbb{P}^1(K)$ of logarithmic Weil height bounded above by $N_\\ell$, by (\\ref{eq:constant-rate-of-return-inequality}) there would be such an algebraic number $t$ with \n\\[\n\\#\\{n\\le \\kappa c_4^{N_\\ell}:n\\in S \\textrm{\\ and\\ } f(\\Phi^n(x))=t\\}>\\frac{c_4^{N_\\ell-1}}{c_1^{N_\\ell}}>c_3\\log(\\kappa c_4^{N_\\ell})\n\\]\nand this violates inequality (\\ref{eq:c3-bound}). We have therefore proven our claim that for all $\\ell$ sufficiently large, there exists a positive integer $n_\\ell\\leq \\kappa c_4^{N_\\ell}$ with $h(f(\\Phi^{n_\\ell}(x)))\\ge N_\\ell$. So, \n\\[\n\\limsup_{n\\to\\infty}\\frac{h(f(\\Phi^n(x)))}{\\log(n)} \\geq \\lim_{\\ell\\to\\infty}\\frac{N_\\ell}{\\log(\\kappa)+N_\\ell\\log(c_4)}=\\frac{1}{\\log(c_4)}>0\n\\]\nas desired in the conclusion of Theorem~\\ref{thm:gaps}.\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{lem:approx_2}\nIf $H(z)$ is a constant, then $\\limsup_{n\\to\\infty}\\frac{h(f(\\Phi^n(x)))}{\\log(n)}=\\infty$.\n\\end{lemma}\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:approx_2}.]\nBy property \\ref{constant-rate-of-return::infinite} of Lemma \\ref{l:constant-rate-of-return}, we can find a sequence $n_10$ such that for all sufficiently large $k$, we have\n\\[\nC'>\\frac{h(f(\\Phi^{n_{2k}}(x)))}{\\log(n_{2k})}\\geq \\frac{1}{2\\log(n_{2k})}(c_5 n_{2k-1} - \\log(2)),\n\\]\nwhere we have made use here of inequality (\\ref{eq:one height is large}). In particular, there is a constant $C>1$ such that for all $k$ sufficiently large,\n\\begin{equation}\n\\label{eqn:big-gaps}\nn_{2k}>C^{\\hspace{0.1em}n_{2k-1}}.\n\\end{equation}\nRecalling that $S$ does not contain any positive integers between $n_{2k-1}$ and $n_{2k}$, inequality (\\ref{eqn:big-gaps}) implies that $\\mathbb N\\smallsetminus S$ has positive upper Banach density. This contradicts property \\ref{constant-rate-of-return::Banach-density} of Lemma \\ref{l:constant-rate-of-return}, and so our initial assumption that $C'>\\frac{h(f(\\Phi^{n_{2k}}(x)))}{\\log(n_{2k})}$ is incorrect. This proves equation (\\ref{eq:even-term-subseq-limsup-infty}), and hence Lemma \\ref{lem:approx_2}.\n\\end{proof}\n\nClearly, Lemmas~\\ref{lem:approx_1} and \\ref{lem:approx_2} finish the proof of Theorem~\\ref{thm:gaps}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nSince the early work of Bloch and P{\\'o}lya \\cite{bloch} in the 30's, the study of random algebraic equations has now a long story \\cite{bharucha, farahmand}. In the last few years, it attracted a renewed interest in the context of probability and number theory \\cite{edelman}, as well as in the field of quantum chaos\n\\cite{bogo}. Recently, we showed that there are also interesting\nconnections between random polynomials and persistence properties of\nphysical systems \\cite{us_short, us_long}. \n\nHere we consider real random polynomials, {\\it i.e.} polynomials with\nreal random coefficients, of degree $n$. While these polynomials have exactly $n$ roots in the complex plane, \nthe number of roots on the {\\it real} line $N_n$ is a random variable. One would like to characterize the\nstatistics of this random variable and a natural question is thus :\nwhat is the mean number $\\langle N_n \\rangle$ of real roots and how\ndoes it behave with $n$ for large $n$ \\cite{edelman}? This question \nhas been widely studied in the past for Kac's polynomials $K_n(x) =\n\\sum_{k=0}^n a_k\\, x^k$ where $a_k$ are independent and identically\ndistributed (i.i.d.) random variables of\nfinite variance $\\langle a_k^2 \\rangle = \\sigma^2$. In that case it is\nwell known \nthat $\\langle N_n \\rangle \\sim \\frac{2}{\\pi}\\log \nn$, independently of $\\sigma$. This result was first obtained by\nKac \\cite{kac} for Gaussian \nrandom variables and it was later shown to hold also for a wider class\nof distributions of the coefficients $a_k$ \\cite{bharucha,\n farahmand}. Interesting generalizations of Kac's polynomials have\nbeen studied in the literature where $a_k$ are independent Gaussian variables but\nnon identical, such that \n$\\langle a_k^2\\rangle = k^{d-2}$, where $d>0$ is a real number,\nleading to $\\langle N_n \\rangle \\sim \\pi^{-1}(1+\\sqrt{d\/2})\\log{n}$\n\\cite{us_long, das}. Given the robustness of this asymptotic\nlogarithmic behavior of $\\langle N_n \\rangle$, it is natural to search for random\npolynomials for which $\\langle N_n \\rangle$ increases faster than $\\log{n}$, for instance algebraically. \n\nOne such instance is provided by the real Weyl polynomials $W_n(x)$\ndefined by \n\\begin{eqnarray}\\label{weyl}\nW_n(x) = \\sum_{k=0}^n \\epsilon_k \\frac{x^k}{\\sqrt{k!}} \\;,\n\\end{eqnarray}\nwhere $\\epsilon_k$ are i.i.d. random variables of zero mean and unit\nvariance. Thus here, $a_k = \\epsilon_k\/\\sqrt{k!}$ and the variance is\n$\\langle a_k^2 \\rangle = 1\/k!$, which for large $k$ behaves as\n$\\langle a_k^2 \\rangle \\propto e^{-k \\log k}$. For these real polynomials\nin Eq. (\\ref{weyl}), it is known that $\\langle N_n \\rangle \\propto\nn^{1\/2}$. For instance, in the special case where $\\epsilon_k$ are\nGaussian random \nvariables of unit variance, one has $\\langle N_n \\rangle \\sim\n\\frac{2}{\\pi} \\sqrt{n}$ \\cite{us_long, leboeuf}. Another interesting\nand intriguing instance of real random polynomials was introduced a\nlong time ago by Littlewood and Offord \\cite{littlewood} who studied\nthe random polynomials $L_n(x)$ given by \n\\begin{eqnarray}\\label{little}\nL_n(x) = \\frac{1}{2}+\\sum_{k=1}^n \\epsilon_k \\frac{x^k}{(k!)^{k}} \\;,\n\\end{eqnarray}\nwhere $\\epsilon_k = \\pm 1$ with equal probability. Thus in this case\n$a_k = \\epsilon_k \/(k!)^{k}$ and the variance is $\\langle a_k^2\n\\rangle = 1\/(k!)^{2k}$, which behaves for large $k$ as $\\langle a_k^2\n\\rangle \\propto e^{-2k^2 \\log k}$. Using algebraic\nmethods, they showed that such polynomials $L_n(x)$ have all their\nroots real and therefore $\\langle N_n \\rangle = n$. \n\nWe thus have here two examples of real random polynomials in\nEq.~(\\ref{weyl}) and Eq.~(\\ref{little}) where, at variance with Kac's\npolynomials, $\\langle N_n \\rangle$ grows algebraically with $n$. In the\nsecond example (\\ref{little}), the number of real roots is\n``macroscopic'' in the sense that, for large $n$, there is a finite\nfraction $\\langle \nN_n \\rangle\/n$ of the roots which are on the real axis. For $L_n(x)$\nin Eq. (\\ref{little}) this fraction is exactly one. We thus say\nthat there is a {\\it condensation} of the roots on the real line,\nsimilar to a Bose-Einstein condensation where a finite fraction of the\nparticles of a quantum-mechanical system (Bosons) condense into the lowest\nenergy level. In the case of random polynomials, the roots play the\nrole of the particles and the equivalent of the ground state is the real line.\n\n\nThe purpose of this paper is to understand what types of polynomials\nlead to this condensation phenomenon. Of course, it is very difficult to\naddress this question for any random coefficients $a_k$. However,\nguided by the two examples above in Eq.~(\\ref{weyl}) and\nEq.~(\\ref{little}), and in particular by the large $k$ behavior of \n$\\langle a_k^2 \\rangle$, we introduce a family of random polynomials\n$P_n(x)$ indexed by a real $\\alpha \\geq 0$ defined by \n\\begin{eqnarray}\\label{def_poly} \nP_n(x) = \\sum_{k=0}^{n} a_k \\, x^k \\;, \\; \\langle a_k^2 \\rangle = e^{-k^\\alpha} \\;,\n\\end{eqnarray} \nwhere $a_k$ are real independent Gaussian random variables of zero\nmean. While $\\alpha=0$ corresponds to Kac's polynomials, we recall that, for $W_n(x)$ in Eq. (\\ref{weyl}), $\\langle\na_k^2 \\rangle \\propto e^{-k \\log k}$ and for $L_n(x)$ in\nEq. (\\ref{little}), $\\langle a_k^2 \\rangle \\propto e^{-2 k^2 \\log\n k}$. Therefore, due to the extra \nlogarithmic factor, these random polynomials are not exactly of the form\nintroduced above (\\ref{def_poly}). However, for $\\alpha \\to 1^+$, one\nexpects to recover the behavior of $W_n(x)$ in Eq. (\\ref{weyl}) while\nfor $\\alpha \\to 2^+$, one\nexpects $P_n(x)$ to behave similarly to $L_n(x)$ in\nEq. (\\ref{little}) : this is depicted schematically in Fig. \\ref{fig1}. \n\nOur main results can be summarized as follows. As $\\alpha \\geq 0$ is varied\none finds three different {\\it phases}. The first phase corresponds to $0 \\leq\n\\alpha < 1 $, where one finds that $\\langle N_n \\rangle \\sim (2\/\\pi) \\log{n}$. In the second one,\ncorresponding to $1 < \\alpha < 2$, one has $\\langle N_n \\rangle \\sim \n\\frac{2}{\\pi}\\sqrt{\\frac{\\alpha-1}{\\alpha}} \\, n^{\\alpha\/2}$. And in the third phase, for $\\alpha > 2$, one\nfinds $\\langle N_n \\rangle \\sim n$. The condensation of the roots\non the real axis thus happens for $\\alpha \\geq 2$ and as one increases\n$\\alpha$, the condensation transition sets in at the critical value\n$\\alpha_c = 2$. Furthermore, one finds that these real\nroots condense into a quasi-periodic structure such that there is, on\naverage, one root in the interval \n$[-x_{m+1},-x_m] \\cup [x_m,x_{m+1}]$, with $x_m =\ne^{\\frac{\\alpha}{2}m^{\\alpha-1}}$, with $1 \\ll m 2$\ncorresponds to the low-temperature (ordered) phase. \n\\begin{figure}\n\\includegraphics[angle=0,scale=0.6]{recap.eps}\n\\caption{Asymptotic behavior of the mean number of real roots $\\langle\n N_n \\rangle$ of $P_n(x)$ in Eq. (\\ref{def_poly}) as a function of\n $\\alpha$. These polynomials exhibit a condensation of their roots on\n the real axis for $\\alpha \\geq 2$.}\\label{fig1}\n\\end{figure}\nRoughly speaking, one can consider our results as an interesting example where the transition from the\nhigh temperature where $\\langle N_n \\rangle \\propto \\log{n}$ (governed\nby a ``$\\alpha = 0$ fixed point'') to the\nlow temperature phase where $\\langle N_n \\rangle \\propto n$ (governed\nby ``$\\alpha = \\infty$'' fixed point) happens through a \n{\\it marginal phase}, for $1< \\alpha < 2 $, where $\\langle N_n \\rangle\n\\sim n^{\\phi}$ with an exponent $\\phi = \\alpha\/2$ which depends\ncontinuously on $\\alpha$. \n\nThe paper is organized as follows. In section 2, we describe the general\nframework to compute the local density of real roots, which directly\nleads to $\\langle N_n \\rangle$. In section 3 to 6 we then analyse\nseparately the \ncases $0 \\leq \\alpha < 1$, $\\alpha < 2$, $\\alpha > 2$ and the\n''critical case'' $\\alpha = 2$. In section 7, we\ngive a qualitative argument to explain the condensation transition\noccurring at \n$\\alpha_c = 2$ before we conclude in section 8. The Appendix contains\nsome useful technical details. \n \n\\section{General framework}\n\n\nFirst we notice that given that $P_n(x)$, as a function of\n$x$, is a Gaussian process, it is completely characterized by its\ntwo-point correlation function $C_n(x,y)$\n\\begin{equation}\\label{def_correl}\nC_n(x,y) = \\langle P_n(x) P_n(y) \\rangle = \\sum_{k=0}^n e^{-k^\\alpha} \\,x^k\\,y^k \\;,\n\\end{equation}\nwhere we used the notation $\\langle ... \\rangle$ to denote an average over the random variables $a_k$. \nA central object involved in the calculation of $\\langle N_n \\rangle$\nis $\\rho_n(x)$, the mean density of real roots at point $x$. If we denote $\\lambda_1, \\lambda_2, ..., \\lambda_p$ the $p$ real\nroots (if any) of $P_n(x)$, one has $\\delta(P_n(x)) = \\sum_{i=1}^p \\delta(x-\\lambda_i)\/|P_n'(\\lambda_i)|$ such that \n$\\rho_n(x)$ can be written as\n\\begin{eqnarray}\n\\rho_n(x) &=& \\sum_{i=1}^p \\langle \\delta(x-\\lambda_i) \\rangle = \n\\langle |P_n'(x)|\\delta(P_n(x)) \\nonumber \\\\\n&=& \\int_{-\\infty}^\\infty dy |y| \\langle \\delta(P_n'(x)-y) \\delta(P_n(x)) \\rangle \\;. \\label{def_density}\n\\end{eqnarray}\nUnder this form (\\ref{def_density}), one observes that the computation of \nthe mean density involves the joint distribution of the polynomial\n$P_n(x)$ and its derivative $P'_n(x)$ which is simply a bivariate\nGaussian distribution. After Gaussian integration over $y$, one obtains \n\\begin{eqnarray}\\label{def_density_inter}\n&&\\rho_n(x) = \\frac{\\sqrt{c_n(x) (c_n'(x)\/x + c_n''(x)) - [c_n'(x)]^2\n}}{2 \\pi c_n(x)} \\;, \\\\\n&&c_n(x) = C_n(x,x) = \\sum_{k=0}^n e^{-k^\\alpha} x^{2k}\\;. \\nonumber\n\\end{eqnarray}\nThis formula (\\ref{def_density_inter}) can be written in a very\ncompact way \\cite{edelman} : \n\\begin{eqnarray}\n\\rho_n(x) = \\frac{1}{\\pi} \\sqrt{\\partial_u \\partial_v \\log C_n(u,v)}\n\\bigg \n|_{u=v=x} \\;.\\label{ek_formula}\n\\end{eqnarray} \nGiven that the random coefficients $a_k$ are drawn from a symmetric distribution, we can restrict our study of $\\rho_n(x)$ on ${\\mathbb {R}}^+$ from which one obtains the mean number of real roots $\\langle N_n \\rangle$ as\n\\begin{equation}\n\\langle N_n \\rangle = 2 \\int_0^\\infty \\rho_n(x) dx \\;.\n\\end{equation}\n\n\n{\\bf An important change of variable.} We will see below that it is useful to consider these polynomials $P_n(x)$ in terms of another variable\n$Y$ defined as\n\\begin{equation}\\label{new_variable}\nY = \\left(\\frac{2}{\\alpha} \\log{x} \\right)^{\\frac{1}{\\alpha-1}} \\;.\n\\end{equation}\nWe denote $\\hat \\rho_n(Y)$ the mean density of the real roots in terms of this new variable such that one \nhas also $\\langle N_n \\rangle = \\int_0^\\infty \\hat \\rho_n(Y) dY$. For $0 < \\alpha < 1$ we will see that, for large $n$, most of the real roots of $P_n$ are located close to $Y = n$ while for $\\alpha > 1$, the density extends over the whole interval $Y \\in [1,n]$. This change of variable (\\ref{new_variable}) is motivated by the following analysis. \n\nFirst we notice that $C_n(x,y) = \\sum_{k=0}^n e^{-k^\\alpha} x^k y^k$ in Eq. (\\ref{def_correl}) is of the form $C_n(x,y)=\nc_n(\\sqrt{x y})$. Anticipating a saddle point analysis, one writes $c_n(x)$ as \n\\begin{eqnarray}\\label{def_series}\n&&c_n(x) = \\sum_{k=0}^n e^{-k^\\alpha} x^{2k} = \\sum_{k=0}^n\n\\exp{\\left(-\\phi(k,x) \\right)} \\;, \\;\\phi(k,x) = k^\\alpha - 2 k\n\\log{x} \\;. \n\\end{eqnarray}\nAlthough $\\phi(k,x)$ is defined for integers $k = 0, 1, 2, \\cdots, n$,\nit is readily extended to the real axis and denoted $\\phi(u,x) =\nu^\\alpha - 2u\\log{x}$ for $u \\in\n\\mathbb{R}^+$. The behavior of $c_n(x)$ is essentially governed by the\nbehavior of $\\phi(u,x)$ as a function of $u$ (and fixed $x$). In\nparticular, for $\\alpha < 1$, $\\phi(u,x)$ has a single maximum while\nfor $\\alpha > 1$, it has a single minimum for $u = u^*(x)$ given by\n\\begin{eqnarray}\\label{def_ustar} \n&&\\partial_u \\phi(u^*(x),x) = 0 \\; , \\; \\partial^2_u \\phi(u^*(x),x) =\n\\alpha(\\alpha-1) u^*(x)^{\\alpha-2} > 0 \\;, \\nonumber \\\\\n&& u^*(x) = \\left(\\frac{2}{\\alpha} \\log{x} \\right)^{\\frac{1}{\\alpha-1}} \\;.\n\\end{eqnarray} \nThe new variable $Y$ introduced above in Eq. (\\ref{new_variable}) is\nthus precisely $Y = u^*(x)$. As a consequence, the density behaves\nquite differently in both cases $\\alpha < 1$ and $\\alpha > 1$.\n\nFor $\\alpha < 1$, most of the real roots on $\\mathbb{R}^+$ are located\nin $[1, \\infty]$. For \nfixed $x>1$, $\\phi(u,x)$ as a function of $u$ in the interval $[0,n]$ has a\nglobal minimum for $u=n$. Therefore, the sum entering in the expression of \n$c_n(x)$ in Eq. (\\ref{def_series}) will be dominated by the terms with $k \\sim n$. The expansion\nof $\\phi(k,x)$ in Taylor series around $k=n$ yields\n\\begin{eqnarray}\n\\phi(k,x) &=& \\phi(n,x) + (k-n) (\\alpha n^{\\alpha-1} - 2 \\log{x} ) + \\cdots \\nonumber \\\\\n&=& (1-\\alpha)n^{\\alpha} - k(\\alpha n^{\\alpha-1}-2\\log{x}) + \\cdots \\;,\n\\end{eqnarray}\nwhere the higher order terms can be neglected in the large $n$ limit because $\\partial^j \\phi(n,x)\/\\partial u^j = {\\cal O}(n^{\\alpha - j})$ for $j \\geq 2$. Thus, for $\\alpha < 1$ one has \n\\begin{eqnarray}\\label{kac_sim}\nc_n(x) \\sim e^{-(1-\\alpha)n^\\alpha} \\sum_{k=0}^n (x e^{-\\frac{\\alpha}{2} n^{\\alpha-1}})^{2k} \\;,\n\\end{eqnarray}\nwhich, in terms of the rescaled variable $\\tilde x = x \\,e^{-\\frac{\\alpha}{2} n^{\\alpha-1}}$, is the correlator of Kac's polynomials. From this observation (\\ref{kac_sim}), one can straightforwardly obtain the mean number of real roots $\\langle N_n \\rangle$, this will be done in section 3. \n\nFor $\\alpha > 1$, the situation is quite different and in that case, $\\phi(u,x)$ has a single\nminimum for $u = u^*(x) = (\\frac{2}{\\alpha}\n\\log{x})^{\\frac{1}{\\alpha-1}}$ (\\ref{def_ustar}). Besides, we will see \nbelow that the main contribution \nto $\\langle N_n \\rangle$ on $\\mathbb{R}^+$ comes from the interval $1 < x < \\exp{\\left(\\frac{\\alpha}{2} n^{\\alpha-1} \\right)}$ where $1<\nu^*(x) < n$. In that case the sum entering in the definition of\n$c_n(x)$ in Eq. (\\ref{def_series}) is indeed dominated by $k \\sim\nu^*(x)$ and $c_n(x)$ can be evaluated by a saddle point\ncalculation. For this purpose, one obtains after some algebra explained in the Appendix, a convenient expression of $\\rho_n(x)$ as\n\\begin{eqnarray}\\label{start_expr_rho}\n\\hspace*{-1cm}\\rho_n(x) = \\frac{1}{\\pi x} \\left( \\frac{ \\sum_{k=0}^n (k-u^*(x))^2 e^{-\\phi(k,x)}}{\\sum_{k=0}^n e^{-\\phi(k,x)}} - \n \\left[ \\frac{ \\sum_{k=0}^n (k-u^*(x)) e^{-\\phi(k,x)}}{\\sum_{k=0}^n e^{-\\phi(k,x)}} \\right]^2\n\\right)^{\\frac{1}{2}} \\;,\n\\end{eqnarray}\nwhich is the starting point of our analysis for $\\alpha > 1$. For $1 < x < \\exp{\\left(\\frac{\\alpha}{2} n^{\\alpha-1} \\right)}$, one has $u^*(x)2$ and $\\alpha=2$ separately. This will be done in section 4, 5 and 6 respectively. \n\n\n\n\\section{The case $0 < \\alpha < 1$}\n\nIn that case, from the expression for $c_n(x)$ in Eq. (\\ref{kac_sim}),\nwe can use the results of Kac's polynomials to obtain that most of the\nreal roots will be such that, for large $n$, $x\ne^{-\\frac{\\alpha}{2}n^{\\alpha-1}} -1 = {\\cal O}(n^{-1})$\n\\cite{fyodorov}. \nIn other words, the real roots are distributed in a region of width\n$1\/n$ around $e^{\\frac{\\alpha}{2}n^{\\alpha-1}} = 1 + \\frac{\\alpha}{2}\nn^{\\alpha-1} + {\\cal O}(n^{\\alpha-2})$ and this distribution is\nexactly the same as the one for Kac's polynomials (corresponding to\n$\\alpha=0$). The number of real roots is thus also the same and given\nby \n\\begin{equation}\\label{last_kac}\n\\langle N_n \\rangle \\sim \\frac{2}{\\pi} \\log{n} \\;,\n\\end{equation}\nindependently of $\\alpha < 1$.\n\n\n\n\\section{The case $1 < \\alpha < 2$}\n\nIn that case $[u^*(x)]^{\\alpha-2} \\to 0$ for large $u^*(x)$ and one thus sees \non the asymptotic expression in\nEq. (\\ref{asympt_largex}) that the discrete sum can be replaced by an\nintegral. This yields, for large $n$ and large $x$ with $x <\n\\exp{(\\frac{\\alpha}{2} n^{\\alpha-1}})$ \n\\begin{equation}\\label{discrete_integral}\n\\hspace*{-0.5cm}\\sum_{k=0}^n\n g\\left( k-u^*(x)\\right) \\exp{(- \\phi(k,x))} \\sim e^{-\\phi(u^*(x),x)}\\int_{-\\infty}^\\infty g(y)\n e^{-\\frac{\\alpha(\\alpha-1)}{2} y^2 u^*(x)^{{\\alpha-2}} } \\, dy \\;.\n\\end{equation}\nNote that the prefactor $e^{-\\phi(u^*(x),x)}$ is unimportant for the computation of $\\rho_n(x)$ because it disappears between the numerator and the denominator in Eq. (\\ref{start_expr_rho}) and it will be omitted below. In particular, setting $g(z) = 1$ in Eq. (\\ref{discrete_integral}) one has\n\\begin{eqnarray}\\label{eq_g1}\n\\sum_{k=0}^n \\exp{(-\\phi(k,x))} \\propto \\sqrt{2 \\pi}\n\\left[\\frac{u^*(x)^{2-\\alpha}}{\\alpha(\\alpha-1)} \\right]^{\\frac{1}{2}} \\;,\n\\end{eqnarray}\nand similarly, setting $g(z)=z^2$ in Eq. (\\ref{discrete_integral}) one has\n\\begin{eqnarray}\\label{eq_gx2}\n\\sum_{k=0}^n (k-u^*(x))^2 \\exp{(-\\phi(k,x))} \\propto \\sqrt{2 \\pi}\n\\left[\\frac{u^*(x)^{2-\\alpha}}{\\alpha(\\alpha-1)} \\right]^{\\frac{3}{2}} \\;,\n\\end{eqnarray}\nwhile $\\sum_{k=0}^n (k-u^*(x))\\exp{(-\\phi(k,x))} \\sim 0$ to\nlowest order in $n$. Therefore using the exact expression given in\nEq. (\\ref{start_expr_rho}) together with the \nasymptotic behaviors given in Eq.~(\\ref{eq_g1}, \\ref{eq_gx2}), one obtains\nthe large $x$ behavior of \n$\\rho_n(x)$ as\n\\begin{equation}\\label{asympt_largex_alleq2}\n\\rho_n(x) \\sim \\frac{1}{\\pi x}\n\\frac{1}{\\sqrt{\\alpha(\\alpha-1)}} \\left(\\frac{2}{\\alpha} \\log{x}\n\\right)^{\\frac{2-\\alpha}{2(\\alpha-1)}} \\;.\n\\end{equation} \nFor a clear comparison with the case $\\alpha > 2$ (which will be analysed in\nthe next section), it is convenient to\nwrite the density $\\hat \\rho_n(Y)$, in terms of the variable $Y =\n\\left(\\frac{2}{\\alpha} \\log{x} \\right)^{\\frac{1}{\\alpha-1}}$, \nwhich reads, for $1 \\ll Y < n$ \n\\begin{eqnarray}\\label{asympt_largeX_alleq2}\n\\hat \\rho_n(Y) \\sim \\frac{\\sqrt{\\alpha(\\alpha-1)}}{2 \\pi} Y^{-\\frac{1}{2}(2-\\alpha)} \\;,\n\\end{eqnarray}\nand in Fig. \\ref{fig2} a), we show a sketch of this asymptotic\nbehavior (\\ref{asympt_largeX_alleq2}) of $\\hat \\rho_n(Y)$ for $1 \\ll Y\n< n$. \n\nWe can now compute $\\langle N_n \\rangle = \\int_{-\\infty}^\\infty \\rho_n(x) \\, dx$. First, one notices that for $\\alpha > 1$, the series entering in the definition of $c_n(x)$\nin Eq. (\\ref{def_series}) has an infinite radius of convergence so\nthat one readily obtains that $\\int_{-1}^{+1} \\rho_n(x) \\, dx$ is of\norder ${\\cal O}(1)$ in the \nlimit $n \\to \\infty$. Besides, for large $x \\gg e^{\\frac{\\alpha}{2}\n n^{\\alpha-1}}$, one has (see also Ref. \\cite{us_long}) \n\\begin{equation}\\label{very_largeX}\n\\rho_n(x) \\sim \\sqrt{\\frac{\\langle a_{n-1}^2\\rangle}{\\langle a_{n}^2\n \\rangle}}\\frac{1}{\\pi x^2} \\sim \\frac{e^{\\frac{\\alpha}{2}\n n^{\\alpha-1}}}{\\pi x^2} \\;,\n\\end{equation} \nwhich implies that $\\int_{e^{\\frac{\\alpha}{2}\n n^{\\alpha-1}}}^\\infty \\rho_n(x) \\, dx$ is also of order ${\\cal\n O}(1)$ in the limit $n\\to \\infty$. From these properties, it follows\nthat the main contributions to $\\langle N_n \\rangle$ on ${\\mathbb R}^+$ comes from the\ninterval $[1, e^{\\frac{\\alpha}{2} n^{\\alpha-1}}]$ where the asymptotic\nbehavior of $\\rho_n(x)$ is given in\nEq. (\\ref{asympt_largex_alleq2}). Therefore one has\n\\begin{equation}\\label{N_alleq2}\n\\langle N_n \\rangle \\sim 2 \\int_1^{e^{\\frac{\\alpha}{2} n^{\\alpha-1}}} \\rho_n(x) \\, dx\n\\sim \\frac{2}{\\pi} \\sqrt{\\frac{\\alpha-1}{\\alpha}} \\,n^{\\alpha\/2} \\;,\n\\end{equation}\nwhere the factor $2$ comes from the additional contribution coming from $[-e^{\\frac{\\alpha}{2} n^{\\alpha-1}},-1]$. We thus have here an algebraic growth\n$\\langle N_n \\rangle \\propto n^{\\alpha\/2}$ with a continuously varying exponent $\\alpha\/2$. This exponent tends to $1\/2$ as $\\alpha \\to 1^+$, which is\nexpected from the analysis of Weyl polynomials $W_n(x)$ in Eq. (\\ref{weyl}) for which $\\langle a_k^2 \\rangle \\propto e^{-k \\log k}$ (although the variance is not exactly of the form $\\langle a_k^2 \\rangle = e^{-k^\\alpha}$). Besides, from Eq. (\\ref{N_alleq2}), one also obtains that \nthe amplitude of this term proportional to $n^{\\alpha\/2}$ vanishes when $\\alpha \\to 1$. We recall that for $\\alpha \\leq 1$, one has instead $\\langle N_n \\rangle \\propto (\\frac{2}{\\pi}) \\log{n}$ (\\ref{last_kac}), characteristic for Kac's polynomials. This suggests that this limit $\\alpha \\to 1$ is rather singular in the sense that the asymptotic behavior of $\\langle N_n \\rangle$\nfor large $n$ changes \"discontinuously\" from $\\log{n}$ to $\\sqrt{n}$.\n\n\n\n\\section{The case $\\alpha > 2$}\n\nIn that case, the behavior of the discrete sum in\nEq. (\\ref{asympt_largex}), which \nenters in the computation of $\\rho_n(x)$ (\\ref{start_expr_rho}) is quite\ndifferent. Indeed, in that case $[u^*(x)]^{\\alpha-2} \\propto (\\log{x})^{(\\alpha-2)\/(\\alpha-1)}\\to \\infty$ for large\n$x$ and therefore the leading term for large $x$ in\nEq.~(\\ref{asympt_largex}) corresponds to $m=0$ if $b < 1\/2$ or $m=1$ in\n$b>1\/2$. \nKeeping these leading contributions, one has\n\\begin{eqnarray}\n&&\\sum_{k=0}^n\n g\\left( k-u^*(x)\\right) \\exp{(-\\phi(k,x))} \\propto g(-b) \\exp{\\left[-\\frac{\\alpha (\\alpha-1)}{2} b^2 u^*(x)^{{\\alpha-2}}\\right]} \\nonumber \\\\\n&&+ g(1-b)\\exp{\\left[-\\frac{\\alpha (\\alpha-1)}{2} \n (1-b)^2 u^*(x)^{{\\alpha-2}}\\right]}\\label{asympt_largex_alphaleq2} \\;.\n\\end{eqnarray}\nwhere, again, we have omitted the unimportant prefactor $e^{-\\phi(u^*(x),x)}$. Using this large $x$ expansion (\\ref{asympt_largex_alphaleq2}), one obtains $\\rho_n(x)$ in Eq. (\\ref{start_expr_rho}) as\n\\begin{eqnarray}\n \\rho_n(x) \\sim \\frac{2}{(\\pi x)\\cosh{\\left[\\frac{\\alpha(\\alpha-1)}{2} Y^{\\alpha-2}(1-2b) \\right]} } \\, , \\, Y = \\left(\\frac{2}{\\alpha} \\log{x} \\right)^{\\frac{1}{\\alpha-1}} \\;.\n\\end{eqnarray}\nIn terms of the variable $Y$, the density $\\hat \\rho_n(Y)$ reads, \n\\begin{eqnarray}\\label{pseudo_periodic}\n\\hat \\rho_n(Y = \\lfloor Y \\rfloor + b) \\sim \\frac{\\alpha(\\alpha-1)\n Y^{\\alpha-2}}{2\\pi\\cosh{\\left[ \\frac{\\alpha(\\alpha-1)}{2}\n Y^{\\alpha-2}(1-2b) \\right]}} \\;. \n\\end{eqnarray}\nIn Fig. \\ref{fig2} c), one shows a sketch of $\\hat \\rho_n(Y)$ for\nlarge $Y < n$ given by Eq. (\\ref{pseudo_periodic}) : it is\nqualitatively very different from the case\n$\\alpha < 2$ (see Fig. \\ref{fig2} a)). Indeed, $\\hat \\rho_n(Y)$\nexhibits peaks centered around $k + \\frac{1}{2}$ for large integers\n$1 \\ll k < n$. The height of these peaks is given by $\\alpha(\\alpha-1)\nk^{\\alpha-2}\/(2 \\pi)$ whereas its width scales like $k^{2 - \\alpha}$. \n\nFrom $\\rho_n(x)$, one can now compute the mean number of real\nroots. As in the case $\\alpha < 2$ (see Eq. (\\ref{very_largeX}) and above), one\ncan show that the main contribution to $\\langle N_n \\rangle$ comes\nfrom the intervals $[-e^{\\frac{\\alpha}{2} n^{\\alpha-1}},-1]$ and \n$[1, e^{\\frac{\\alpha}{2} n^{\\alpha-1}}]$. One thus\nhas from Eq. (\\ref{pseudo_periodic})\n\\begin{eqnarray}\\label{condensation}\n\\langle N_n \\rangle &=& 2 \\int_0^\\infty \\rho_n(x) \\, dx \\sim 2 \\int_0^n \\hat\n\\rho_n(Y) \\, dY \\\\\n&\\sim&\n \\sum_{k \\gg 1}^n \\int_0^1 \\frac{\\alpha(\\alpha-1)\n k^{\\alpha-2}}{\\pi\\cosh{\\left[ \\frac{\\alpha(\\alpha-1)}{2}\n k^{\\alpha-2}(1-2b) \\right]}} \\,db \\sim \\sum_{k \\gg 1}^n\n \\int_{-\\frac{\\alpha(\\alpha-1)}{2}k^{\\alpha-2}}^{\\frac{\\alpha(\\alpha-1)}{2}k^{\\alpha-2}} \\frac{dz}{\\pi \\cosh{z}} \\,, \\nonumber\n\\end{eqnarray}\nand finally \n\\begin{eqnarray}\n\\langle N_n \\rangle \\sim n \\;,\n\\end{eqnarray}\nwhere we have used $\\int_{-\\infty}^\\infty dz\/\\cosh{z} = \\pi$. This\ncondensation of the roots on the real axis, characterized by the fact\nthat $\\langle N_n \\rangle \\sim n$ thus occurs via the\nformation of this quasi-periodic structure (see Fig. \\ref{fig2}\nc)). More precisely, this computation in Eq. (\\ref{condensation})\nshows that for large $k$, $2 \\int_k^{k+1} \\hat \\rho_n(Y) \\,dY \\sim 1$ which\nmeans, going back to the original variable $x$, that there is, on\naverage, one root in the interval $[-x_{k+1},-x_k] \\cup [x_k,x_{k+1}]$,\nwith $x_k = e^{\\frac{\\alpha}{2} k^{\\alpha-1}}$. \n\n\n\\section{The special case $\\alpha = 2$}\n\nIn view of the previous analysis, it is tempting to consider the fraction of real roots $\\Phi = \\lim_{n\n \\to \\infty} \\langle N_n \\rangle \/ n$ as an ``order paramater''. For\n $\\alpha < 2$, \n one has $\\Phi = 0$ whereas $\\Phi = 1$ for $\\alpha > 2$. One can\n however interpolate smoothly between these two limiting cases by\n considering the case $\\alpha = 2$ and introducing an additional real\nparameter $\\mu$ such that\n\\begin{equation}\\label{def_mu}\n\\langle a_k^2 \\rangle = e^{-\\mu k^2} \\;.\n\\end{equation}\nPerforming the same algebra as explained in the Appendix, one obtains the same formula as given in Eq. (\\ref{start_expr_rho}) with $u^*(x) = \\mu^{-1} \\log{x}$. The new variable is thus here $Y = \\mu^{-1} \\log{x}$ and, setting $Y = \\lfloor Y \\rfloor + b$ it is easy to see that the\ndensity $\\hat \\rho_n(Y)$ is given by for $1 \\ll Y < n$ \n\\begin{equation}\\label{start_expr_rho_mu}\n\\hspace*{-2cm}\\hat \\rho_n(Y) = \\frac{\\mu}{\\pi} \\left[ \\frac{\\sum_{m=-\\infty}^\\infty (m-b)^2 e^{-\\mu(m-b)^2} }{\\sum_{m=-\\infty}^\\infty e^{-\\mu(m-b)^2}} - \\left[ \n\\frac{\\sum_{m=-\\infty}^\\infty (m-b) e^{-\\mu(m-b)^2} }{\\sum_{m=-\\infty}^\\infty e^{-\\mu(m-b)^2}}\\right]^2 \\right]^{1\/2} \\;,\n\\end{equation}\nwhich is thus 1-periodic for all $\\mu$. In Fig. \\ref{fig2} c), one\nshows a sketch of $\\hat \\rho(Y)$ for $\\alpha = 2$ given by\nEq. (\\ref{start_expr_rho_mu}). For $\\mu \\to 0$, the\ndensity is almost constant and $\\hat \\rho_n(Y) \\sim\n\\pi^{-1}\\sqrt{\\mu\/2}$ and the modulation of the density increases\nwith $\\mu$. For large $\\mu$, the sum in Eq.~(\\ref{start_expr_rho_mu})\nis dominated by the terms corresponding to $m=0$ and $m=1$ and $\\hat\n\\rho_n(Y)$ is thus given by a formula similar to Eq. (\\ref{pseudo_periodic})\nsetting $\\alpha=2$ and replacing $Y^{\\alpha-2}$ by $\\mu$. For the\naverage number of real roots one has \n\\begin{eqnarray}\n\\langle N_n \\rangle \\propto \n\\cases{\n\\frac{\\sqrt{2\\mu}}{\\pi} n \\;,\\; \\mu \\ll 1 \\\\\nn \\;,\\; \\mu \\gg 1\\;,}\n\\end{eqnarray}\nwhich shows that this family of real random polynomials (\\ref{def_mu})\ninterpolate smoothly between the cases $\\alpha < 2$\n(\\ref{N_alleq2}) and $\\alpha > 2$ (\\ref{condensation}). \n\n\\begin{figure}\n\\includegraphics[angle=0,scale=0.8]{combined.ps}\n\\caption{{\\bf a)} : Sketch of $\\hat \\rho_n(Y)$ (in arbitrary units)\n given in Eq. (\\ref{asympt_largeX_alleq2}) as \n a function of $Y$ for $1 \\ll Y < n$ for $\\alpha < 2$. {\\bf b)} : Sketch of $\\hat \\rho_n(Y)$ (in arbitrary units)\n given in Eq. (\\ref{start_expr_rho_mu}) as \n a function of $Y$ for $1 \\ll Y < n$ for $\\alpha = 2$. {\\bf c)} : Sketch of $\\hat \\rho_n(Y)$ (in arbitrary units)\n given in Eq. (\\ref{pseudo_periodic}) as \n a function of $Y$ for $1 \\ll Y < n$ for $\\alpha > 2$. Here $k$\n denotes an integer with $1 \\ll k < n$.}\\label{fig2} \n\\end{figure}\n\n\n\\section{A qualitative argument for the transition at $\\alpha=2$}\n\nThis condensation of the roots on the real axis can be qualitatively\nunderstood if one considers the random polynomials (for $x >0$) $\\hat P_n(Y) =\nP_n(x)$ of the variable\n$Y$, which one writes as \n\\begin{eqnarray}\\label{P_hat}\n\\hat P_n(Y) = \\sum_{k=0}^n \\hat a_k w(k,Y) \\;, \\; w(k,Y) =\n\\exp{\\left[-\\frac{1}{2}(k^\\alpha - \\alpha k Y^{\\alpha-1})\\right]} \\;,\n\\end{eqnarray}\nand $\\hat a_k$ are i.i.d. Gaussian variables of unit variance. It is\neasy to see that the weights $w(k,Y)$, as a function of $k$, have a\nsingle maximum for $k = Y$ where the second derivative is proportional to\n$k^{\\alpha - 2}$. Thus for $\\alpha > 2$, the weights get more and\nmore peaked around this maximum for large $k$, whereas $\\hat a_k$ is typically of order ${\\cal O}(1)$. \nTherefore, given a large integer $m$, $\\hat P_n(m)$ is, for $\\alpha > 2$, \ndominated by a single term corresponding to $k=m$. Consequently, the sign\nof $\\hat P_n(m)$ is essentially the sign of $\\hat a_m$. This in turn implies that, if $\\hat\na_m$ and $\\hat a_{m+1}$ have an opposite \nsign, $P_n(x)$ has, with a probability close to $1$, a root in the interval\n$[e^{\\frac{\\alpha}{2}m^{\\alpha-1}},e^{\\frac{\\alpha}{2}(m+1)^{\\alpha-1}}]$.\nIn the case where $\\hat a_m$ and $\\hat a_{m+1}$ have the same sign, the same\nargument shows that $P_n(x)$ has, with a probability close to $1$, a root in the interval\n$[-e^{\\frac{\\alpha}{2}(m+1)^{\\alpha-1}},-e^{\\frac{\\alpha}{2}(m)^{\\alpha-1}}]$.\nOne thus recovers qualitatively the result we had found from the\ncomputation of $\\hat \\rho_n(Y)$ in Eq. (\\ref{condensation}) where we\nhave shown that $P_n(x)$ has, on average, one root in the interval\n$[-e^{\\frac{\\alpha}{2}(m+1)^{\\alpha-1}},-e^{\\frac{\\alpha}{2}(m)^{\\alpha-1}}]\n\\cup\n [e^{\\frac{\\alpha}{2}m^{\\alpha-1}},e^{\\frac{\\alpha}{2}(m+1)^{\\alpha-1}}]$. This shows finally that $P_n(x)$ has, on average, $\\langle N_n \\rangle \\propto n$real \nroots. \n\nWe also point out that our argument explains in a\nrather intuitive way the result obtained \nby Littlewood and Offord \\cite{littlewood} for the random polynomials\n$L_n(x)$ (\\ref{little}). For these specific polynomials, \ndefining $x_0 = 0$, $x_{m} = m^m m !$, they rigorously proved, using\nalgebraic (and rather cumbersome) methods, that $L_n(x)$ has a root either on $[x_m,x_{m+1}]$\nif $\\epsilon_m \\epsilon_{m+1} =-1$ or in $[-x_{m+1},-x_{m}]$ if\n$\\epsilon_{m} \\epsilon_{m+1} = 1$. Our argument gives some insight on their intriguing result and allows to understand it in a rather simple way. \n\n \n \n\n\\section{Conclusion}\n\nTo conclude we have introduced a new family of random polynomials\n(\\ref{def_poly}), indexed by a real $\\alpha$. For these random\npolynomials, we have computed the mean density of real roots\n$\\rho_n(x)$ from which we computed the mean number of real roots\n$\\langle N_n \\rangle$\nfor large $n$. We have shown that, while for $0 \\leq \\alpha < 1$,\n$\\langle N_n \\rangle \\sim (\\frac{2}{\\pi}) \\log{n}$, the behavior of\n$\\langle N_n \\rangle$ for $\\alpha > 1$ deviates \nsignificantly from the logarithmic behavior characteristic for \nKac's polynomials. For $1< \\alpha < 2$, we have shown that $\\langle\nN_n \\rangle\n\\propto n^{\\alpha\/2}$ whereas for $\\alpha > 2$, $\\langle N_n \\rangle\n\\sim n$. This \nfamily of real random polynomials thus displays an interesting\ncondensation phenomenon \nof their roots on the real axis, which is accompanied by an ordering\nof the roots in \na quasi periodic structure : this is depicted in Fig. \\ref{fig2}. \n\nOf course, the occurrence of this transition raises several interesting\nquestions like the behavior of the variance of the number of real\nroots for large $n$ as $\\alpha$ is varied. It would be also interesting to\ncompute the two-point correlation function of the \nreal roots, which is a rather natural tool to characterize this periodic\nstructure we have found. In view of this, we hope that this interesting\nphenomenon will stimulate further research on random polynomials. \n\n\n\\begin{appendix}\n\\section{A useful expression for the mean density $\\rho_n(x)$}\n\nIn this appendix, we derive the expression for the mean density $\\rho_n(x)$ as given in Eq. (\\ref{start_expr_rho}) \nstarting from Eq. (\\ref{ek_formula}). We first write $c_n(x) = \\langle P_n(x) P_n(x)\\rangle$ as\n\\begin{eqnarray}\\label{c_app1}\nc_n(x) = e^{-\\phi(u^*(x),x)} \\sum_{k=0}^n e^{-\\tilde \\phi(k,x)} \\;,\n\\end{eqnarray}\nwhere $u^*(x)$ is the location of the minimum of $\\phi(u,x)$ given in\nEq. (\\ref{def_ustar}) \n\\begin{equation}\\label{def_ustar_app}\nu^*(x) = \\left(\\frac{2}{\\alpha} \\log{x}\\right)^{\\frac{1}{\\alpha-1}} \\;,\n\\end{equation}\nand\n\\begin{eqnarray}\\label{phi_tilde}\n&& \\phi(u^*(x),x) = (1-\\alpha)u^*(x)^\\alpha \\\\\n&&\\tilde \\phi(k,x) = \\phi(k,x) - \\phi(u^*(x),x) = k^\\alpha - \\alpha k\n\t [u^*(x)]^{\\alpha-1} + (\\alpha-1) [u^*(x)]^\\alpha \\;. \\nonumber\n\\end{eqnarray}\nThe correlator $C_n(x,y) = c_n(\\sqrt{xy})$ is given by\nEq. (\\ref{c_app1}) together with Eq. (\\ref{phi_tilde}) where $x$\nis replaced by $\\sqrt{xy}$. All the dependence of $C_n(x,y)$ in\n$x,y$ is thus contained in $u^*(\\sqrt{xy})$ only. From its definition\nin Eq.~(\\ref{def_ustar_app}) one has immediately\n\\begin{equation}\n\\partial_x u^*(\\sqrt{xy}) = \\frac{1}{\\alpha(\\alpha-1)} \\frac{1}{x}\n\t[u^*(\\sqrt{xy})]^{2-\\alpha} \\;,\n\\end{equation}\nfrom which we obtain a set of useful relations\n\\begin{eqnarray}\\label{relations}\n&&\\partial_{x,y}^2 \\phi(u^*(\\sqrt{xy}),\\sqrt{xy}) = -\\frac{1}{\\alpha(\\alpha-1)} \\frac{1}{xy} [u^*(\\sqrt{xy})]^{2-\\alpha}\\\\\n&& \\partial_x \\tilde \\phi(k,\\sqrt{xy}) = \\frac{1}{x}\n(u^*(\\sqrt{xy})-k) \\nonumber \\\\\n&& \\partial_{x,y}^2 \\tilde \\phi(k,\\sqrt{xy}) =\n\\frac{1}{\\alpha(\\alpha-1)} \\frac{1}{xy} [u^*(\\sqrt{xy})]^{\\alpha-2}\n\\;. \\nonumber\n\\end{eqnarray}\nFor the computation of $\\rho_n(x)$ from Eq. (\\ref{ek_formula}), it is\nuseful to introduce the notation, for any function $g(k)$ \n\\begin{equation}\n\\langle g(k) \\rangle_Z = \\frac{\\sum_{k=0}^n g(k)\\exp{(-\\tilde\n \\phi(k,\\sqrt{xy}))} \n}{\\sum_{k=0}^n \\exp{(-\\tilde \\phi(k,\\sqrt{xy}))}} \\;.\n\\end{equation}\nFrom $C_n(x,y) = c_n(\\sqrt{xy})$ and\n$c_n(x)$ given in Eq.(\\ref{c_app1}) one obtains \n\\begin{eqnarray}\\label{last_eq_app}\n&&\\partial_x \\partial_y \\log{C_n(\\sqrt{xy})} = - \\partial_{x,y}^2\n\\phi(u^*(\\sqrt{xy}),\\sqrt{xy}) - \\langle \\partial_x \\tilde\n\\phi(k,\\sqrt{xy}) \\partial_y \\tilde \\phi(k,\\sqrt{xy}) \\rangle_Z\n\\nonumber \\\\\n&&-\n\\langle \\partial_{x}\\tilde \\phi(k,\\sqrt{xy}) \\rangle_Z \\langle\n\\partial_{x}\\tilde \\phi(k,\\sqrt{xy}) \\rangle_Z \n- \\langle \\partial^2_{x,y}\\tilde \\phi(k,\\sqrt{xy}) \\rangle_Z \\;.\n\\end{eqnarray}\nFrom the above relations in Eq. (\\ref{relations}), it is readily seen\nthat the first and the last term in Eq. (\\ref{last_eq_app}) cancel\neach other. Using the relation in Eq. (\\ref{ek_formula}), one finally obtains\nthe relation given in the text in Eq. (\\ref{start_expr_rho}).\n\n\n\\end{appendix}\n\n\n\\section*{References}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section[#1]{\\centering\\normalfont\\scshape #1}}\n\\newcommand{\\ssubsection}[1]{%\n \\subsection[#1]{\\raggedright\\normalfont\\itshape #1}}\n\\newcommand{\\mathfrak{p}}{\\mathfrak{p}}\n\\newcommand{\\mathfrak{q}}{\\mathfrak{q}}\n\\begin{document}\n\\maketitle\n\n\n\\begin{abstract}\nThis paper has the following main results. Let $S$ be a polynomial ring in $n$ variables, over an arbitrary field. Let $\\mathscr{M}$ be the family of all monomial ideals in $S$.\n\\begin{enumerate}[(i)]\n \\item We give an explicit characterization of all $M\\in \\mathscr{M}$, such that $\\pd(S\/M)=n$.\n \\item We give the total, graded, and multigraded Betti numbers of $S\/M$, in homological degree $n$, for all $M\\in \\mathscr{M}$.\n \\item Let $M\\in \\mathscr{M}$. If $\\pd(S\/M)=n$, then $\\sum\\limits_{i=0}^n \\betti_i(S\/M)\\geq 2^n$.\n \\item Let $M\\in \\mathscr{M}$. If $M$ is Artinian and $\\betti_n(S\/M)=1$, then $M$ is a complete intersection. \n \\end{enumerate}\n \\end{abstract}\n \n\n \\section{Introduction}\n Let $S=k[x_1,\\ldots,x_n]$ be a polynomial ring in $n$ variables, over a field $k$. The title of this paper makes reference to those monomial ideals $M$ in $S$, for which the quotient module $S\/M$ has projective dimension $n$, and the present work is entirely concerned with the study of such ideals.\n \n We begin to examine projective dimension $n$ in the context of squarefree monomial ideals. We show that the only squarefree monomial ideal $M$ for which the projective dimension of $S\/M$ equals $n$ is the maximal ideal $M=(x_1,\\ldots,x_n)$. This result turns out to be instrumental in the proof of a later theorem, where we characterize the class of all monomial ideals with large projective dimension. This characterization, in turn, is an avenue to three results that we discuss below.\n \n General consensus says that the problem of describing the Betti numbers of an arbitrary monomial ideal of $S$ is utopian. In homological degree $n$, however, such description is particularly simple. In fact, we give the total, graded, and multigraded Betti numbers of $S\/M$, in homological degree $n$, for every monomial ideal $M$ of $S$.\n \n Another theorem proven in this article states that when the quotient $S\/M$ has projective dimension $n$, the sum of its Betti numbers is at least $2^n$. This result, already known for Artinian monomial ideals [Ch, CE], is related to the Buchsbaum-Eisenbud, Horrocks conjecture, which has been investigated and generalized over the course of the years [CE], [PS, Conjectures 6.5, 6.6, and 6.7]. The proof of our theorem has strong combinatorial flavor.\n \n Finally, we show that when $M$ is Artinian and the $n^{th}$ Betti number of $S\/M$ is 1, $M$ must be of the form $M=(x_1^{\\alpha_1},\\ldots,x_n^{\\alpha_n})$, where the $\\alpha_i$ are\n positive integers. Combining this result with [Pe, Theorem 25.7] (a criterion for $S\/M$ to be Gorenstein), we obtain the following. If $\\betti_n(S\/M)=1$, then $S\/M$ is Cohen-Macaulay if and only if $S\/M$ is Gorenstein if and only if $M=(x_1^{\\alpha_1},\\ldots,x_n^{\\alpha_n})$, for some ${\\alpha_1},\\ldots,{\\alpha_n}\\geq 1$.\n \n The organization of the article is as follows. Section 2 is about background and notation. Sections 3 and 4 prepare the ground to characterize all monomial ideals with large projective dimension. \n This characterization is the content of section 5. Section 6 is the heart of this work; it is in this section that we prove the three theorems advertised above.\n \n \\section{Background and Notation\n Throughout this paper $k$ is an arbitrary field, and $S$ represents a polynomial ring over $k$, in a finite number variables. The letter $n$ is always used to denote the number of variables of $S$. The letter $M$ represents a monomial ideal\nin $S$. With minor modifications, the construction that we give below can be found in [Me].\n \n\\begin{construction}\nLet $M$ be generated by a set of monomials $\\{l_1,\\ldots,l_q\\}$. For every subset $\\{l_{i_1},\\ldots,l_{i_s}\\}$ of $\\{l_1,\\ldots,l_q\\}$, with $1\\leq i_1<\\ldots1$, perhaps favoring an episodic nuclear obscuration and blowout governed by radiation pressure. \n\nThe largest limitation for the $\\it Swift$\/BAT survey, which is relatively unbiased and complete for local AGNs, is that its shallow sensitivity misses luminous quasars in the distant universe. Here, we explore high-luminosity\/redshift samples of optical quasars (e.g., \\citealt{Sch10}), optical--IR red quasars with large color excess, where $E(B-V)\\lesssim1.5$ mag (e.g., \\citealt{Gli07}; \\citealt{Ban12}; \\citealt{Ros15}; \\citealt{Ham17}), dust-obscured galaxies (DOGs, \\citealt{Dey08}; Hot DOGs, \\citealt{Eis12}), and submillimeter galaxies (SMGs, \\citealt{Bla02}), where the latter two are likely subsets and distant analogs of local ultraluminous infrared galaxies (ULIRGs, $\\log L_{\\rm IR}>10^{12} L_{\\odot}$; \\citealt{San88}) and their higher-luminosity cousins (e.g., hyLIRGs, \\citealt{San96}; ELIRGs, \\citealt{Tsa15}). We also add the highest-obscuration Compton-thick AGNs, that is, AGNs with X-ray-obscuring column densities of $N_{\\rm H}\\gtrsim10^{24}\\, \\rm cm^{-2}$, observed with $\\it NuSTAR$ \\citep{Har13}.\n\n\\begingroup\n\\begin{deluxetable*}{ccccccc}\n\\tablecolumns{7}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{AGN samples}\n\\tablehead{\n\\colhead{Name} & \\colhead{Sample} & \\colhead{Selection} & \\colhead{Obscuration} & \\colhead{$z$} & \\colhead{$\\log L_{\\rm bol}\\,(\\ergs)$} & \\colhead{N}}\n\\startdata\nB15b; B16 & Compton-thick & Hard X-ray & $N_{\\rm H}$ & 0.001--0.051 & 42.5--45.8 & 16\\\\\nR17c & {\\it Swift}\/BAT & Hard X-ray & $N_{\\rm H}$, $E(B-V)_{\\rm nl}$ & 0.00--0.27 & 40.8--46.9 & 366\\\\\nY09; M17; V18b & Type 1 & Optical & $N_{\\rm H}$, $E(B-V)_{\\rm cont}$ & 0.15--4.26 & 44.8--48.7 & 174\\\\\nJ13 & Type 1 & Optical & $E(B-V)_{\\rm cont}$ & 0.14--4.13 & 45.7--48.2 & 14,531\\\\ \nL16; L17; G17& Red & Optical--IR & $N_{\\rm H}$, $E(B-V)_{\\rm cont}$ & 0.14--2.48 & 45.2--46.9 & 12\\\\ \nU12; K18 & Red & Optical--IR & $E(B-V)_{\\rm cont}$ & 0.29--0.96 & 45.8--47.1 & 23\\\\ \nG18 & Extremely red & Optical--IR & $N_{\\rm H}$ & 2.32 & 47.5 & 1\\\\ \nP19 & Extremely red & Optical--IR & $E(B-V)_{\\rm cont}$ & 2.24--2.95 & 46.7--47.8 & 28\\\\ \nL20 & Heavily reddened & Optical--IR & $N_{\\rm H}$ & 2.09--2.66 & 45.7--46.9 & 7\\\\ \nB12; B13; B15a; T19 & Heavily reddened & Optical--IR & $E(B-V)_{\\rm cont}$ & 1.46--2.66 & 46.0--48.6 & 51\\\\\nC16; T20 & DOG & Optical--IR & $N_{\\rm H}$ & 1.22--5.22 & 43.8--48.2 & 15\\\\\nS14; A16; R17a; V18a; Z18; A20 & Hot DOG & IR & $N_{\\rm H}$ & 1.01--4.60 & 46.2--48.1 & 9\\\\ \nA15 & Hot DOG & IR & $E(B-V)_{\\rm cont}$ & 0.29--4.59 & 45.4--48.9 & 129\\\\\nA08 & ULIRG, SMG & IR\/submm & $N_{\\rm H}$ & 0.04--2.05 & 44.6--45.6 & 3\\\\\n\\enddata\n\\tablecomments{AGN samples used in this work. We refer to each reference as having the objects with values or constraints on obscuration or accretion rate, while the original catalog paper is provided in the text (\\S2). Here, $N$ denotes the number of objects for each set of references having obscuration and accretion rate information used in \\S5, so they could be smaller than the number of sources from the references. The subscripts ``cont'' and ``nl'' under $E(B-V)$ values are derived using the continuum SED and narow-line ratios, respectively. The abbreviated references are \\citet{Bri15} (B15b); \\citet{Bri16} (B16); \\citet{Ric17c} (R17c); \\citet{You09} (Y09); \\citet{Mar17} (M17); \\citet{Vie18} (V18b); \\citet{Jun13} (J13); \\citet{Lam16,Lam17} (L16, L17); \\citet{Gli17a} (G17); \\citet{Urr12} (U12); \\citet{Kim18} (K18); \\citet{Gou18} (G18); \\citet{Per19} (P19); \\citet{Lan20} (L20); \\citet{Ban12,Ban13,Ban15} (B12, B13, B15a); \\citet{Tem19} (T19); \\citet{Cor16} (C16); \\citet{Tob20} (T20); \\citet{Ste14} (S14); \\citet{Ass16} (A16); \\citet{Ric17a} (R17a); \\citet{Vit18} (V18a); \\citet{Zap18} (Z18); \\citet{Ass20} (A20); \\citet{Ass15} (A15); \\citet{Ale08} (A08).} \n\\end{deluxetable*} \n\\endgroup \n\nThe latest studies of obscured quasars with large $E(B-V)$ values, through careful analysis to quantify and minimize the effect of obscuration, have reported near-Eddington to Eddington-limited accretion ($f_{\\rm Edd}\\sim$\\,0.1--1, e.g., \\citealt{Ale08}; \\citealt{Urr12}; \\citealt{Kim15}; \\citealt{Ass20}; \\citealt{Jun20}). Furthermore, \\citet{Gli17b} and \\citet{Lan20} find many obscured quasars with $f_{\\rm Edd, dust}>1$ at high $N_{\\rm H}$. These observations suggest that radiation pressure on dusty gas is effective, but is potentially less effective for obscured, luminous quasars since the length of time that luminous quasars are active is shorter than the length of time that less-luminous AGN are active (e.g., \\citealt{Hop05, Hop06}). Alternatively, luminous, obscured quasars are thought to be observed in a short phase in which they are blowing out the material through outflows stronger at higher luminosities (e.g., \\citealt{Lam17}; \\citealt{Per19}; \\citealt{Tem19}; \\citealt{Jun20}), perhaps requiring a different nuclear or galactic environment from less-luminous, obscured AGNs. Hence, there is growing interest in which AGN property drives radiation-pressure feedback, and in which temporal and spatial scales is it effective.\n\nIn this work, we attempt to constrain the $N_{\\rm H}$--$f_{\\rm Edd}$ and $E(B-V)$--$f_{\\rm Edd}$ planes for quasars from multiwavelength AGN samples (\\S2) and through a consistent method to estimate $N_{\\rm H}$, $E(B-V)$ (\\S3), and $f_{\\rm Edd}$ values (\\S4). We present (\\S5) and discuss (\\S6) the $N_{\\rm H}$--$f_{\\rm Edd}$ and $E(B-V)$--$f_{\\rm Edd}$ distributions for quasars in terms of various feedback mechanisms. Throughout, including the luminosities from the literature, we use a flat $\\Lambda$CDM cosmology with $H_{0}=\\mathrm{70\\,km\\,s^{-1}\\,Mpc^{-1}}$, $\\Omega_{m}=0.3$, and $\\Omega_{\\Lambda}=0.7$. \n\n\n\\section{The sample}\nProbing the distribution of $N_{\\rm H}$--$f_{\\rm Edd}$ and $E(B-V)$--$f_{\\rm Edd}$ values from a statistically complete AGN sample is complicated for several reasons. AGNs radiate across almost the entire electromagnetic spectrum, but show a wide range of spectral energy distributions (SEDs) due to physical processes governing the radiation, host galaxy contamination, and obscuration on various scales around the accreting BHs (e.g., \\citealt{Lan17}; \\citealt{Hic18}). Therefore, we found it beneficial to compile quasar samples selected at various wavelengths over a wide range of luminosity and redshift. Still, we chose to add only the data from the literature that meaningfully increase the sample size for a given wavelength selection.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=.95]{f2.eps}\n\\caption{Bolometric luminosity ($L_{\\rm bol}$) as a function of redshift (left: $z<0.3$, right: $z>0.3$) for AGNs having $f_{\\rm Edd}$ and measurements of either $N_{\\rm H}$ (filled symbols) or $E(B-V)$ (open symbols, plotted together if they have both $N_{\\rm H}$ and $E(B-V)_{\\rm cont}$). The samples plotted are ULIRGs\/SMGs (A08, red circles), Hot DOGs (S14; A15; A16; R17a; V18a; Z18; A20, red stars), optical--IR red AGN samples named red quasars (U12; L16, L17; G17; K18, yellow circles), extremely red quasars (P19, yellow squares), heavily reddened quasars (B12; B13; B15a; T19; L20, yellow stars), Compton-thick AGNs (B15b; B16, black stars), $\\it Swift$\/BAT AGNs (R17c, black circles), and optically selected SDSS type 1 quasars (J13, blue squares for $z<0.3$, density plot for $z>0.3$ (due to a large sample), matched with Y09 in blue circles; WISSH quasars from M17 and V18b as blue stars). The $L_{\\rm bol} \\ge 10^{45.7}\\ergs$ boundary is marked (dashed line), and a constant 14--195\\,keV flux of $10^{-11} \\ergs \\rm cm^{-2}$ with bolometric correction applied (\\S4), roughly denotes the detection limit of {\\it Swift}\/BAT X-ray data (dotted line, drawn up to $z=1$).}\n\\end{figure*}\n\nIn Figure 2 we plot the AGN samples from X-ray, optically blue, optical--IR red, and IR\/submillimeter-bright populations, also summarized in Table 1. At $z<0.3$, the $\\it Swift$\/BAT AGN in R17c (with $N_{\\rm H}$ from \\citealt{Ric17d}; $E(B-V)$ from \\citealt{Kos17}) has the advantage of minimal obscuration bias from the hard X-ray selection and covers a wide range of luminosities, reaching down to low-luminosity AGNs and up to quasar luminosities ($10^{41}\\lesssim L_{\\rm bol} \\lesssim 10^{47} \\ergs$). However, the $\\it Swift$\/BAT survey lacks the sensitivity to probe the distant or luminous AGN populations marked in Figure 2. We complemented the highest obscurations using Compton-thick ($N_{\\rm H} \\gtrsim 10^{24}\\, \\rm cm^{-2}$) AGNs observed with $\\it NuSTAR$ (B15b; B16), and the higher luminosities\/redshifts from optical Sloan Digital Sky Survey (SDSS) quasars (e.g., \\citealt{Sch10}; $N_{\\rm H}$ from Y09; $E(B-V)$ from J13), red quasars (L16, originally from \\citealt{Gli12}; K18, averaged between \\citealt{Gli07} and \\citealt{Urr09}), and quasars from ULIRGs (broad-line\\footnote{Throughout, we classify narrow-line AGNs to be type $\\ge1.8$ (weak broad H$\\alpha$ and H$\\beta$, [\\ion{O}{3}]\/H$\\beta>3$, \\citealt{Win92}), and broad-line AGNs to be type $\\le1.5$ (comparable broad-to-narrow H$\\beta$ and [\\ion{O}{3}]\/H$\\beta<3$). When the AGN types are not specified, we follow the visual classifications from the literature.} ULIRGs in A08, with $N_{\\rm H}$ from \\citealt{Sev01}). \n\nWe further include $z>0.3$ quasars to search for luminous quasars, adding type 1 quasars from the SDSS (i.e., WISSH quasars, $N_{\\rm H}$ from M17 and $E(B-V)$ from V18b), and a variety of quasars with red optical-to-infrared colors, that is, heavily reddened quasars (B12; B13, B15a; T19), red quasars (U12; G17; L17; K18), extremely red quasars (P19 with $N_{\\rm H}$ from G18 and $E(B-V)$ from \\citealt{Ham17}), and DOGs (C16; T20). Hot DOGs (S14; A15\\footnote{This sample is being updated by P. R. M. Eisenhardt et al. (in preparation), but we simply refer to the numbers from A15 at this time.}; A16; R17a; V18a; Z18; A20), and broad-line (AGN-like) SMGs (A08 with $N_{\\rm H}$ from \\citealt{Ale05}) were also included, adding part of some samples at $z<0.3$ that extend to $z>0.3$ (A08; Y09; J13; L16; R17c; K18). \n\nDuplication among the samples was found in Compton-thick AGNs (B15b; B16), heavily reddened quasars (B12; T19) and red quasars (U12; K18), where we used the most recent values, except for those between B15b\/B16 and R17c, where we kept both the $N_{\\rm H}$ and $f_{\\rm Edd}$ estimates as they were based on multiple X-ray observations. The samples based on follow-up studies of SDSS quasars were separated into those with $N_{\\rm H}$ (Y09; M17) and those with $E(B-V)$ (J13; V18b), where the $f_{\\rm Edd}$ values from signal-to-noise ratio (S\/N) $>$20 spectra in \\citet{She11} and \\citet{Par12} were added. We removed beamed sources (R17c, flagged by \\citealt{Kos17} using the blazar catalog from \\citealt{Mas15}) for reliable $N_{\\rm H}$ and $f_{\\rm Edd}$ values (but see also, e.g., \\citealt{Bae19}, for estimation of $f_{\\rm Edd}$ in radio-bright AGNs). We used line widths corrected for instrumental resolution in estimating $M_{\\rm BH}$ (\\S4).\n\n\n\\section{Gas and Dust Obscuration}\nWe compiled $N_{\\rm H}$ and $E(B-V)$ values, representing gas and dust obscuration, for the AGN samples. For $N_{\\rm H}$, we used the line-of-sight X-ray obscuration from sources with enough X-ray counts to model the spectra ($\\gtrsim$40--60, defined by the respective references). Exceptions are obviously large absorption ($N_{\\rm H}\\ge10^{24}\\, \\rm cm^{-2}$) constraints in S14, V18a, and A20, where the exposure times are longer than 20 ks but have a relatively smaller number of X-ray counts due to Compton-thick absorption. We add these values to our analysis. The choice of models to fit or estimate the X-ray obscuration varies in the literature: \\citet{Mur09} (S14; L16; G17; V18a; Z18), \\citet{Bri11} (B15b; A16; B16; C16; A20), hardness-ratio-based $N_{\\rm H}$ conversion (L17), (absorbed) power-law fit (Y09; C16; M17; G18; L20), and a combination of models (A08; R17c; T20). Still, when the $N_{\\rm H}$ values are compared between various models (e.g., B15b; B16; G17; Z18; L20), they are mostly consistent within the uncertainties (but see also B15b and \\citealt{Liu15}, for the limitations of the models at Compton-thick column densities).\n\nFor $E(B-V)$, we used the UV\/optical--IR continuum SED-based $E(B-V)_{\\rm cont}$\\footnote{Throughout, $E(B-V)_{\\rm cont\/bl\/nl}$ are those derived from the continuum SED and broad\/narrow-line ratios, respectively, and the $E(B-V)_{\\rm nl}$ values are only mentioned as lower limits to $E(B-V)_{\\rm cont}$. We used the Milky Way extinction curve with total-to-selective extinction of 3.1 when transforming extinction to $E(B-V)$.} from the literature. Lower limits in $E(B-V)$ were given to the P19 data from \\citet{Ham17} because of likely underestimation using a narrow range of wavelengths to determine $E(B-V)$. For optical quasars, we determined the rest-frame $>0.3\\mu$m power-law continuum slope ${\\alpha}$ following $F_{\\nu} \\propto \\nu^{\\alpha}$, fit to the photometric SED. We assumed an intrinsic slope of ${\\alpha}=0.1\\pm0.2$ from the most blue (hot dust-poor, $\\sim3\\sigma$ outliers) quasars in J13, consistent with accretion disk models and polarized observations of quasar SEDs ($\\alpha \\approx 1\/3$, e.g., \\citealt{Sha73}; \\citealt{Kis08}). We limited the sample to quasars with at least three SDSS optical or UKIDSS near-IR \\citep{Law07} photometric detections at rest-frame $0.3-1\\mu$m and rest-frame near-IR detections at up to at least 2.3$\\mu$m to decompose the SED into the power-law continuum and dust emission (see J13 for details). We converted $\\alpha$ into $E(B-V)$ by reddening the intrinsic slope using a Milky Way extinction curve at $0.3-1\\mu$m to match the observed value of $\\alpha$, while fixing $E(B-V)=0$ when $\\alpha>0.1$. We checked if the $E(B-V)$ estimates from J13 are consistent with the literature by comparing the values cross-matched with 17 sources in V18b and 277 sources in C. Carroll et al. (submitted) at $E(B-V)<0.5$ mag. The J13 $E(B-V)$ values have a median offset and scatter of $0.05\\pm0.05$ and $0.04\\pm0.06$ mag, respectively, consistent within the uncertainties. We adopt the J13 values for the cross-matched sources (Y09; M17; V18b).\n\nThe $E(B-V)_{\\rm cont}$ values can suffer from host galaxy contamination in the rest-frame optical\/near-IR. We limited the samples with the SEDs decomposed into an AGN and a host galaxy (U12; A15; L17) to $L_{\\rm bol} \\ge 10^{45.2} \\ergs$ for the decomposition to contain a sufficient AGN contribution, and the samples without an SED decomposition (the remaining samples with $E(B-V)_{\\rm cont}$ in Table 1) to $L_{\\rm bol} \\ge 10^{45.7} \\ergs$, to minimize host galaxy contamination. Above the luminosity limits, the average host contamination at 5100\\AA\\ drops below 50\\% and 10\\%, respectively, for type 1 quasars \\citep{She11} and is consistent with the growing AGN contribution to the observed SEDs for red quasars at higher $L_{\\rm bol}$ (L17). The $L_{\\rm bol} \\ge 10^{45.7} \\ergs$ limit corresponds to $L_{\\rm bol} =10^{12.1} L_{\\odot}$, selecting ULIRG luminosities for IR-bright AGNs (e.g., \\citealt{Fan16}; \\citealt{Tob17}). The majority of the hard X-ray selected AGNs from R17c are less luminous than the $L_{\\rm bol}$ limits for $E(B-V)_{\\rm cont}$, but instead have robust measurements of $N_{\\rm H}$ from their hard X-ray spectra.\n\nThe nuclear dust-to-gas ratio traced by the $\\log E(B-V)_{\\rm cont\/bl}\/N_{\\rm H}$ (mag cm$^{2}$) values in Figure 3 is constant if the gas and dust obscuration are proportional (e.g., \\citealt{Usm14}). The values are overall smaller than the Galactic value ($-$21.8, e.g., \\citealt{Boh78}),\\footnote{One of the reasons for the offset may be that the majority of the AGNs have an excess of dust-free gas within the sublimation radius (e.g., \\citealt{Ris02, Ris07}; \\citealt{Mai10}; \\citealt{Bur16}; \\citealt{Ich19}), but we focus here on the overall value when including the more luminous AGNs.} with reported average values ranging between $-$22.8 \\citep{Mai01} and $-$22.3 (L20). L20 find relatively higher $E(B-V)\/N_{\\rm H}$ values for a sample of heavily reddened broad-line quasars at high luminosity, but there are similarly luminous quasars with relatively smaller $E(B-V)\/N_{\\rm H}$ values (i.e., the Hot DOGs or some optical--IR red quasars in Figure 3). Apart from the type 1 AGNs where the large scatter in $E(B-V)\/N_{\\rm H}$ could in part arise from the uncertainty constraining the lowest values in either quantity, we find the mean and scatter of $\\log E(B-V)\/N_{\\rm H} \\,(\\rm mag\\,\\,cm^{2})=-22.77 \\pm 0.51$ (observed) or $\\pm 0.41$\\footnote{We refer to the intrinsic scatter of the quantity $x=\\log E(B-V)\/N_{\\rm H}$, $\\sigma_{\\rm int}$, as the observed scatter with measurement error $\\Delta x$ subtracted in quadrature, that is, $\\sigma_{\\rm int}^{2}=\\Sigma_{i=1}^{n}\\{(x_{i}+22.77)^{2}-\\Delta x_{i}^{2}\\}\/(n-1)$. The errors on $E(B-V)$ values are missing for the L16 and G17 samples, but the intrinsic scatter of $\\log E(B-V)\/N_{\\rm H}$ decreases by only 0.01 dex if we assign the mean error of $\\Delta E(B-V)=0.12$ mag from the L17 sample used here.} (intrinsic) from 31 obscured AGNs (type 2 AGNs, optical--IR red quasars, and Hot DOGs) without upper\/lower limits in Figure 3, spanning absorption-corrected $L_{\\rm 2-10 keV}=10^{42.4-45.6}\\ergs$. The ratios are close to the \\citet{Mai01} value, but are highly scattered for any combination of AGN type over the luminosity probed, complicating a simple correspondence between dust and gas. We thus refer to both $E(B-V)$ and $N_{\\rm H}$ when selecting AGNs with dusty gas, using a conversion of $\\log E(B-V)\/N_{\\rm H}=-$22.8.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.95]{f3.eps}\n\\caption{Log ratio between $E(B-V)_{\\rm cont\/bl}$ and $N_{\\rm H}$, plotted against intrinsic 2--10 keV luminosity. We plot only the data with $N_{\\rm H}$ uncertainty values based on sufficient X-ray counts. The data come from \\citet{Mai01} (marked M01, blue and red squares for type 1--1.5 and 1.8--2 classifications, respectively), red quasars (L16; G17; L17, yellow circles), heavily reddened quasars (L20, yellow stars), and Hot DOGs (S14; A16; R17a; Z18; A20, red stars). We show Galactic, L20, and \\citet{Mai01} $\\log E(B-V)\/N_{\\rm H}$ values and the range of their applicable luminosities for obscured AGNs (dashed line), and likewise from this work (solid line).}\n\\end{figure}\n\nThe $N_{\\rm H}$ and $E(B-V)$ values are thought to be nuclear obscuration close to the AGN center, but as the AGN geometry consists of an extended, kpc order narrow line region outside the central dusty structure (e.g., \\citealt{Kan18}; \\citealt{Min19}), we expect the narrow-line-based $E(B-V)_{\\rm nl}$ values to be smaller than the $E(B-V)_{\\rm cont\/bl}$ values measured closer to the nucleus (e.g., \\citealt{Zak03}; \\citealt{Zak05}; \\citealt{Gre14}; \\citealt{Jun20}). For the R17c sample providing narrow Balmer decrements, we find that the $E(B-V)_{\\rm nl}\/N_{\\rm H}$ values for obscured (type 1.8--2.0) AGNs are about an order of magnitude smaller than the $E(B-V)_{\\rm cont}\/N_{\\rm H}$ values in Figure 3, although showing an even larger scatter. This demonstrates that $E(B-V)_{\\rm nl}$ is simply much lower than $E(B-V)_{\\rm cont}$. Furthermore, the dust-to-gas ratios may decrease when using the global $N_{\\rm H}$, as it is larger than the nuclear line-of-sight $N_{\\rm H}$, such as for red quasars in L16, implying extended gas. The extended obscuration in obscured quasars will be considered to assess the effect of radiation pressure (\\S6.2), but for a better comparison of nuclear dust and gas obscuration, we remove $E(B-V)_{\\rm nl}$ values from further analysis, and we use $E(B-V)_{\\rm cont}$ as the fiducial estimate of $E(B-V)$ hereafter.\n\n\n\\section{Eddington Ratio}\nEstimating $f_{\\rm Edd}$ throughout the samples relies on several bolometric correction methods and black hole scaling relations. For the bolometric correction, we primarily relied on the hard X-ray (2--10\\,keV) luminosity to minimize the absorption correction for the X-ray samples in Table 1. The 2--10 keV intrinsic luminosities are based on a simple conversion of the 14--195 keV luminosities from R17c using a typical X-ray spectral slope; namely, $L_{\\rm 2-10 keV} = 2.67 L_{\\rm 14-195 keV}$ \\citep{Rig09}. The absorption-corrected X-ray-to-bolometric correction depends on $L_{\\rm bol}$ or $f_{\\rm Edd}$ (e.g., \\citealt{Marc04}; \\citealt{Vas07}; \\citealt{Lus12}). We used the \\citet{Marc04} bolometric correction as a function of $L_{\\rm bol}$, as the dynamic range of $L_{\\rm bol}$ ($\\sim$3--4 dex) is wider than that of $f_{\\rm Edd}$ ($\\sim$2 dex). When the X-ray luminosity was absent, we adopted the monochromatic bolometric correction from IR or extinction-corrected UV\/optical continuum or line luminosities, which are relatively insensitive to $L_{\\rm bol}$ (e.g., \\citealt{Ric06}; \\citealt{Lus12}). We used the corrections from $L_{\\rm 1350}$, $L_{\\rm 3000}$, and $L_{\\rm 5100}$\\footnote{Throughout, subscripts of $L$ indicate monochromatic continuum luminosity at that wavelength, measured in units of \\AA.} (3.81, 5.15, and 9.26, respectively, \\citealt{She11}) for optical quasars (J13; V18b) and obscured AGNs with extinction-corrected continuum luminosities (B12; B13; A15; B15a; objects in L17 without $N_{\\rm H}$; T19; A20), $L_{\\rm P\\beta}$ ($\\log L_{\\rm bol}\/10^{44} \\ergs = 1.29 +0.969 \\log L_{\\rm P\\beta}\/10^{42} \\ergs$, \\citealt{Kim15}) for K18, $L_{\\rm 3.4\\mu m}$ (8, \\citealt{Ham17}) for P19, $L_{\\rm 15\\mu m}$ (9, \\citealt{Ric06}) for U12, with each correction having systematic uncertainties of a few tens of percent up to a factor of few (e.g., \\citealt{Hec04}; \\citealt{Ric06}; \\citealt{Lus12}).\n\nWe estimated $M_{\\rm BH}$ mostly through stellar absorption or broad emission lines, using a mutually consistent methodology. The mass constant of single-epoch estimators for AGNs ($f$-factor), is determined assuming that the reverberation mapped AGNs lie on the $M_{\\rm BH}$--$\\sigma_{*}$ relation for inactive galaxies.\nWe thus use the same $M_{\\rm BH}$--$\\sigma_{*}$ relation (e.g., \\citealt{Woo15}), \n\\begin{eqnarray}\\begin{aligned}\n\\log &\\Big(\\frac{M_{\\rm BH}}{M_{\\odot}}\\Big)=(8.34\\pm0.05)\\\\\n&+(5.04\\pm0.28)\\log \\Big(\\frac{\\sigma_{*}}{200\\kms}\\Big),\n\\end{aligned}\\end{eqnarray}\nto estimate $\\sigma_{*}$-based $M_{\\rm BH}$ values for narrow-line AGNs where the host absorption lines are better seen, and to derive the $f$-factor in the broad FWHM-based single-epoch $M_{\\rm BH}$ estimators for broad-line AGNs where AGN emission dominates over the host galaxy. The $M_{\\rm BH}$($\\sigma_{*}$) estimates based on other $M_{\\rm BH}$--$\\sigma_{*}$ relations with a shallower slope, e.g., \\citet{Kor13}, are systematicaly offset to Equation (2) by 0.35 and $-$0.05 dex at $\\sigma_{*}=100$ and 400\\kms, respectively.\n\n\\begin{deluxetable*}{ccccc}\n\\tablecolumns{5}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{AGN $M_{\\rm BH}$ estimators}\n\\tablehead{\n\\colhead{Type} & \\colhead{$a$} & \\colhead{$b$} & \\colhead{$c$} & \\colhead{$d$}}\n\\startdata\n$M_{\\rm BH}$($L_{1350}$, FWHM$_{\\rm C\\,IV}$) & 6.99$\\pm$0.16 & 0.547$\\pm$0.037 & 2.11$\\pm$0.11 & 0\\\\\n$M_{\\rm BH}$($L_{1350}$, FWHM$_{\\rm C\\,IV}$, $\\Delta \\rm v_{C\\,IV}$) & 6.62$\\pm$0.16 & 0.547$\\pm$0.037 & 2.11$\\pm$0.11 & 0.335$\\pm$0.022\\\\\n$M_{\\rm BH}$($L_{3000}$, FWHM$_{\\rm Mg\\,II}$) & 6.57$\\pm$0.13 & 0.548$\\pm$0.035 & 2.45$\\pm$0.06 & 0\\\\\n$M_{\\rm BH}$($L_{5100}$, FWHM$_{\\rm H\\beta}$) & 6.88$\\pm$0.12 & 0.533$\\pm$0.034 & 2 & 0\\\\ \n$M_{\\rm BH}$($L_{5100}$, FWHM$_{\\rm H\\alpha}$) & 6.99$\\pm$0.12 & 0.533$\\pm$0.034 & 2.12$\\pm$0.03 & 0\\\\\n$M_{\\rm BH}$($L_{\\rm P\\beta}$, FWHM$_{\\rm P\\beta}$) & 7.24$\\pm$0.16 & 0.45$\\pm$0.03 & 1.69$\\pm$0.16 & 0\\\\ \n$M_{\\rm BH}$($L_{\\rm P\\alpha}$, FWHM$_{\\rm P\\alpha}$) & 7.20$\\pm$0.16 & 0.43$\\pm$0.03 & 1.92$\\pm$0.18 & 0\\\\\n\\enddata\n\\tablecomments{$L$ is the continuum or broad-line luminosity, FWHM is the full width at half maximum of the best-fit broad-line model, and $\\Delta \\rm v_{CIV}$ is the broad \\ion{C}{4} line offset to the systemic redshift (\\citealt{She11} in J13, negative for blueshifts). $a$, $b$, $c$, $d$ are the coefficients in Equation (3).} \n\\end{deluxetable*} \n\nThe single-epoch $M_{\\rm BH} (L, \\rm FWHM)$ estimators were empirically calibrated between H$\\beta$ and H$\\alpha$, \\ion{Mg}{2}, or \\ion{C}{4} (\\citealt{Jun15} using the \\citealt{Ben13} $R_{\\rm BLR}$--$L$ relation) with \\ion{C}{4} blueshift correction when broad-line shifts were available (\\citealt{Jun17}), or were calibrated between hydrogen Balmer and Paschen series (\\citealt{Kim10}\\footnote{Using our adopted cosmology, we find that the $R_{\\rm BLR}$ values from \\citet{Ben13} are higher than from \\citet{Ben09} by 0.00--0.03 dex for the luminosity range used to derive Paschen line $M_{\\rm BH}$ values ($L_{5100}=10^{43.5-46}\\ergs$, A08; K18). K18 also note that using a single Gaussian to fit the broad Paschen lines will underestimate the $M_{\\rm BH}$ values by 0.06--0.07 dex, but these amounts are negligible compared to the significance of the results (\\S5).}, using the \\citealt{Ben09} $R_{\\rm BLR}$--$L$ relation), over a wide range of redshift and luminosity. This approach reduces the systematic offset from the choice of emission line by up to an order of magnitude\\footnote{We note that a nonlinear relation between $\\sigma$ and FWHM values could further result in positive\/negative biases in the FWHM-based $M_{\\rm BH}$ estimate at notably high and low FWHM values (e.g., \\citealt{Pet04}; \\citealt{Col06}), as well as whether to construct the UV or IR mass estimators to match the $M_{\\rm BH}$ values to the Balmer line based, or to match the UV or IR broad-line widths and the luminosities to the optical values separately. Our choice of $M_{\\rm BH}$ estimators has its own merits and limitations, and we test the systematic uncertainty of $M_{\\rm BH}$ in \\S5.} at extreme $M_{\\rm BH}$ values \\citep{Jun15}, or at extreme \\ion{C}{4} blueshifts \\citep{Jun17}. The estimators were updated using a common $f$-factor and uncertainty of $1.12 \\pm 0.31$ for the FWHM-based $M_{\\rm BH}$ \\citep{Woo15}, as shown below:\n\\begin{eqnarray}\\begin{aligned}\n\\log \\Big(&\\frac{M_{\\rm BH}}{M_{\\odot}}\\Big)=a+\\log\\Big(\\frac{f}{1.12}\\Big)+b\\,\\log\\Big(\\frac{L}{\\rm 10^{44} \\ergs}\\Big)\\\\\n&+c\\,\\log\\Big(\\frac{\\mathrm{FWHM}}{\\rm10^{3}\\kms}\\Big)+d\\,\\log\\Big(\\frac{\\Delta \\mathrm{v_{C\\,IV}}}{\\rm10^{3}\\kms}\\Big).\n\\end{aligned}\\end{eqnarray}\nThe set of $(a, b, c, d)$ values for the combination of $M_{\\rm BH}$($L$, FWHM, $\\Delta \\rm v$) are given in Table 2. For broad-line AGNs with X-ray observations and single-epoch UV\/optical $M_{\\rm BH}$ estimates, we converted the X-ray-based $L_{\\rm bol}$ into $L_{1350}$, $L_{3000}$, $L_{5100}$ using the aforementioned bolometric corrections, to minimize host galaxy contamination in 1350--5100\\AA. We removed sources with Balmer line widths similar to [\\ion{O}{3}] (A08) to distinguish broad lines from broadening by ionized gas outflows. We also limited the FWHM values to $\\le$10,000\\kms\\ where values otherwise (e.g., 4\\%\\ of the J13 sample) are potentially affected by rotating accretion disks and show double-peaked lines (e.g., \\citealt{Che89}; \\citealt{Era94}; Table 4 in \\citealt{Jun17}). Meanwhile, R17c removed single-epoch $M_{\\rm BH}$ estimates for $N_{\\rm H}\\ge10^{22}\\, \\rm cm^{-2}$ AGNs as the emission line profiles could be modified by obscuration or are dominated by the narrow component \\citep{Kos17}. However, as we already removed type $\\ge$1.8 sources when estimating $M_{\\rm BH}$ for broad-line AGNs, we keep the $N_{\\rm H}\\ge10^{22}\\, \\rm cm^{-2}$ sources. These obscured type $\\le$1.5 AGNs with $M_{\\rm BH}$(FWHM) in R17c do not significantly change the distribution of $N_{\\rm H}$--$f_{\\rm Edd}$ with respect to using $M_{\\rm BH}$($\\sigma_{*}$) values. This hints that obscuration does not significantly bias the single-epoch $M_{\\rm BH}$ estimates for broad-line AGNs, also consistent with the independence of broad \\ion{C}{4}-to-H$\\beta$ line width ratios with respect to the continuum slope for type 1 quasars (e.g., \\citealt{Jun17}). We thus carefully selected only the type $\\le$1.5 sources when using rest-frame UV--optical spectra to estimate $M_{\\rm BH}$(FWHM) for AGNs.\n\nAmong single-epoch $M_{\\rm BH}$ estimates with multiple broad-line detections, we adopted the estimators in the order of decreasing rest wavelength, while direct dynamical (B15b; B16; R17c) or reverberation-mapped (\\citealt{Ben15} in R17c) $M_{\\rm BH}$ values were adopted over other methods. Hot DOGs, which are heavily obscured AGNs typically showing strong, narrow lines, often display signatures of narrow-line outflows instead of ordinary broad emission lines (e.g., \\citealt{Wu18}; \\citealt{Jun20}). Unless the sources are thought to show scattered or leaked light from the broad-line region (A16; A20), we utilized the SED fit from A15 when deriving the $M_{\\rm BH}$ constraints. Applying their maximal stellar mass ($M_{*}$) estimates from the SED fit, we gave upper limits to the $M_{\\rm BH}$ values using the $M_{\\rm BH}$--$M_{*}$ relation. The $M_{\\rm BH}\/M_{*}$ values are thought to evolve less with redshift ($\\propto (1+z)^{\\gamma}$, $\\gamma\\lesssim1$) than $M_{\\rm BH}\/M_{\\rm bulge}$ (e.g., \\citealt{Ben11}; \\citealt{Din20}; \\citealt{Suh20}). We adopt $M_{\\rm BH}\/M_{*}\\sim0.003$ from the $z\\sim$1--2 AGNs in \\citet{Din20} and \\citet{Suh20}. The same relation was used to estimate $M_{\\rm BH}$ for the DOGs in C16 and T20. \n\nAlthough this analysis attempted to consistently estimate $M_{\\rm BH}$ for the various samples, systematic uncertainties of a factor of several are expected from the intrinsic scatter in the BH--host mass scaling relations (e.g., \\citealt{Kor13}) and the $R_{\\rm BLR}$--$L$ relation (e.g., \\citealt{Ben13}; \\citealt{Du14}). Overall, the compiled $L_{\\rm bol}$ and $M_{\\rm BH}$ estimates each have systematic uncertainties of up to a factor of several or more, and although the AGN $M_{\\rm BH}$ estimators include the $\\sim L^{0.5}$ dependence, reducing uncertainty from the bolometric correction going into $f_{\\rm Edd}\\propto L_{\\rm bol}\/M_{\\rm BH}$, we still expect systematic uncertainties of a factor of several in $f_{\\rm Edd}$. We thus will interpret only the group behavior of each AGN sample within the uncertainties in $f_{\\rm Edd}$. \n\n\n\\section{Results}\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=0.95]{f4.eps}\n\\caption{The $N_{\\rm H}$--$f_{\\rm Edd}$ plane showing AGNs selected at different wavelengths. Horizontal lines separate obscured\/unobscured AGNs at $N_{\\rm H}=10^{22}\\, \\rm cm^{-2}$, and the effective Eddington ratio curves are plotted as solid\/dashed lines with respect to $N_{\\rm H}=10^{22}\\, \\rm cm^{-2}$. The symbols and color format follow that of Figure 2, except that the symbols are now filled when $L_{\\rm bol} \\ge 10^{45.7}\\ergs$ and open otherwise. Data outside the plotted region are shown along the boundary.}\n\\end{figure*}\nIn Figure 4 we plot the distributions of $N_{\\rm H}$--$f_{\\rm Edd}$ and $E(B-V)$--$f_{\\rm Edd}$ for the collection of AGN samples. It is clear that the forbidden region for dusty gas (Figure 1), previously less occupied by the AGNs from R17c, is well populated with IR\/submillimeter-selected and optical--IR red quasars, with a minor fraction of type 1 quasars. This is seen in both the $N_{\\rm H}$--$f_{\\rm Edd}$ and the $E(B-V)$--$f_{\\rm Edd}$ diagrams. We investigate this further in Figure 5 where we show the fraction of sources in the forbidden zone (i.e., $N_{\\rm H} \\ge 10^{22}\\, \\rm cm^{-2}$, $f_{\\rm Edd, dust}>1$) per bolometric luminosity bin as a function of $L_{\\rm bol}$. In the following, this fraction is referred to as $\\varphi$. It is clear that $\\varphi$ is minimal among the X-ray-selected AGNs with $\\varphi_{N_{\\rm H}}\\lesssim$10\\%\\ at $L_{\\rm bol} \\sim 10^{42-47}\\ergs$. Similarly, optically selected quasars have $\\varphi_{N_{\\rm H}}$ and $\\varphi_{E(B-V)}\\lesssim$20\\%\\ at $L_{\\rm bol} \\sim 10^{44-48}\\ergs$, with some uncertainty for $\\varphi_{N_{\\rm H}}$ at $L_{\\rm bol} \\sim 10^{47-48}\\ergs$. In contrast, the optical--IR red and IR\/submillimeter-bright quasars (hereafter referred to together as luminous, obscured quasars) commonly lie mostly in the forbidden region over a wide range of $N_{\\rm H}$ and $E(B-V)$ values, and we combined their statistics.\\footnote{A potential caveat is the difference in the dust-to-gas ratio observed between optical--IR red and IR\/submillimeter-bright quasars, which may bias the $\\varphi$ value between the populations. The average dust-to-gas ratios for each population from \\S3, are $\\langle\\log E(B-V)\/N_{\\rm H}\\rangle=-22.33$ and $-22.88$, respectively. Using the separate ratios, however, $\\varphi_{E(B-V)}$ (Figure 5 right) still remains consistent between the two populations.} The luminous, obscured quasars show $\\varphi_{N_{\\rm H}}$ and $\\varphi_{E(B-V)}\\gtrsim$\\,60\\%\\ at $L_{\\rm bol} \\sim 10^{46-48}\\ergs$, significantly higher than the less-luminous X-ray AGNs at comparable obscuration, or the comparably luminous but less obscured optical quasars. These findings confirm earlier studies by \\citet{Gli17b} and L20 on optical--IR red quasars, with our results applicable to general luminous, obscured quasars.\n\nSystematic uncertainties of a factor of several in $f_{\\rm Edd}$ (\\S4) may change the fraction of the samples in the forbidden region. We test this by giving a $\\pm$0.5 dex offset to the $f_{\\rm Edd, dust} (N_{\\rm H})=1$ curve and recalculating $\\varphi$. The $\\varphi$ values are nearly unchanged for the X-ray AGNs and optical quasars, whereas for the luminous, obscured quasars, $\\varphi$ may drop down to 40\\%--50\\% at $L_{\\rm bol} \\sim 10^{46-48}\\ergs$ if the observed $f_{\\rm Edd}$ values are overestimated by 0.5 dex. Still, the $\\varphi$ values for the luminous, obscured quasars are several times the X-ray AGNs or optical quasars at a given luminosity, and the main trend in Figure 5 remains unchanged. Modifications to the $f_{\\rm Edd, dust}$ curve may also occur when considering the effect of dust-to-gas ratios closer to the Milky Way value than the value adopted in this work, or radiation trapping. The enhanced absorption of the incident radiation by dust or trapping of reprocessed radiation lowers the $f_{\\rm Edd, dust}$ curve, at $N_{\\rm H} \\ge 10^{22}\\, \\rm cm^{-2}$ \\citep{Ish18}. Still, we note that both effects simply increase $\\varphi$ for luminous, obscured quasars, reinforcing our findings in Figures 4 and 5.\n\nThe $f_{\\rm Edd, dust}$ values can be further shifted by nuclear stars.\nAdopting the sphere of influence from the BH, we have\n\\begin{eqnarray}\\begin{aligned}\nr_{\\rm BH}=GM_{\\rm BH}&\/\\sigma_{*}^{2}\\\\\n=107\\,{\\rm pc}\\,&\\Big(\\frac{M_{\\rm BH}}{10^{9}\\,M_{\\odot}}\\Big)\\Big(\\frac{\\sigma_{*}}{200\\,\\kms}\\Big)^{-2},\n\\end{aligned}\\end{eqnarray} \nand the enclosed mass $M(10^{46} \\ergs$ quasars in this work are similar ($\\sim10^{9}M_{\\odot}$) for both the obscured and unobscured populations, the timescale of the luminous, obscured quasar phase with $f_{\\rm Edd, dust}>1$ is presumed to be similar to that of the $f_{\\rm Edd, dust}<1$ quasars. In contrast, the nearly complete absence of lower-luminosity AGNs with $f_{\\rm Edd, dust}>1$ (\\S5) suggests a much shorter obscured phase for lower-luminosity AGNs. This appears as a challenge for the radiation-pressure feedback in regulating the nuclear obscuration for luminous quasars, and we next consider possible evolutionary scenarios to achieve a coherent picture of dust obscuration in luminous quasars. \n\n\\subsection{Active timescale}\nFirst, the AGN timescale (hereafter $t_{\\rm AGN}$) is thought to be shorter for more luminous quasars, and this may explain higher $\\varphi$ values for luminous quasars. The nearby AGN fraction is measured to be tens of percent of the galaxy lifetime (e.g., \\citealt{Ho97}; \\citealt{Kau03}), with a corresponding $t_{\\rm AGN}$ of $\\sim10^{9}$\\,yr assuming typical galaxy lifetimes of $\\sim10^{10}$\\,yr. More luminous quasars are more rare, with expected $t_{\\rm QSO}\\sim10^{7-8}$\\,yr (e.g., \\citealt{Mart04}; \\citealt{Hop05}; \\citealt{Hop09}). To explain $\\varphi\\lesssim$1--10\\% for $L_{\\rm bol} \\lesssim 10^{44}\\ergs$ AGNs in Figure 5, we constrain the timescale for radiation pressure to clear the nuclear obscuration (hereafter the radiation feedback timescale, $t_{\\rm rad}=t_{\\rm AGN}\\,\\varphi$) to be $t_{\\rm rad}\\lesssim10^{9}\\,(0.01-0.1)\\sim10^{7-8}$\\,yr. Assuming that luminous, obscured quasars will evolve into comparably luminous unobscured quasars through radiation-pressure feedback, so as to explain the comparable number density between the populations, $t_{\\rm rad}$ for luminous ($L_{\\rm bol} \\gtrsim 10^{46}\\ergs$) quasars with $\\varphi\\gtrsim$\\,60\\%\\ would be $t_{\\rm rad}\\sim0.5t_{\\rm QSO}\\,\\varphi \\sim0.5(10^{7-8})(0.6-1)=(3-5)\\times10^{6-7}$\\,yr, roughly comparable to $t_{\\rm rad}$ for less-luminous AGNs. We note that if AGN activity is more episodic (e.g., \\citealt{Par11}; \\citealt{Yaj17}), feedback timescales may be shortened accordingly, although it seems more likely that luminous quasars have few episodes of vigorous accretion (e.g., \\citealt{Hop09}). Luminous, obscured quasars may thus appear to show higher $\\varphi$ values due to a shorter $t_{\\rm AGN}$ than less-luminous, obscured AGNs, even if they feel the same radiation pressure.\n\nWe have referred to $t_{\\rm QSO}\\lesssim10^{7-8}$\\,yr for quasars as a whole (e.g., $M_{\\rm B}\\lesssim-23$ mag or $L_{\\rm bol}\\gtrsim10^{45}\\ergs$), but if more luminous, obscured quasars are in a shorter phase of AGN evolution (shorter $t_{\\rm AGN}$), it better explains the highest $\\varphi$ values observed at $L_{\\rm bol} \\gtrsim 10^{46}\\ergs$. L20 note outflow timescales ($t_{\\rm out}$) for nuclear obscuration to clear away in an expanding shell by radiation pressure on dust,\n\\begin{equation}\nt_{\\rm out} \\approx 2\\times10^{5}\\,{\\rm yr}\\,\\Big(\\frac{r_0}{30\\rm pc}\\Big) \\Big(\\frac{v_{\\rm out}}{1000\\kms}\\Big)^{-1}\n\\end{equation}\nfinding $t_{\\rm out} \\approx 2\\times 10^{5}$ yr for Compton-thick gas expanding from an initial distance of $r_{0}=$30 pc until it reaches $N_{\\rm H}=10^{22}\\, \\rm cm^{-2}$, assuming $v_{\\rm out}=10^{3}$\\kms. If the dusty gas outflows are triggered by radiation pressure, we expect $t_{\\rm out}$ to be equal to $t_{\\rm rad}$. However, it is shorter than our estimated $t_{\\rm rad}$ values for luminous quasars, by $\\sim \\{(3-5)\\times10^{6-7}\\}\/(2\\times 10^{5})$ or $\\sim$1--2 orders of magnitude. This can be explained if $t_{\\rm AGN}$ for $L_{\\rm bol}\\gtrsim10^{46}\\ergs$ quasars are $\\sim$1--2 orders of magnitude shorter than the $t_{\\rm QSO}\\sim10^{7-8}$\\,yr we adopt, qualitatively consistent with the drop of $t_{\\rm AGN}$ for more luminous quasars in simulations (e.g., \\citealt{Hop05,Hop06}).\n\n\n\\subsection{Extended obscuration}\nAn alternative description is that it takes a longer $t_{\\rm rad}$ for luminous, obscured quasars to clear their obscuration than at lower luminosity. Radiation pressure from luminous, obscured quasars should effectively reach larger distances in the galaxy according to the decreasing small-scale dust covering factor observed in high $L_{\\rm bol}$ or $f_{\\rm Edd}$ AGNs (e.g., \\citealt{Mai07}; \\citealt{Tob14}; \\citealt{Ezh17}). Thus, observing high $\\varphi$ values in luminous, obscured quasars implies that dusty gas may be spatially extended into their hosts, in contrast to lower-luminosity AGNs. This is supported by observations of obscured quasars showing an extended distribution of disturbed emission (e.g., \\citealt{San88}; \\citealt{San96}; \\citealt{Urr08}; \\citealt{Gli15}; \\citealt{Fan16}; \\citealt{Ric17b}). An increased fraction of obscured yet broad-line AGNs (e.g., \\citealt{Lac15}; \\citealt{Gli18}) or extended dust extinction through lines of sight kiloparsecs away from narrow-line AGNs are seen (\\S3) at quasar luminosities, also in agreement with global column densities much larger than the line-of-sight $N_{\\rm H}$ for red quasars (\\S3).\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.95]{f6.eps}\n\\caption{A schematic diagram showing the maximally allowed accretion rate and radiation pressure\/outflow track on the $N_{\\rm H}$--$f_{\\rm Edd}$ plane for $N_{\\rm H}=10^{23}, 10^{24}\\, \\rm cm^{-2}$ quasars. At $L_{\\rm bol}\\lesssim10^{45}$\\ergs, the lack of objects in the forbidden region suggests that radiation pressure on dusty gas controls nuclear ($\\lesssim$10 pc order) obscuration and quickly drops the obscuration down to $N_{\\rm H}\\lesssim10^{22}\\, \\rm cm^{-2}$, while extended AGN outflows are less observed. At $L_{\\rm bol}\\gtrsim10^{46}$\\ergs, $f_{\\rm Edd, dust}>1$ accretion (solid line) may likewise clear the nuclear obscuration, but the high fraction of obscured quasars in the forbidden region suggests a short luminous quasar timescale and\/or an extended, $\\sim10^{2-3}$ pc-scale obscuration being cleared slowly by outflows (dotted line).}\n\\end{figure}\n\nAccording to simplified models for AGNs and host galaxy obscuration at multiple scales (e.g., \\citealt{Buc17}; \\citealt{Hic18}), the host galaxy kiloparsecs away from the nucleus is responsible for $N_{\\rm H} \\sim 10^{22}\\, \\rm cm^{-2}$, whereas obscuration from the inner AGN structure ($\\lesssim$10\\,pc) or circumnuclear starbursts ($\\sim$10--100\\,pc) can reach Compton-thick column densities. In addition, obscuration from gas-rich mergers (e.g., \\citealt{Hop08}) or higher gas fractions in high-redshift galaxies (e.g., \\citealt{Tac10}; \\citealt{Buc17}) may enhance the obscuration up to kiloparsec scales. Coming back to Equation (5), we find that $t_{\\rm out}$ for luminous, obscured quasars will be extended by 1--2 orders of magnitude ($t_{\\rm out}\\sim10^{6-7}$ yr) if extended obscuration due to mergers is spread over $\\sim10^{2-3}$ pc, closing the gap between the timescale arguments (\\S6.1) without even changing $t_{\\rm AGN}$. This scenario is also consistent with the $t_{\\rm out}$ values estimated by modeling expanding shells of dusty gas located $\\sim10^{2-3}$ pc away from luminous quasars \\citep{Ish17}. We direct readers to the theoretical and observational studies on how the nuclear outflows triggered by radiation pressure extend to the host galaxy (e.g., \\citealt{Har14}; \\citealt{Ish15}; \\citealt{Tho15}; \\citealt{Ish17}; \\citealt{Kan18}).\n\nAlthough the impact of radiation pressure from the AGN itself is weaker at extended regions of the galaxy, and R17c separate radiation-pressure feedback from inflows and outflows, radiation pressure has still been considered to launch outflows that may reach large distances (e.g., \\citealt{Hop10} and the discussion in L20). In this work, we separately considered radiation pressure to regulate $\\lesssim10$ pc order obscuration and outflows $\\sim10^{2-3}$pc scales, but note that radiation pressure is thought to cause extended outflows that eventually clear obscured quasars, according to gas-rich, merger-driven quasar evolution models (e.g., \\citealt{Hop08}; \\citealt{Hic09}). Not only are the highly ionized gas outflows on the order of $\\sim10^{3}\\kms$ found in the majority of quasars with $L_{[\\rm O\\,III]}\\gtrsim10^{42}$\\ergs\\ (or $L_{\\rm bol}\\gtrsim10^{45.5}$\\ergs), or $f_{\\rm Edd} \\gtrsim 0.1$ (e.g., \\citealt{Woo16}; \\citealt{Rak18}; \\citealt{Shi19}; \\citealt{Jun20}), they extend over kiloparsec scales together with Balmer line outflows with a weaker ionization potential or molecular outflows (e.g., \\citealt{Fio17}; \\citealt{Kan17}; \\citealt{Fle19}). This is in line with higher merger fractions seen in $L_{\\rm bol}\\gtrsim10^{46}$\\ergs\\ quasars (e.g., \\citealt{Tre12}; \\citealt{Fan16}; \\citealt{Dia18}), which is also the transitional luminosity where radiation-pressure feedback appears less effective (Figure 5). \n\nWe thus consider radiation pressure to be responsible for regulating not only the $\\lesssim$10 pc-order dusty structure (e.g., \\citealt{Law91}; R17c), but also the host galaxy environment in obscured $L_{\\rm bol} \\gtrsim 10^{46}$\\ergs\\ quasars where the triggered nuclear outflows may reach and clear $\\sim10^{2-3}$ parsec-scale material, slowly over a timescale of $\\sim10^{6-7}$\\,yr. This is consistent with the high-$f_{\\rm Edd}$ AGN outflows discussed in R17c, though their sample lacked the luminous quasars that we argue are responsible for producing extended outflows at $f_{\\rm Edd, dust}>1$ values. We summarize our discussion in Figure 6.\\\\\n\n\\section{Summary}\nUsing a collection of AGN samples spanning a wide dynamic range of luminosity, obscuration, and redshift, we probed the distribution of obscuration and accretion rate values to comparatively examine the role of radiation pressure in blowing out obscured quasars. We summarize our findings below:\n \n1. The fraction of AGNs in the forbidden zone for radiation pressure, $\\varphi$, is kept to $\\lesssim$20\\%\\ for all of the multi-wavelength-selected AGN samples compiled over a wide range of luminosity and redshift, consistent with previous findings that nuclear obscuration is quickly blown away by radiation pressure once the accretion rate exceeds the Eddington limit for dusty gas.\n\n2. This radiation-pressure feedback, that is, the acceleration of nuclear dusty gas, appears limited for luminous, obscured quasars at $N_{\\rm H}\\gtrsim10^{22}\\, \\rm cm^{-2}$ or $E(B-V) \\gtrsim 0.2$ mag, and $L_{\\rm bol}\\gtrsim10^{46}$\\ergs, where they show $\\varphi\\gtrsim60\\%$ over a wide range of AGN selection wavelengths or amount of obscuration. This may be explained by a combination of shorter luminous quasar lifetimes and extended obscuration cleared by outflows over a longer timescale than to clear the nuclear obscuration. \n\nUltimately, we expect to see the $M_{\\rm BH}$ values grow while luminous, obscured quasars become unobscured if extended outflows, slower than radiation pressure clearing the nuclear obscuration, are the bottleneck for AGN feedback. Ongoing hard X-ray surveys probing fainter sources (e.g., \\citealt{Lan17}; \\citealt{Oh18}) will confirm if distant, gas-obscured quasars are going through similar strengths of radiation-pressure feedback as dust-obscured quasars. Spatially resolved or global $N_{\\rm H}$ and $E(B-V)$ estimates for luminous, obscured quasars will better tell whether obscuration is indeed more extended in luminous quasars and will quantify the relative effect of radiation pressure and outflows to their parsec-to-kiloparsec scale gas and dust environments. \n\n\\acknowledgments\nWe thank the anonymous referee for the comments that greatly improved the paper and Andrew Fabian for kindly providing the $f_{\\rm Edd, dust}(N_{\\rm H})=1$ curves.\nThis research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2017R1A6A3A04005158). R.J.A. was supported by FONDECYT grant No. 1191124. R.C.H. and C.M.C. acknowledge support from the National Science Foundation under CAREER award no. 1554584. C.R. acknowledges support from the Fondecyt Iniciacion grant 11190831. \n\nThis work makes use of data from the $\\it NuSTAR$ mission, a project led by Caltech, managed by the Jet Propulsion Laboratory, and funded by NASA.\nThis research has made use of data and\/or software provided by the High Energy Astrophysics Science Archive Research Center (HEASARC), which is a service of the Astrophysics Science Division at NASA\/GSFC and the High Energy Astrophysics Division of the Smithsonian Astrophysical Observatory.\nThis publication makes use of data products from the United Kingdom Infrared Deep Sky Survey. UKIRT is owned by the University of Hawaii (UH) and operated by the UH Institute for Astronomy; operations are enabled through the cooperation of the East Asian Observatory. When the data reported here were acquired, UKIRT was operated by the Joint Astronomy Centre on behalf of the Science and Technology Facilities Council of the U.K.\nThis publication makes use of data products from the Wide-field Infrared Survey Explorer, \nwhich is a joint project of the University of California, Los Angeles, and the Jet Propulsion \nLaboratory\/California Institute of Technology, funded by the National Aeronautics and Space Administration.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}