diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzqhnj" "b/data_all_eng_slimpj/shuffled/split2/finalzzqhnj" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzqhnj" @@ -0,0 +1,5 @@ +{"text":"\n\\section{Introduction}\n\nWith the rapid development of digital technologies, online education has become increasingly popular in the past decade, which provides students with easy access to various online learning materials such as massive open online courses (MOOCs) and different online question pools.\nThe availability of these online learning materials and student activity logs has made it possible to model student learning and further predict their performance~\\cite{kaser2017modeling}. It has become a well-established task in computer-aided education, since student performance prediction can help course designers to better handle the possible dropout, improve the retention rate of online learning platforms~\\cite{manrique2019analysis}, and provide students with personalized education to enhance student learning by recommending different learning resources to them based on their different needs~\\cite{piech2015deep,yeung2018addressing,thaker2019comprehension,ding2018effective}.\n\nOnline question pools, consisting of a collection of questions, have become increasingly popular for helping students to practice their knowledge and skills~\\footnote{\\href{https:\/\/help.blackboard.com\/Learn\/Instructor\/Tests_Pools_Surveys\/Orig_Reuse_Questions\/Question_Pools}{https:\/\/help.blackboard.com\/}}. For example, there have been many popular online question pools that provide students hands-on exercises to practice their programming skills (e.g., LeetCode\\footnote{\\href{https:\/\/leetcode.com\/}{\\url{https:\/\/leetcode.com\/}}}, Topcoder\\footnote{\\href{https:\/\/www.topcoder.com\/}{\\url{https:\/\/www.topcoder.com\/}}}) and mathematics skills (e.g., Learnlex\\footnote{\\href{https:\/\/mad.learnlex.com}{\\url{https:\/\/mad.learnlex.com\/login}}}, Math Playground\\footnote{\\href{https:\/\/www.mathplayground.com\/}{https:\/\/www.mathplayground.com\/}}).\nThe \\textit{interactive online question pool}, as one kind of the online question pools, comprises interactive questions where interactions such as mouse movement and drag-and-drop are often needed. For example, some online educational games have been designed to involve user interactions to make it a fun way to practice important skills~\\cite{irene2010multimedia}~\\footnote{\\href{https:\/\/www.weareteachers.com\/the-best-online-interactive-math-games-for-every-grade-level\/}{https:\/\/www.weareteachers.com\/}}.\nDespite the popularity and importance of interactive online question pools,\nlittle work has been done to model student learning and predict their performance in online question pools.\n\n\nStudent performance prediction has been widely explored in education community and is a critical step for downstream tasks, such as recommending an adaptive learning path~\\cite{xia2019peerlens} or assisting students at an early stage~\\cite{chen2016dropoutseer}. \nHowever, most of them focus on student performance prediction on the MOOCs platforms (e.g, Coursera, EdX, Khan Academy),\nand little work has been done on interactive online question pools.\nCompared with student performance prediction on MOOCs platforms, it is more challenging to predict student performance on interactive online question pools due to two major reasons.\nFirst, there is \\textbf{\\textit{no predefined question order}} that users can follow; students need to explore the questions by themselves when working on the interactive online question pools.\nMoreover, for some interactive online question pools, their questions only have \\textbf{\\textit{rough knowledge tags}} annotated by domain experts, which are not accurate enough to evaluate the similarity among questions.\nAs will be introduced in Section~\\ref{sec_performance_prediction}, the major models for student performance prediction include Bayesian knowledge tracing ~\\cite{corbett1994knowledge,pardos2011kt}, deep knowledge tracing ~\\cite{piech2015deep,yeung2018addressing} and traditional machine learning models ~\\cite{chen2016dropoutseer, manrique2019analysis,o2018student}.\nHowever, almost all of these models intrinsically depend on the course curriculum or the predefined question order and Bayesian knowledge tracing models also require knowledge tags, making them inapplicable to student performance prediction on interactive online question pools.\n\nTo handle the above challenges, we propose a novel method by introducing a set of new features based on interactions between students and questions to perform student performance prediction in interactive online question pools. \nSpecifically, we utilize the mouse movement trajectories, which consists of the mouse interaction timestamp, mouse event type and mouse coordinates, to predict student performance. Such mouse movement trajectories represent the problem-solving path of a student for each question.\nInspired by prior researchers in education and psychology fields~\\cite{stahl1994using,row1974wait}, which shows that student's ``think'' time on questions affect their performance,\nwe propose a set of novel features (e.g., think time, first attempt, first drag-and-drop) based on mouse movement trajectories to predict student performance in interactive online question pools. Specifically, we define and measure the time between when students see the question and the time that they start solving the questions as the ``think time''. In addition, attributes related to the first attempt and first drag-and-drop are also extracted.\nFurther, since there is no predefined question order in interactive online question pools, different students can work on the questions in different orders. We apply a heterogeneous information network (HIN) to calculate the similarity among questions, in an effort to incorporate students' problem-solving history to enhance performance prediction in online question pools.\nWe evaluated our approach on a real-world dataset that are collected from a K-12 interactive mathematical question pool \\textit{Learnlex}. We tested our new feature set on four typical multiple-classification machine learning models -- Logistic Regression (LR), Gradient Boosting Decision Tree (GBDT), Support Vector Machine (SVM) and Random Forest (RF). \n\nThe contributions of this paper can be summarized as follows.\n\n\\begin{itemize}\n \\item We introduce novel features based on student mouse movement trajectories to predict student performance in interactive online question pools. Features like the ``think time'' can reveal students' thinking details when working on a specific question.\n \n \n \\item We propose a novel approach based on HIN to incorporate students' historical problem-solving information on similar questions into the performance prediction on a new question.\n \n \\item We evaluate our approach using real-world dataset and compare with state-of-the-art baseline features on typical multiple-classification machine learning algorithms.\n The 4-class prediction result shows that our approach achieves a much higher performance prediction accuracy than the baseline features in various models, demonstrating the effectiveness of the proposed approach.\n \n\\end{itemize}\n\n\n\\iffalse\n\\yong{Huan, please double check the contributions above.}\n\n\n\n\n\n\n\n\n\nThe existing methods, like DKT, can only work for the prediction of questions with predefined order.\n\nThaker \\emph{et al.}~\\cite{thaker2019comprehension} incorporated student reading behaviors in modeling students' learning and performance.\n\n\n\nThe interactive math question is one kind of k-12 educational games~\\cite{irene2010multimedia}. Students solve these questions by using a mouse to drag or drop web items to appropriate locations, which provides them with an effective and enjoyable way to learn or practice knowledge. \nA question pool of interactive math questions is a collection of many different kinds of interactive math questions that is stored for repeated use ~\\cite{question}.\nComparing with questions in traditional printed material or other e-learning platforms, these interactive questions are more intuitive, entertaining, and expressive in presenting abstract concepts for k-12 students.\nOne key goal of online question pool, such as the interactive math question pool, is to be adaptive, which means resources can be suggested to students based on their individual needs, and content which is predicted to be too easy or too hard can be skipped or delayed ~\\cite{basu2007multimedia, mayempowering}.\nStudent's performance prediction is extremely important because the earlier we know students' future potential performance, the earlier we can take action to help them.\n\n\nHowever, it is difficult to predict students' performance in such environment, i.e., interactive math question pools~\\cite{liu2013predicting}. \n\\yong{This reference is about educational game, NOT math questions.----The main idea is interactive math question, not math question, I can't find any reference about prediction in interactive math question except educational game....}\nFirst, the consecutive problems are of different difficulty levels, which blocks the way to predict the performance according to the recent performance on physically close questions. Second, Students can freely explore materials and select questions to practice their knowledge in any order, which cause some question may have little history records. There may be no enough data in training a single question.\n\nThe prediction approaches to predict students' scores usually lie in two aspects. One branch of work mainly focus on model or algorithm rather than feature's improvements or innovations ~\\cite{piech2015deep, ding2018effective}. It can not work in our scenario because 1. score 0-100 not True or False, model not suitable. 2. they didn't take mouse movement interaction into consideration(key information). \n\\yong{because of what?}\nAnother branch of work attempts to predict students' performance according to their interaction data. Reading rate is defined as the speed with which\nthe student is reading through the material~\\cite{thaker2019comprehension}. We utilized this feature and incorporate more interaction behaviors in the proposed method.\n\\yong{Then, what?}\nWe consider to make use of the potential information in interaction data -- mouse movement trajectories from all the questions in the question pool to help with prediction. Extracting useful features from mouse movement trajectory, which is a kind of time series data, is not a easy task.\nO'Connell \\cite{o2018student} conducted a typical multiple regression model in predicting student's final grade (A-F), which got an accuracy of 57.7\\% based on the AICc value. As well as Rovira et al.\\cite{rovira2017data}, they only extract features from question labels, students' personal information and historical data rather than figure out insightful features(e.g, Reading Speed) in problem-solving process.\nTraditional machine learning models' performance severely affected by data source features in prediction tasks ~\\cite{heaton2016empirical, rovira2017data}, which also means efficient data features may contribute a lot in prediction task regardless of models or algorithms.\n\nWe propose a novel algorithm that integrates students' mouse movement trajectories to predict their score for each question in the online interactive math question pool.\nSpecifically, we transfer the concept of \"wait-time\" -- a period that teachers provide students to think during class interaction to students' \"wait-time\" before act in online interactive problem-solving process. In order to detect the end of \"wait-time\" in mouse movement trajectories, we apply the concept of \"change point\" in time series analysis, which represents abrupt changes in time series data. We utilize the two concepts to extract \"wait-time\" related features(e.g. wait-time length) in mouse movement trajectory. There are other features like student's historical record feature and first drag-and-drop features(e.g., first drag-and-drop trajectory length) in mouse movement trajectory.\nInitially, we trained models with two parts of data features, students' historical record features and mouse movement features on questions. For the purpose of score prediction on a candidate question, we must have one student's historical records and mouse movement features of the question, the latter of which doesn't exist before the student finish solving it. Secondly, to infer the student's mouse movement features on the candidate question, we construct a correlation matrix among questions with HIN using student-question solving information in the online interactive math question pool. With the help of the correlation matrix, we find the most similar question that the student has solved to the candidate question. Combining the student's historical record features and similar question's mouse movement features in models, we achieve a higher prediction accuracy in our real-world data test.\n\nWe are the first to integrate wait-time and change point in students' problem-solving process and we develop a \"wait-time\" ending detection method for general mouse movement based interactive circumstances.\nThe correlation analysis between interactive data and performance often varies in different questions or tasks ~\\cite{seelye2015computer, hagler2014assessing, brown2014finding}. But we break the one fit one analysis barrier and extend the single question to a set of similar questions.\nWe evaluate our approach using real-world data and compared with state-of-the-art baselines. The 4-class prediction result shows we achieve an accuracy of 76\\% which is better than prior studies \\cite{o2018student, ren2016predicting}.\nprediction\nstep1: mouse movement; purpose, method, why works\nstep2: similarity;\nstep3: model\n\ncontribution:\n1. model mouse movement\n2. hin similarity\n3. evaluation\n\n\\fi\n\n\n\n\n\n\n\n\\section{Related Work}\nThe related work can be categorized into three groups: student performance prediction, problem-solving feature extraction, and question similarity calculation. \n\n\\subsection{Student Performance Prediction}\n\\label{sec_performance_prediction}\nThere are mainly two ways in performance prediction: the knowledge tracing and the traditional machine learning approach (e.g., Multiple regression). Methods based on or extended from knowledge tracing usually utilize a computational model of the effect of practice on KCs (i.e. knowledge components, which may include skills, concepts or facts) as the way to individually monitor and infer students' learning performance~\\cite{piech2015autonomously}. Bayesian Knowledge Tracing (BKT)~\\cite{corbett1994knowledge} was the most popular approach to model students' learning process, where each learning concept was represented as a binary variable to indicate whether or not the student has mastered the learning concept or not. However, this method gives little consideration to the individual students' learning ability. Learning Factors Analysis (LFA)~\\cite{cen2006learning} extended the basic formulation by adding the factor--learning rate. More factors were taken into consideration by later methods, such as Performance Factors Analysis (PFA)~\\cite{pavlik2009performance}, which further incorporated the students' responses to the questions (correct or incorrect). Additive Factors Analysis Model (AFM)~\\cite{chounta2019square} added the step duration (time between actions). The performance was improving as more factors are considered. However, the drawback of the methods based on the knowledge tracing is that they need a requirement for accurate concept labeling. Though recent studies showed that there is a possibility to make use of deep learning algorithm (i.e., Recurrent Neural Network) to predict students' performance on consecutive questions, the performance highly relied on the predefined question order (i.e., most of the students followed the same order to solve questions)~\\cite{piech2015deep}.\n\nFor questions in question pools, they have no predefined order and no accurate concept labels~\\cite{xia2019peerlens}, which hinders the way to easily adapt methods based on the knowledge tracing. Many studies have used traditional machine learning methods to predict the drop out rate or the course grade~\\cite{manrique2019analysis, chen2016dropoutseer, kabakchieva2013predicting, moreno2018prediction,maldonado2018predicting}. For example, Naive Bayes (NB), RF, GBDT, SVM, k-Nearest Neighbour (KNN) with Dynamic Time Warping (DTW) distance were used to predict college students' dropout~\\cite{manrique2019analysis}. Chen~\\emph{et al.}~\\cite{chen2016dropoutseer} also used Logistic Regression and Nearest-neighbors to predict the dropout in MOOCs. Kabakchieva and Dorina ~\\cite{kabakchieva2013predicting} compared different machine learning models (Decision tree classifier, Bayesian classifiers, KNN classifier) for college students for grade prediction. Regardless of the chosen algorithm, much of the performance of a prediction model depended on the proper selection of feature vectors~\\cite{manrique2019analysis} and the transformation of feature vectors~\\cite{heaton2016empirical}. To achieve a higher accuracy of student performance prediction in the interactive online question pools, we need to extract more meaningful features that are closely related to students problem-solving capabilities.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Problem-Solving Feature Extraction}\nMany problem-solving features have been extracted and applied to the student performance prediction on questions. Among them, the submission record and the clickstream are two data types that are studied most. Xia~\\emph{et al.}~\\cite{xia2019peerlens} used the submission records to model how a student solves a particular problem in online question pools, e.g., solving the problem with one submission or many submissions. Chen and Qu~\\cite{qu2015visual, chen2015peakvizor} extracted features of the clcikstream data of watching MOOC videos and analyzed its correlation with final grades. Chounta and Carvalho~\\cite{chounta2019square} proposed the use of the response time (i.e., the time between seeing the questions and giving a response to tutor's question or task) to predict students' performance for unseen tasks in intelligent tutoring systems. The result showed that the quadratic response time parameter outperformed the linear response time parameter in the prediction tasks.\n\nHowever, students' problem-solving behavior is a complex process involving different stages -- problem decomposition, abstraction, and execution~\\cite{stahl1994using, yadav2016computational}. It is critical to figure out different stages to better understand and predict their influence on individuals' performance ~\\cite{xiong2011analysis} \nrather than treating it as a whole solving period. In the interactive question pool we study, more detailed mouse movement sequences (i.e., the trajectories with both position and timestamp information) are collected, which reflect the students' problem-solving ability in extensive details. \nStahl~\\emph{et al.}~\\cite{stahl1994using} introduced \\textit{``think time''}, which is a period of uninterrupted silence time given by the teacher in class and all students are asked to complete appropriate information processing tasks. Different students may have different abilities in processing the question information before they start to solve it~\\cite{thaker2019comprehension}.\nInspired by these previous work, we propose extracting ``think time'' from students' mouse movement trajectories to delineate the thinking process and problem-solving capabilities of different students.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Question Similarity Calculation}\nQuestion similarity is one of the key features to infer students' performance in previous researches. For example, When students repeatedly solve questions under the same topic, they may improve the mastery on this learning concept ~\\cite{corbett1994knowledge,cen2006learning,pavlik2009performance}. \nA wide range of work has been done to calculate question similarity when there is no accurate expert annotation of the problems. One branch of work calculated the semantic similarity of questions based on Natural Language Processing (NLP). For example, Song~\\emph{et al.}~\\cite{song2007question} first identified keywords in questions, and then calculated semantic similarity between questions based on the keywords. They also extended the semantic similarity to statistic similarity, which was calculated using the cosine similarity between two question vectors. Each question vector was a string of binary bits with each bit indicating the existence of a certain word.\nTheir results showed that the combination of semantic similarity and statistic similarity could achieve a better performance than each individual algorithm. Similarly, textual similarities~\\cite{charlet2017simbow} and question type similarity~\\cite{achananuparp2008utilizing} were also proposed to further improve the performance prediction.\n\n\n\n\n\n\n\nHowever, all these methods require accurate knowledge tags and abundant text information about the questions. The questions in the interactive math question pools usually use simple description to describe the problem background without showing the knowledge tested explicitly. Another branch of work tried to calculate the question similarity according to the interaction data between students and problems. For example, Xia~\\emph{et al.}~\\cite{xia2019peerlens} calculated the question similarity based on the submission types (e.g., successful submission after one submission or many submissions). \nHowever, more detailed student interaction information (e.g., mouse movement trajectories), which intrinsically reflects students' problem-solving habits and capabilities, is not considered.\nHIN is defined as the information network with more than one type of objects or relationship between objects, which is used to describe the complex structure of information in the real world. Bibliographic information network\nand Twitter information network are examples of HIN \\cite{sun2013mining}. In this paper, we use the HIN to incorporate different information (e.g., students' historical scores, mouse movement trajectories)\ninto calculating the question similarity.\n\n\n\n\n\n\\section{Context}\nOur study is built on the dataset collected from an interactive online math question pool. This section introduces the interactive math questions, the collected data and the overall student performance prediction framework of our study.\n\n\\subsection{Interactive Math Question}\nThe platform we cooperate with is an interactive question pool used by more than 40,000 students from 30+ primary and junior high schools in Hong Kong. There are 1,720 interactive math questions on the platform and each question has labels from \\textit{math dimension}, \\textit{difficulty}, and \\textit{grade}. The \\textit{math dimension} indicates the general knowledge domain of the question. \\textit{Difficulty} is a five-star rating ranging from 1-5 (easy to hard), which is predefined by several education experts and the question maker. \\textit{Grade} represents the year of study this question is designed for. Figure~\\ref{example} shows an example of the interactive math question in the question pool, students need to use their mouse to drag the red dot to fulfill the requirement and get a score ranging from 0 to 100. Since the scores are discrete and possible scores of each question are not the same, we manage to map the original score ranging from 0 to 100 to 4 score classes (0-3).\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.7\\linewidth]{src\/image\/question_example3.png}\n \\vspace{-1em}\n \\caption{An example of the interactive math question.}\n \n \\label{example}\n\\end{figure}\n\n\\subsection{Data Collection}\nThe data we collected are historical score records and mouse movement trajectories. In total, we collected 858,115 historical score records from September 13, 2017 to June 27, 2019 in total. We developed a tool to collect students' mouse movement trajectories, which included the mouse events (mouse move, mouse down, mouse drag and mouse up), the timestamps, and the positions of the mouse during the whole problem-solving process. We selected two question sets that contains rich mouse interactions from two math dimensions (\\textit{Area} and \\textit{Deductive geometry}).\nTable \\ref{tabledataset} shows the statistical information of these two question sets. There are 61 and 64 questions in these two categories respectively. 724 students produced 1764 mouse movement trajectories in the Area and 562 students made 2610 mouse movement trajectories in Deductive geometry. Note that the system allows students to submit up to two times for each question, our research goal in this paper is to predict students' score on the first submission of the question.\n\n\n\n\n\n\\begin{table}[]\n\\begin{tabular}{@{}cccc@{}}\n\\toprule\nDataset & \\#Questions & \\#Trajectories & \\#Students \\\\ \\midrule\nADD & 61 & 1764 & 724 \\\\\nDGDD & 64 & 2610 & 562 \\\\ \\bottomrule\n\\end{tabular}\n\\caption{The statistics of two datasets: ADD (area dimension dataset) and DGDD (deductive-geometry dimension dataset). Both consist of students' problem-solving records from April 12, 2019 to June 27, 2019.}\n\\label{tabledataset}\n\\vspace{-3em}\n\\end{table}\n\n\\subsection{Overall Prediction Framework}\nIn order to conduct a state-of-the-art machine learning prediction experiment, we survey existing studies for\nthe features used in student performance prediction~\\cite{o2018student, manrique2019analysis},\nmouse trajectory feature definition~\\cite{yamauchi2013mouse,seelye2015computer},\nand possible methods in question similarity calculation ~\\cite{song2007question,charlet2017simbow,achananuparp2008utilizing,sun2013mining,sun2011pathsim}.\nWe summarize this knowledge with our dataset and task, then we propose our prediction framework in Figure~\\ref{framework}. It mainly contains three modules: data collection and preprocessing, feature extraction, and prediction and evaluation. \n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=8cm]{src\/image\/process2.png}\n \\vspace{-1.5em}\n \\caption{The prediction framework contains three modules: data collection \\& preprocessing module, feature extraction module, and prediction \\& evaluation module. The blocks highlighted in yellow and green correspond to the major contributions of this paper.}\n \\label{framework}\n\\end{figure}\nIn the feature extraction module, as suggested by previous studies~\\cite{galyardt2014recent,goldin2015convergent}, students' recent performance may have a great impact on prediction. We then extract the historical performance statistical features and recent performance statistical features from the records. In addition, we summarize each question's basic information (e.g., grade, difficulty) as well as the number of total submissions and second submissions.\nFurther more, mouse movement trajectories are processed to representation features including think-time, first drag-and-drop, first attempt, and other problem-solving related features. \nIn addition, we combine these features and statistical features (e.g., score) and build the problem-solving information network, a HIN, on them to calculate the question similarity matrix.\n\nIn the prediction and evaluation module, we test these features on four typical machine learning models to compare their performances with and without the mouse movement features. Similar to previous research ~\\cite{gardner2019evaluating, manrique2019analysis}, we compare prediction accuracy, weighted F1 and ROC curves and feature importance score in GBDT to evaluate and discuss our method.\n\n\\section{Feature Extraction}\nIn this section, we introduce the feature extraction module in detail. Specifically, we first explain how to detect change points in the mouse movement trajectories, based on which we illustrate how we extract mouse movement features\n(i.e., think time, first drag-and-drop, and first attempt).\nWe then introduce other statistical features like students' historical performance features. Lastly, we describe how to use a HIN to incorporate both statistical features and mouse movement features to calculate the similarities between questions. Assuming that students often have similar performance on similar questions, we integrate features of similar questions to further enhance the performance prediction on a new question.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=8cm]{src\/image\/change_point.png}\n \\vspace{-2em}\n \\caption{A sample mouse movement trajectory and the scheme diagram of change point detection algorithm. The 1st change point is both the think time end point and the first attempt start point. The 2nd change point is the first attempt end point.}\n \\label{change point}\n \\vspace{-1em}\n\\end{figure}\n\\subsection{Change Point Detection}\n\\label{sec_change_point}\nTo infer students' problem-solving capabilities from the mouse movement trajectories, we need to identify different problem-solving stages~\\cite{yadav2016computational,xiong2011analysis}.\nThe change points are the time points where there is an abrupt change in the mouse movement trajectory to distinguish different problem-solving stages in our context.\nAs shown in Figure~\\ref{change point}, we split the mouse movement trajectory into three subparts using two change points: the think time stage, the first attempt stage (i.e., the first action episode after the thinking period) and the following actions stage. Think time is a stage that starts from when a student opens the question and ends at the students use the mouse to solve the question with actual interactions~\\cite{stahl1994using}. The first attempt stage is a series of mouse drag events trying to solve the problem after the think stage and ends when the frequency of mouse drag events becomes low. The first change point can differentiate the think time stage from the first attempt stage and the second change point is the boundary of the first attempt stage and the following actions stage. \nA straightforward way of change point detection is to use the first movement of the mouse as a signal that the student starts to solve the question. However, in the real world, it is not always true that the first mouse movement represents the starting point of solving a problem. After analyzing the mouse movement trajectories, we find that there may exist a short drag-and-drop or click by mistake, which makes it seem like the student starts to solve questions. To detect the most probable start and end time points of each stage, we propose the change point detection algorithm.\n\\vspace{-2em}\n\\begin{figure}[h]\n \\vspace{-1em}\n \\centering\n \\includegraphics[width=9cm]{src\/image\/movement.png}\n \\vspace{-4em}\n \\caption{An example of mouse movement trajectories in problem-solving process.}\n \\label{trajectory}\n \\vspace{-1em}\n\\end{figure}\n\nWe use the sliding window to detect the change points, which is a common method for abrupt change detection in the time series data~\\cite{basseville1993detection}. It has two key parameters: window size and detection threshold.\nThe detection threshold can be observed through the test data or sometimes random selection~\\cite{basseville1993detection}. \nFor a mouse movement trajectory, we define the total mouse events as $C_t$, the total mouse drag events as $C_d$. The event density in trajectory is defined as:\n\\begin{equation}\n \\rho = C_d\/C_t\n\\end{equation}\nWe count the mouse drag events and calculate the mouse drag event density in sliding window and compare the value of density with the threshold. The scheme of change detection is as follows:\n\\begin{itemize}\n\\item Move the window from the beginning of the mouse events sequence, when the first time $\\rho$ is higher than the threshold, the first point of the sliding window is defined as the first change point.\n\\item Continue sliding window until $\\rho$ is lower than the threshold, the last point of the sliding window is defined as the second change point.\n\\end{itemize}\n\\begin{table}[]\n\\small\n\\begin{tabular}{@{}lll@{}}\n\\toprule\nType & Feature & Description \\\\ \\midrule\n\\multirow{4}{*}{Think time} & Time length & \\begin{tabular}[c]{@{}l@{}}Time between opening the web \\\\ page and the first change point.\\end{tabular} \\\\ \\cmidrule(l){2-3} \n & Time percent & \\begin{tabular}[c]{@{}l@{}}Percentage of think time in \\\\ time length of the whole trajectory.\\end{tabular} \\\\ \\cmidrule(l){2-3} \n & Event length & \\begin{tabular}[c]{@{}l@{}}Mouse event number \\\\ in think time.\\end{tabular} \\\\ \\cmidrule(l){2-3} \n & Event percent & \\begin{tabular}[c]{@{}l@{}}Percentage of mouse event\\\\ number in the whole trajectory.\\end{tabular} \\\\ \\midrule\n First attempt & \\begin{tabular}[c]{@{}l@{}}Event end \\\\ index\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Mouse event number during\\\\ the first attempt.\\end{tabular} \\\\\\midrule\n\\multirow{9}{*}{\\begin{tabular}[c]{@{}l@{}}First drag-\\\\ and-drop\\end{tabular}} & Time length & \\begin{tabular}[c]{@{}l@{}}Time between opening the web \\\\ page to the first mouse drag.\\end{tabular} \\\\ \\cmidrule(l){2-3} \n & Time percent & \\begin{tabular}[c]{@{}l@{}}Percentage of first drag-and-drop\\\\ in time length of the whole trajectory.\\end{tabular} \\\\ \\cmidrule(l){2-3} \n & \\begin{tabular}[c]{@{}l@{}}Event start\\\\ index\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Mouse event number before\\\\ first drag-and-drop starts.\\end{tabular} \\\\ \\cmidrule(l){2-3} \n & Event percent & \\begin{tabular}[c]{@{}l@{}}Percentage of event number \\\\ before first drag-and-drop starts\\\\ in the whole trajectory..\\end{tabular} \\\\ \\cmidrule(l){2-3} \n & \\begin{tabular}[c]{@{}l@{}}Event end\\\\ index\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Mouse event number when first\\\\ drag-and-drop ends.\\end{tabular} \\\\ \\cmidrule(l){2-3} \n & K & First drag-and-drop trajectory curvature. \\\\ \\cmidrule(l){2-3} \n & D & First drag-and-drop trajectory length. \\\\ \\cmidrule(l){2-3} \n & Delta & First drag-and-drop chord length. \\\\ \\bottomrule\n\\end{tabular}\n\\caption{TFF feature table. TFF: think time, first attempt, and first drag-and-drop.}\n\\label{table:3}\n\\vspace{-3em}\n\\end{table}\n\\subsection{Mouse Movement Features}\nStudent interaction data, especially mouse movement data, contains massive information~\\cite{hagler2014assessing,seelye2015computer}.\nWe extract two sets of features: TFF (think time, first attempt, and first drag-and-drop) and MDSM (mouse drag statistical measurements) based on previous studies~\\cite{stahl1994using,yamauchi2013mouse,seelye2015computer}.\nWe use TFF and MDSM to model students' initial and overall behaviors in a problem-solving process, respectively.\n\nAs for the TFF, the think time and the first attempt have been introduced in Section 4.1. Drag-and-drops is a series of consecutive mouse drag events that start with the mouse down and end with the mouse up and thus the first drag-and-drop is the first drag event with a mouse down and a mouse up. Table~\\ref{table:3} shows the detailed attributes of the features in TFF.\n\nAs for MDSM, we define the following features to represent mouse drag statistical measurements as Figure 4 shows.\n\\begin{itemize}\n \\item $D$: Drag-and-drop trajectory length.\n \\item $Delta$: Drag-and-drop chord length.\n \\item $K$: Drag-and-drop trajectory curvature (Delta\/D).\n \\item $T_{Drag}$: Drag-and-drop duration.\n \\item $T_{Idle}$: Drag-and-drops interval.\n \\item $t_{Idle}$: Mouse events interval.\n\\end{itemize}\n \nHowever, there may be more than one drag-and-drop in a trajectory. To extract useful information in drag-and-drops, we use three statistical methods median, mean and interquartile range (IQR), to measure it (Table ~\\ref{tableMmeasure}). Besides the features above, we also build other features to measure the whole trajectory, including the number of drag-and-drops, total time, total mouse events and mean, median, IQR of mouse events in every second.\n\n\n\n\\begin{table}[]\n\\small\n\\begin{tabular}{@{}clcccccc@{}}\n\\toprule\n & \\multicolumn{1}{c}{$K$} & $D$ & $T_{drag}$ & $Delta$ & $t_{Idle}$ & $T_{Idle}$ &\\\\ \\midrule\nMedian & \\multicolumn{7}{l}{The feature's median value in its value list of the whole trajectory.} \\\\\nIQR & \\multicolumn{7}{l}{The feature's IQR in its value list of the whole trajectory.} \\\\\n\\multicolumn{1}{l}{Mean} & \\multicolumn{7}{l}{The feature's mean value in its value list of the whole trajectory.} \\\\ \\bottomrule\n\\end{tabular}\n\\caption{Three statistical measures for six features. IQR: Interquartile range. $D$: Drag-and-drop trajectory length, $Delta$: Drag-and-drop chord length, $K$: Drag-and-drop trajectory curvature, $T_{drag}$: Drag-and-drop duration, $T_{Idle}$: Drag-and-drops interval and $t_{Idle}$: Mouse events interval.}\n\\label{tableMmeasure}\n\\vspace{-3em}\n\\end{table}\n\n\n\n\n\n\\subsection{Other Statistical Features}\nInformation such as score records and score distribution on questions can also reflect a student's ability and the question difficulty for all the students. Features extracted from such records are widely applied in prediction of students' performance, for example, Yu \\textit{et al.}~\\cite{yu2010feature} used students' recently solved questions to construct temporal features and Manrique \\textit{et al.} introduced average grades of students to predict their dropout rates~\\cite{manrique2019analysis}.\nTo make full use of the information, we take the cross feature approach. A cross feature is a synthetic feature formed by multiplying (crossing) two or more features. Crossing combinations of features can provide predictive abilities beyond what those features can provide individually~\\cite{crossfeature}. Thus, we apply the cross feature method to questions' and students' basic statistics. As Table~\\ref{table_other_fea} shows, we have three parts of statistical features: question statistics, student statistics, and the recent statistics of a student. \n\nWe use the expression A $\\times$ B to represent the cross feature of A and B, expression \\#C in $[$ E $]$ to represent the numbers of C for each category or dimension in E and expression \\%C in $[$ E $]$ to represent the proportion of C for each category or dimension in E. For example, in students statistics, \\%Submission in $[$math dimension $\\times$ grade $\\times$ difficulty$]$ represents the proportion of a student's submission number in each math dimension with a specific grade and difficulty level.\n\n\n\n\\begin{table}[]\n\\small\n\\begin{tabular}{l|l}\n\\hline Feature & Description \\\\ \\hline\n\\multicolumn{2}{c}{\\textbf{Question statistics}} \\\\ \\hline \\hline\nMath dimension & Question's domain knowledge (e.g, area). \\\\ \\hline\n Grade & \\begin{tabular}[c]{@{}l@{}} Student's grade that the question suggests.\\end{tabular} \\\\ \\hline\n Difficulty & \\begin{tabular}[c]{@{}l@{}}Question's difficulty given by experts.\\\\ \\end{tabular} \\\\ \\hline\n \\#Total submissions & \\begin{tabular}[c]{@{}l@{}}Total number of submissions in question.\\end{tabular} \\\\ \\hline\n \\#2nd submissions & \\begin{tabular}[c]{@{}l@{}}Total number of second submissions in question.\\end{tabular} \\\\ \\hline \\begin{tabular}[c]{@{}l@{}}\n \\%Submissions in \\\\ $[$score class$]$\n \\end{tabular} \n & \\begin{tabular}[c]{@{}l@{}}Proportion of submissions in each score class.\\end{tabular} \\\\ \\hline \\hline\n\\multicolumn{2}{c}{\\textbf{Student statistics}} \\\\ \\hline \\hline\n\\#Total submissions & \\begin{tabular}[c]{@{}l@{}}Student's total submissions in history.\\end{tabular} \\\\ \\hline\n \\#2nd submissions & \\begin{tabular}[c]{@{}l@{}}Student 's total second submissions in history.\\end{tabular} \\\\ \\hline \\begin{tabular}[c]{@{}l@{}} \\%Submissions in \\\\$[$math dimension $\\times$ \\\\ grade $\\times$ difficulty$]$\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Student's proportion of submissions in each \\\\specific math dimension, grade and \\\\ difficulty.\\end{tabular} \\\\ \\hline \\begin{tabular}[c]{@{}l@{}} 1stAvgScore in \\\\ $[$math dimension $\\times$ \\\\ grade $\\times$ difficulty$]$\n \\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Student's first submission average score in \\\\ each specific math dimension with assigned \\\\specific grade and difficulty.\\end{tabular} \\\\ \\hline \\hline\n\n\\multicolumn{2}{c}{\\textbf{Student recent statistics}} \\\\ \\hline \\hline\n\\begin{tabular}[c]{@{}l@{}}\\#Submissions in \\\\ $[$math dimension$]$ \\end{tabular} &\n \\begin{tabular}[c]{@{}l@{}}Number of submissions in each math dimension \\\\in past N days.\\end{tabular} \\\\ \\hline \n \\begin{tabular}[c]{@{}l@{}}\\#Submissions in \\\\ $[$grade $\\times$ difficulty$]$\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Number of submissions in each grade and \\\\difficulty in past N days. \\end{tabular} \\\\ \\hline\n \\begin{tabular}[c]{@{}l@{}}Average score in \\\\ $[$math dimension$]$\\\\ \\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Average score in each math dimension in past\\\\ N days.\\end{tabular} \\\\ \\hline \n \\begin{tabular}[c]{@{}l@{}}Average score in \\\\ $[$grade $\\times$ difficulty$]$\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Average score in each grade and difficulty in\\\\ past N days.\\end{tabular} \\\\ \\hline \n \\begin{tabular}[c]{@{}l@{}}Score std in \\\\ $[$math dimension$]$ \\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Score standard deviation of each math \\\\ dimension in past N days.\\end{tabular} \\\\ \\hline\n \\begin{tabular}[c]{@{}l@{}}Score std in\\\\ $[$grade $\\times$ difficulty$]$\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Score standard deviation of each grade and \\\\ difficulty in past N days.\\end{tabular} \\\\ \\hline\n\\end{tabular}\n\\caption{Other statistical features: questions statistics, student statistics and student recent statistics. Symbol: \\#: Number of records, \\%: Proportion of records. Expression A $\\times$ B: cross features of A and B. Expression C in $[$ E $]$: calculate C for each category or dimension in E.}\n\\label{table_other_fea}\n\\vspace{-3em}\n\\end{table}\n\n\\subsection{Problem-solving Information Network and Similarity Calculation} \\label{QIN}\nTo predict the performance of a student $s_i$ on a question $q_x$ using mouse movement features, an intrinsic requirement is that we should have the mouse movement trajectories. \nHowever, they are not available before \na student actually finishes the question.\nPrior studies~\\cite{yera2018recommender, sanchez2017case} have shown that a student's performances on similar questions are often similar.\nTherefore, we propose finding a question $q_y$ similar to the question $q_x$ and\nusing its mouse movement features extracted from the trajectories as part of the feature vectors to predict a student's performance on $q_x$.\n\nThe similarity between questions can be evaluated from different perspectives (e.g., difficulty level, question content, student mouse movement interactions, etc.). For example, \nfor two questions that examine the same knowledge, their difficulty levels and the required problem-solving skills can be significantly different for different students. To more accurately delineate the similarity between any two questions, we use students' interactions (e.g., mouse movement trajectories) as the bridge from one question to another question. Specifically, we build a network consisting of both interactions between students and questions and the intrinsic attributes of questions to calculate similarity for each pair of questions. Such a network is called \\textit{problem-solving information network} in this paper.\n\n\n\n\n\n\\iffalse\nThe only valuable information can be used to calculate question similarity other than the rough math dimension and score distribution about questions is the mouse movement trajectory data of different students under our scenario.\nMoreover, these trajectories involve both students and questions so they cannot be directly used to compute the similarity matrix of questions.\nThus, we use students as the bridge from one question to another question and build a network between students and questions to calculate similarity for each pair of questions.\nBase on the similarity score, we then use the mouse movement features of the most similar question as part of the features of the target question for prediction.\n\\fi\n\n\n\n\n\n\n\\subsubsection{Problem-solving Information Network Structure}\nThe problem-solving information network, a typical bipartite HIN, is established between two kinds of objects, students (S) and questions (Q). For each question $q\\in Q$, it has links to several students and the link between a question $q$ and a student $s$ is defined as solving ($s$ solves $q$) or solved by ($q$ is solved by $s$). The network schema of the problem-solving network is shown in Figure~\\ref{fig1} (a) and the symmetric meta-path in our network is defined as Question-Student-Question ($QSQ$), which denotes the relationship between two questions that have been solved by the same student.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=8cm]{src\/image\/HIN.png}\n \\vspace{-2em}\n \\caption{The schematic diagrams of problem-solving information network. (a) Network schema, (b) Meta path: $QSQ$, and (c) A sample of problem-solving information network.}\n \\label{fig1}\n\\end{figure}\n\n\n\n\n\\subsubsection{Similarity Calculation}\n\n\n\n\n\nSun \\textit{et al.}~\\cite{sun2011pathsim} proposed a meta-path based similarity framework and the framework on meta path $\\mathcal{P}$ is defined as:\n\\begin{equation} \\label{eq_sim1}\n s(a, b)=\\sum_{p \\in \\mathcal{P}} f(p)\n\\end{equation}\nwhere $f(p)$ denotes a measure defined on a path instance $p \\in \\mathcal{P}$ from an object $a$ to another object $b$. \n\nFollowing this guideline, we propose applying such a meta-path based similarity framework to our application scenario and further measure the similarity of a question $q_x$ and another question $q_y$ under a meta path $QSQ$ in the problem-solving network (Figure \\ref{fig1}). Each path instance $q_{x}s_{i}q_{y} \\in QSQ$ passing a student $s_i$ is measured using the cosine similarity:\n\\begin{equation} \\label{eq_sim2}\n f(q_{x}s_{i}q_{y})= \\frac{\\mathbf{Feature_{ix} }\\cdot \\mathbf{Feature_{iy} }}{\\|\\mathbf{Feature_{ix} }\\|\\|\\mathbf{Feature_{iy}}\\|}\n\\end{equation}\nwhere $\\mathbf{Feature_{ix}}$ is the feature vector, consisting of both mouse movement features generated based on the mouse trajectory of the submission on $q_x$ by a student $s_i$ (as introduced in Section 4.2) and the score of his\/her submission.\n\nWe normalize the similarity score of each path instance $q_{x}s_{i}q_{y}$ by using the sum of the similarity scores of all the students who have finished both questions $q_{x},q_{y}$,\nguaranteeing that every score in our problem-solving network is between 0 and 1. Thus, the final similarity between $q_x$ and $q_y$ on meta path $QSQ$ is defined as:\n\\begin{equation}\n s(q_{x}, q_{y})=\\frac{\\sum_{q_{x}s_{i}q_{y} \\in QSQ} \\frac{\\mathbf{Feature_{ix} }\\cdot \\mathbf{Feature_{iy}}}{\\|\\mathbf{Feature_{ix} }\\|\\|\\mathbf{Feature_{iy}}\\|}}{\\left|\\{q_{x}s_{i}q_{y}: q_{x}s_{i}q_{y} \\in QSQ \\}\\right|}\n\\end{equation}\n\n\n\n\n\n\n\n\\section{Experiments}\n\\subsection{Experiment Setup}\nOur experiment was conducted on the two datasets introduced in Table~\\ref{tabledataset}.\nTo make a 4-class classification, we applied four classical multi-class classification machine learning models, i.e., GBDT, RF, SVM, and LR, on our datasets.\nFirst, we built question statistics and student statistics (Table~\\ref{table_other_fea}) with all records before April 12, 2019. We then calculated the student recent statistics for each record after April 12, 2019, using the data in the past 14 days of that record. \nFor those submissions without recent records, we assigned -1 to all recent performance features. The features listed in Table~\\ref{table_other_fea} served as features in baseline method. \n\n\\begin{table}[]\n\\small\n\\begin{tabular}{@{}cccccc@{}}\n\\toprule\nDataset & Method & GBDT & RF & SVM & LR \\\\ \\midrule\n\\multirow{3}{*}{ADD} & Ours & 0.88 & 0.87 & 0.85 & 0.87 \\\\ \\cmidrule(l){2-6} \n & baseline & 0.79 & 0.85 & 0.77 & 0.83 \\\\ \\cmidrule(l){2-6} \n & ABROCA & \\textbf{0.09} & \\textbf{0.02} & \\textbf{0.08} & \\textbf{0.04} \\\\ \\midrule\n \\multirow{3}{*}{DGDD} & Ours & 0.94 & 0.90 & 0.91 & 0.91 \\\\ \\cmidrule(l){2-6} \n & baseline & 0.88 & 0.88 & 0.89 & 0.89 \\\\ \\cmidrule(l){2-6} \n & ABROCA & \\textbf{0.06} & \\textbf{0.02} & \\textbf{0.02} & \\textbf{0.02} \\\\ \\bottomrule\n\\end{tabular}\n\\caption{AUC and ABROCA value in two datasets. ADD: area dimension question dataset. DGDD: deductive geometry dimension question dataset. Ours: our proposed method. ABROCA: Area between baseline curve and Ours curve.}\n\\label{tableAUC}\n\\vspace{-3em}\n\\end{table}\n\nBased on the proposed mouse movement features, similarity matrix $M_{sim}$ of each dataset was extracted according to the problem-solving network and related similarity calculation algorithms mentioned in Section 4.4. Then for each first submission $q_{x}-s_{i}$ in the dataset, we searched $M_{sim}$ to find question $q_y$ that was most similar to $q_x$ (with a similarity threshold of 0.8 in the experiment on ADD, 0.7 in the experiment on DGDD) and is solved by the same student $s_m$. If there was no similar question with $q_x$ done by $s_{i}$, this submission record would be discarded. Finally, the feature vector of $q_{x}-s_{i}$ in our proposed method was composed of $q_{x}$'s and $s_{i}$'s historical statistical features, $s_{i}$'s recent performance feature in Table \\ref{table_other_fea}, $q_y$'s mouse movement features including features in Table \\ref{table:3} and Table \\ref{tableMmeasure}, $q_y$'s score class and the similarity score between $q_x$ and $q_y$.\n\nSince we discarded submissions with no similar questions, the dataset of proposed method for training and testing is not of the same size as the original dataset. We used the same submission records in the baseline and the proposed methods. In addition, to reduce the effect of imbalanced class distribution, we applied SMOTE over-sampling algorithm to increase the size of minority classes in the training set \\cite{smote}.\nAs for hyperparameter tuning in the four algorithms, grid-search was conducted in model training on our datasets~\\cite{manrique2019analysis}. The following parameters have been tested with the best values in bold:\n\\begin{itemize}\n\\item Number of trees for GBDT and RF: 50, 100, 150, 200, \\textbf{250}, 300, 350\n\\item Max depth of trees for GBDT and RF: \\textbf{5}, 10, 15, 20, 25\n\\item Learning rate of GBDT: 1e-4, \\textbf{1e-3}, 1e-2, 5e-2, 0.1, 0.2\n\\item Penalty parameter of SVM (C): 0.1, 1, \\textbf{5}, 10\n\\end{itemize}\n\n\nAfter fixing our hyperparameters, each model performed the task of predicting students' performance on two datasets using the baseline method and our proposed method for 10 times each and the average accuracy scores and weighted F1 scores are in Table \\ref{table_acc}.\n\\subsection{Performance Comparison}\nThe results for our dataset ADD and DGDD are in Table \\ref{table_acc}. From the perspectives of accuracy and weighted F1 scores, we can see that in both two datasets, GBDT, RF, and SVM performed better in our proposed method than in the baseline. In addition, the performances of our method and the baseline are similar in both datasets.\n\\begin{table*}[]\n\\small\n\\begin{tabular}{@{}cccccccccc@{}}\n\\toprule\n\\multirow{2}{*}{Dataset} & \\multirow{2}{*}{Method} & \\multicolumn{2}{c}{GBDT} & \\multicolumn{2}{c}{RF} & \\multicolumn{2}{c}{SVM} & \\multicolumn{2}{c}{LR} \\\\ \\cmidrule(l){3-10} \n & & Accuracy & Weighted F1 & Accuracy & Weighted F1 & Accuracy & Weighted F1 & Accuracy & Weighted F1 \\\\ \\midrule\n\\multirow{2}{*}{ADD} & Baseline & 0.555 & 0.555 & 0.669 & 0.659 & 0.430 & 0.492 & \\textbf{0.650} & \\textbf{0.670} \\\\ \\cmidrule(l){2-10} \n & Ours & \\textbf{0.753} & \\textbf{0.749} & \\textbf{0.690} & \\textbf{0.667} & \\textbf{0.650} & \\textbf{0.677} & 0.649 & 0.659 \\\\ \\midrule\n\\multirow{2}{*}{DGDD} & Baseline & 0.600 & 0.597 & 0.664 & 0.663 & 0.600 & 0.611 & 0.720 & 0.731 \\\\ \\cmidrule(l){2-10} \n & Ours & \\textbf{0.833} & \\textbf{0.805} & \\textbf{0.780} & \\textbf{0.767} & \\textbf{0.633} & \\textbf{0.643} & \\textbf{0.733} & \\textbf{0.744} \\\\ \\bottomrule\n\\end{tabular}\n\\caption{Results of the accuracy and weighted F1 over four typical machine learning algorithms (GBDT, RF, SVM, and LR) on the proposed method and the baseline method. ADD: area dimension question dataset. DGDD: deductive geometry dimension question dataset. Ours: our proposed method.}\n\\label{table_acc}\n\\end{table*}\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{src\/image\/roc_final.png}\n\\vspace{-1em}\n\\caption{ROC-AUC curve of two methods on ADD (Area dimension question dataset).Gray borders represent mean+std or mean-std. Area between the two curves is ABROCA.}\n\\label{ROC}\n\\vspace{-2em}\n\\end{figure}\nBesides the overall accuracy, we further evaluate the results based on the Receiver Operating Characteristic (ROC) curve. The ROC curve is a graph showing the performance of a classification model at all classification thresholds.\nThe ROC curve is a plot of the false positive rate and true positive rate. \nThe area under the ROC curve (AUC) measures the entire two-dimensional area underneath the entire ROC curve from $(0,0)$ to $(1,1)$ \\cite{rocauc,gardner2019evaluating}, which means AUC ranges from 0 to 1. AUC provides an aggregate measure of performance across all possible classification, so we use the area between the proposed method's ROC curve and the baseline ROC curve to further compare the performance of our proposed method with baseline's, which is called ABROCA in prior work~\\cite{gardner2019evaluating}. To eliminate the randomness of the algorithms, we ran the program 10 times and made a $mean+std$ and $mean-std$ ROC-AUC graph (Figure \\ref{ROC}). As Table \\ref{tableAUC} shows, the ABROCA value is always positive, this confirms that the aggregate performance of our proposed method is consistently better than the baseline across different models.\nFurthermore, we extended the student score prediction from a binary classification problem (correct or wrong) to a multiple classification problem (0-3). To further evaluate our method's performance in every score class, we selected ADD (area dimension question dataset) and drew a real-predicted heatmap (Figure \\ref{heatmap}) for each algorithm.\n\\begin{figure*}\n \\begin{subfigure}[b]{0.45\\textwidth} \\includegraphics[width=\\textwidth]{src\/image\/heatmap\/GBDT_h.png}\n \\vspace{-3em}\n \\caption{GBDT}\n \\label{heatmap_GBDT}\n \\end{subfigure}\n %\n \\begin{subfigure}[b]{0.45\\textwidth}\n \\includegraphics[width=\\textwidth]{src\/image\/heatmap\/RF_h.png}\n \\vspace{-3em}\n \\caption{RF}\n \\label{heatmap_RF}\n \\end{subfigure}\n\n \\begin{subfigure}[b]{0.45\\textwidth}\n \\includegraphics[width=\\textwidth]{src\/image\/heatmap\/SVM_h.png}\n \\vspace{-3em}\n \\caption{SVM}\n \\label{heatmap_SVM}\n \\end{subfigure}\n %\n \\begin{subfigure}[b]{0.45\\textwidth}\n \\includegraphics[width=\\textwidth]{src\/image\/heatmap\/LR_h.png}\n \\vspace{-3em}\n \\caption{LR}\n \\label{heatmap_LR}\n \\end{subfigure}\n \\vspace{-1em}\n \\caption{Predicted distribution heatmap of each score class with four algorithms on dataset ADD. The proportion of True Positive samples in each score class is in red box. In (a)-(d), the heatmap on the left shows the result of our proposed method and the right one shows the baseline method. ADD: area dimension question dataset. \n \n }\n \\label{heatmap}\n\\end{figure*}\nSpecially, the GBDT model provides an importance score for each feature, which is computed by the normalized summation of reduction in loss function of each feature and it is also called Gini importance\\cite{scikit-learn}. The importance score of each feature ranges from 0 to 1 and a larger importance score indicates that the corresponding feature is more important in training the model. This directly helps to learn the importance distribution of our feature sets. In our method, the importance score of mouse movement feature set, most similar question's score class and the similarity score (39 features) reaches 27.4\\% among 483 features, which indicates the mouse movement feature set has significant contribution in GBDT model.\n\\vspace{-1em}\n\\iffalse\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{src\/image\/treemap.png}\n\\caption{The treemap of feature importance in algorithm GBDT on the question set Area-dimension. The color encoding of each area: yellow ( question statistics), green (mouse movement), red (student statistics), blue (student recent statistics). Top 4 important features are labeled in the graph as (1)-(4): (1) Proportion of submissions in score class 1 (2) Score class of the most similar question, (3) Proportion of submissions in score class 2, (4) Proportion of submissions in numeric questions of grade 2 and difficulty 5}\n\\label{feature_importance}\n\\end{figure}\n\\fi\n\n\n\n\n\n\\section{Discussion}\nThe above experiment results demonstrate that our approach can achieve higher accuracy for predicting student performance prediction in interactive online question pools than using the baseline features. However, there are still some issues that need further discussions.\n\n\n\\textbf{Parameter Configurations} Some parameters that need to be set in the proposed approach and some of these parameters are empirically chosen after considering different factors. For example, the sliding window is used to detect the change points (Section~\\ref{sec_change_point}), where the value of the window size and threshold need to be determined first.\nToo large a window size will make the mouse drag event histogram too smooth, while too small a window size may make the mouse drag event frequency histogram too steep. Both have a negative effect on change point detection. \nThe corresponding threshold is also important. We empirically set it as the average mouse drag density to guarantee that the detected change points are exactly the actual ones.\nWith similar considerations, we choose only the question with the highest similarity score and require that the similarity score should be at least $0.7$, \nwhen we try to incorporate the information of similar questions.\nOur experiment results provide support for the effectiveness of the current parameter settings. \n\n\\textbf{Prediction Models} This paper focuses on proposing novel methods to extract features for student performance prediction. These features are further combined with existing well-established traditional machine-learning based prediction models to predict student future performance. Compared with other methods based on deep neural networks (e.g., deep knowledge tracing~\\cite{piech2015deep,yeung2018addressing}), we argue that predictions based on feature extraction have better explainability, as the features are often intuitive and meaningful.\n\n\n\n\\textbf{Data Issues} Since there are no other dataset of interactive online question pools publicly available, the whole study is conducted on the data collected from only one interactive online math question pool. But our proposed approach can be easily extended to other datasets of interactive online question pools, which mainly involve drag-and-drop interactions.\nMoreover, the interaction features based on student mouse movement trajectories rely on the size of the interaction records. When there are too few student interaction records for a specific question, it will be difficult for us to accurately compare the similarity between a certain question and others using student interaction features. With more student interaction data being collected, the reliability of the proposed approach can be further improved. In addition, our method can be easily extended to other devices such as tablets with touch screens since the collected data has the same format (i.e., timestamp, event, and position).\n\\vspace{-1em}\n\\iffalse\n\n\n\n\nThis paper demonstrates a method for score prediction in interactive math question pool and compares the performance with baseline in different aspects. \n\nAdditionally, we utilize our mouse movement features and HIN to get a question similarity matrix in a question pool. We don't know the ground truth of similarity, but at least from the results the similarity shows big influence in the performance prediction. We argue that our method has certain generalizability in finding interactive math questions.\nand the proposed ABROCA statistic in particular, can provide a perspective on model performance beyond mere predictive\naccuracy. ABROCA can provide insights into model discrimination\nacross the entire range of possible thresholds.\n\n2. Change point detection algorithm, we can choose high level one. we use fixed window size and grid search to find proper threshold, other algorithms we can choose, like ... Window size and threshold is a tough task.\\\\\n3. We notice a lot of studies doing on response time. People often confused about response time and think time, discuss the differences. \\\\\n4. K-fold cross validation is very common in evaluating the performance of models on prediction. However, instead of applying it in our experiment, we choose to run each machine learning model to train and test on each feature set of each dataset for 10 times to eliminate the randomness in our models. The reason why we try to use this alternative method could be concluded in two aspects. First, for we propose to build question-solving network use the most similar trial's mouse movement features as part of our feature vector, it is not reasonable if we randomly pick some trials to train regardless of the trajectory of them. Randomly picked trials could be later than the trials that will be predicted and thus break the dependency of trials. Furthermore, K-fold could lead to accuracy inflation in evaluation, which is proved by a simulation test on educational data conducted by Pardos et al.\\cite{Pardos2012TheRW}. To better reflect the performance and improvement of our proposed method, we should avoid to use K-fold cross validation for evaluation.\n6. data limitation\n\n\n\\fi\n\\section{Conclusion}\n\nDifferent from the extensively-studied student performance prediction in MOOCs platforms, student performance prediction in interactive online question pools can be more challenging due to the lack of knowledge tags and predefined question order or course curriculum. \nWe proposed a novel method to boost student performance prediction in interactive online question pools by incorporating student interaction features and similarity between questions. \nWe extracted new features based on student mouse movement trajectories to delineate problem-solving details of students. We also applied HIN to further consider students' historical problem-solving information on similar questions, as students' recent performance on similar questions can also be a good indicator of student future performance on a certain question.\nWe conducted extensive experiments with the proposed method on the dataset collected from a real-world interactive online math question pool.\nCompared with using only the traditional statistic features (e.g., average scores), the proposed method achieved a much higher prediction accuracy across different models in different question classes. The results further confirm the effectiveness of our method.\n\nIn future work, we would like to combine the proposed method with adaptive question recommendation in interactive online question pools, providing different students with personalized online learning and question practice in online question pools. Also, it would be interesting to summarize different problem-solving patterns using mouse trajectories to better model and understand students' learning behaviors.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \nThe COMPASS is a fixed-target experiment at the CERN SPS \nfor investigating the structure and spectrum of hadrons. \nConcerning their hadron spectroscopy program, particular \nattention is paid to light-quark meson systems and one of \nthem, the Primakoff production measurements, \nfrom which the radiative widths of light-quark mesons \ncould be obtained, are going on \\cite{COMPASS:2011,\nCOMPASS:2012-1,COMPASS:2012-2}. \n\nDirect observation of the radiative decays, \nsuch as $X^{\\pm} \\to \\pi^{\\pm}\\gamma$, is often \ndifficult to carry out because of their very small rates, \nwhile their inverse reactions, called Primakoff \nreactions \\cite{Primakoff:1951, Pomeranchuk-Shmushkevich:1961}, \nwhich are described as the scattering of a pion in the Coulomb \nfield of atomic nuclei $(A,Z)$ \n\\begin{eqnarray}\n\\pi^{\\pm} + (A, Z) \n\\to \\pi^{\\pm} \\gamma^{*}+ (A, Z) \\to X^{\\pm} + (A, Z), \\ \n\\label{Eq:0}\n\\end{eqnarray} \nare relatively easy of access. Since the cross section for \nreaction (\\ref{Eq:0}) at very low-$q^2$ regions \n($q$ being the four momentum transfer) is proportional to \nthe radiative decay width \n$\\Gamma (X^{\\pm} \\to \\pi^{\\pm} + \\gamma)$, \nit is possible to determine them by measuring \nthe Coulomb contribution of the absolute cross section \n\\cite{Zielinski:1987}. \nWhile a number of new experimental measurements \nwere performed in recent years \\cite{PDG:2012}, \nit should be noted that some earlier experiments, \nsuch as Ref.~\\cite{Zielinski:1984,Collick:1984}, \nhave been cited as the latest data. Thus there is \nno doubt that the high statistics data from COMPASS \nwill play an important role in the progress of \nlight meson spectroscopy. \n\nTheoretically it has been well known that radiative transitions \nprobe the internal structures of hadrons and hence it offers \na useful tool to investigate their nature. In the transitions of \nexcited states of light-quark mesons, although the final mesons \nhave large kinetic energy at the rest frame of initial ones, \nsuch an effect is neglected in the conventional treatment of \nnaive nonrelativistic quark models (NRQM). \nIn addition, respecting the actual situation that \nall physical observations are made through not quarks but hadrons, \nthe relativistic treatment for the center-of-mass (CM) motion \nof hadrons is absolutely necessary. \n\nThe covariant oscillator quark model (COQM) is \none of the possible covariant extension of NRQM, \nretaining the various success principally restricted to the static problem. \nThe remarkable features of the COQM is \nthat hadrons are treated in a manifestly \ncovariant way and the conserved effective electromagnetic currents\nof hadrons are explicitly given in terms of hadron variables themselves. \n{\\footnote{\nThe COQM has a long history of development.\nIt had been applied to investigate the radiative decays \nof light-quark meson systems {\\cite{Ishida-Yamada-Oda:1989}} and \nheavy quarkonium systems {\\cite{Ishida-Morikawa-Oda:1998}}\nwith considerable success. }\n\nIn this work we shall apply the COQM \nto analyze the one photon couplings in the transitions \n\\{$\\rho(770)({}^{3}S_{1}), ~b_{1}(1235)({}^{1}P_{1}), ~a_{1}(1260) ({}^{3}P_{1}), \n~a_{2}(1320)({}^{3}P_{2}), ~\\pi_{2}(1670)({}^{1}D_{2}), \n~\\rho_{3}(1690)$\\\\$({}^{3}D_{3}), ~\\rho (1700)({}^{3}D_{1})$\\}$^{\\pm}$ $\\to \n\\pi({}^{1}S_{0})^{\\pm}\\gamma$. \nThe obtained radiative decay widths are compared with experiment \nand also other quark model predictions. \nThe rates for excited $D$-wave states are newly \npredicted and these results could provide a useful clue to \nobserve the radiative decays of excited light-quark mesons \nat COMPASS. \n\\section{Basic framework of the COQM}\nLet us briefly summarize the framework of \nthe COQM relevant to the present application. In the COQM \nthe wave function (WF) of $u\\bar{d}$ mesons\\footnote{\nIn the following we restrict ourselves to the case of $u\\bar{d}$ mesons. }\nis given by the bilocal bispinor field \n$\\Psi(x_{1\\mu},x_{2\\mu})_{\\alpha}{}^{\\beta}\n=\\Psi(X_{\\mu},x_{\\mu})_{\\alpha}{}^{\\beta}\n$, where $\\alpha$, $\\beta$ denote the Dirac spinor indices\nof respective constituents and $x_{1}^{}$, $x_{2}^{}$ \nrepresent their space-time coordinates, which are \nrelated with the CM and relative coordinates given by \n$X_{\\mu}=(x_{1\\mu}+x_{2\\mu})\/2$ and \n$x_{\\mu}=x_{1\\mu}-x_{2\\mu}$, respectively. \nThe WF is assumed to satisfy the\nfollowing basic equation {\\cite{Yukawa:1953}} \n\\begin{eqnarray}\n\\label{Eq:1}\n\\left(\\sum_{i=1}^{2}\\frac{-1}{2m}\\frac{\\partial^{2}}{\\partial x^{2}_{i\\mu}}\n+\\frac{K}{2}(x^{}_{1\\mu}-x^{}_{2\\mu})_{}^2 \\right) \n\\Psi(x_{1}^{}, x_{2}^{})_{\\alpha}{}^{\\beta}=0, \n\\end{eqnarray}\nwhich is equivalently rewritten as \n\\begin{eqnarray}\n\\label{Eq:2}\n\\left(\n-\\frac{\\partial^2}{\\partial X_{\\mu}^{2}}+{\\mathcal{M}}^2( x, \n\\frac{\\partial}{\\partial x_{}}) \\right)\n\\Psi(X, x)_{\\alpha}{}^{\\beta}=0, \\ \\ \n{\\mathcal{ M}}^2 =\nd \\left( -\\frac{1}{2\\mu}\\frac{\\partial^2}\n{\\partial x_{\\mu}^2}+\\frac{K}{2}x_{\\mu}^2\\right), \n\\end{eqnarray}\nwhere ${\\mathcal{ M}}^2$ represents the spin-independent \nsquared-mass operator in the pure confining force limit, \n$d=4m$, $\\mu=m\/2$ ($m$ being the effective quark mass) \nand $K$ is the spring constant. \nIn order to freeze the redundant freedom \nof relative time, we adopt the definite-metric type of\nsubsidiary condition for the four dimensional harmonic oscillator (HO),\n{\\footnote{The WF satisfying this condition \nis normalizable and gives the desirable asymptotic \nbehavior of electromagnetic form factors of hadrons {\\cite{Takabayashi}.}}\n} leading to the eigenvalue solutions of \nthe squared-mass operator as \n\\begin{eqnarray}\n\\label{Eq:4}\nM_{N}{}^2=M_{0}{}^2+N\\Omega, \\ \\ N=2N_{r}+L,\n\\end{eqnarray}\nwhere $N_{r}$ and $L$ are the radial and \norbital quantum numbers respectively and \n$\\Omega$ is given by \n$\\Omega=d\\sqrt{K\/\\mu}=\\sqrt{32mK}$. \nThe relation (\\ref{Eq:4}) is \nin accord with the well-known linear rising Regge trajectory \nconcerning the squared mass spectra, which is particularly \nevident in light-quark hadron sectors. \n\nThe WF describing mesons with the CM four momentum $P_{\\mu}$ \ncan be written as \n\\begin{eqnarray}\n\\label{Eq:5}\n\\Psi (x_{1}, x_{2})_{\\alpha}^{(\\pm)}{}^{\\beta} \n=\\frac{1}{\\sqrt{2P_{0}}} e^{\\pm P_{\\mu}X_{\\mu}} \n\\Phi (v, x)_{\\alpha}{}^{\\beta (\\pm)}, \n\\end{eqnarray}\nwhere $v_{\\mu}=P_{\\mu}\/M$ is the four velocity ($M$ being the meson masses).\nThe internal WF $\\Phi (v, x)$ is taken as a form of the\n``$LS$-coupling'' product in an analogous fashion to NRQM, \n\\begin{eqnarray}\n\\label{Eq:6}\n\\Phi (v, x)^{(\\pm)}_{\\alpha}{}^{\\beta} =f(v, x)^{(nL)}_{\\mu\\nu\\cdots} \\otimes\n\\left( W^{(\\pm)}_{\\alpha}{}^{\\beta}(v)\\right)_{\\mu\\nu\\cdots}, \n\\end{eqnarray}\nwhere $f(v, x)^{(nL)}_{\\mu\\nu\\cdots}$ with $n=N+1$ are\nthe definite-metric-type wave functions of the four dimensional HO \nfor the space-time part and $W^{(\\pm)}_{\\alpha}{}^{\\beta}(v)$ \nare the Bargmann-Wigner spinor \nfunctions{\\footnote{They are defined by the direct tensor-product \nof respective constituent \nDirac spinors with the four velocity of mesons as \n$W_{\\alpha}^{(+)\\beta}(v) \\sim u_{\\alpha}(v)\\bar{v}^{\\beta}(v)$, \n$W_{\\alpha}^{(-)\\beta}(v)\\sim v_{\\alpha}(v)\\bar{u}^{\\beta}(v)$, \n}} for the spin part. \nDecomposing the $W^{(+)}_{\\alpha}{}^{\\beta}(v)$ \ninto irreducible components with the definite $J^{PC}$, \nthe detailed expressions for meson states relevant to \nthe present study are obtained as follows: \n\\begin{subequations}\n\\begin{eqnarray}\n\\label{Eq:7}\n\\Phi (v, x)^{(+)}=f^{(1S)}(v, x)W^{(+)}(v)=f_{0}(v,x)\n\\left( \\frac{1+iv_{\\rho}\\gamma_{\\rho}}{2\\sqrt{2}}\\left(\n-\\gamma_{5}+i\\gamma_{\\mu} \\epsilon_{\\mu}^{}(P)\n\\right) \\right)\n\\end{eqnarray}\nfor the $S$-wave states, \n\\begin{eqnarray}\n\\label{Eq:8}\n\\lefteqn{\n\\Phi (v, x)^{(+)}=f^{(1P)}(v, x)_{\\nu} W^{(+)}(v)_{\\nu}}\n\\nonumber\\\\\n&&=\n\\sqrt{2\\beta^2} x_{\\nu} f_{0}(v, x)\n\\left( \\frac{1+i{v}_{\\rho}\\gamma_{\\rho}}{2\\sqrt{2}}\n\\left(-\\gamma_{5} \\epsilon^{}_{\\nu}(P)+i\\gamma_{\\mu} \\epsilon\n_{\\mu\\nu}(P)\\right)\\right) \n\\end{eqnarray}\nfor the $P$-wave states and \n\\begin{eqnarray}\n\\label{Eq:8e}\n\\lefteqn{\\Phi (v, x)^{(+)}\n=f^{(1D)}(v, x)_{\\nu\\lambda} W_{}^{(+)}(v)_{\\nu\\lambda}}\\nonumber\\\\\n&&=2\\beta^2 x_{\\nu} x_{\\lambda} f_{0}(v, x)\n\\left( \\frac{1+iv_{\\rho}\\gamma_{\\rho}}{2\\sqrt{2}}\n\\left(-\\gamma_{5} \\epsilon_{\\nu\\lambda}(P)+i\\gamma_{\\mu} \n\\epsilon_{\\mu\\nu\\lambda}(P)\\right)\\right) \n\\end{eqnarray}\n\\end{subequations}\nfor the $D$-wave states, where $f_{0}(v,x)$ is the ground-state\nWF of the four dimensional HO given by \n\\begin{eqnarray}\nf_{0}(v,x)=\\left(\\frac{\\beta^2}{\\pi}\\right) \\exp \n\\left(-\\frac{\\beta^2}{2}\\left( x_{\\sigma}^2+\n2(v_{\\sigma}x_{\\sigma})^2\\right)\\right)\n\\label{Eq:12}\n\\end{eqnarray}\nwith the parameter $\\beta^2=\\sqrt{\\mu K}$ \nand $\\epsilon_{\\mu\\cdots}$ are the\npolarization tensors for respective meson states. \n\\section{Electromagnetic Meson Current in the COQM} \n\nNext, we introduce the single photon couplings of \n$q\\bar{q}$ meson systems. The relevant \ndecay amplitude is described by \n\\begin{eqnarray}\n\\label{Eq:9}\n\\int d^4 X \\langle f | {\\mathcal{H}}_{\\rm int.} |i \\rangle \n= -\\int d^4 X \\langle f |J_{\\mu}(X) A_{\\mu}(X) |i \\rangle \n=\\sqrt{\\frac{1}{8P_{I0}P_{F0}q_{0}}}\\delta^{4}(P_{I\\mu}-P_{F\\mu}-q_{\\mu})\n\\mathcal{M}_{fi}, \n\\end{eqnarray}\nwhere $P_{I\\mu}$, $P_{F\\mu}$ and $q_{\\mu}(=P_{I\\mu}-P_{F\\mu})$ are \nthe four momenta of initial and final \nmesons and emitted photon, respectively. \nIn the COQM, as is the case in NRQM, we consider that the\none-photon emission proceeds through a single quark transition \nin which the respective constituents couple with a photon. \nIn order to obtain the electromagnetic current of mesons, \nwe perform the minimal substitutions \n\\cite{Feynman-Kislinger-Ravndal, Lipes:1972}\n$\n{\\partial}\/{\\partial x_{i\\mu}}\\to \n{\\partial}\/{\\partial x_{i\\mu}}-ieQ_{i}A_{\\mu}(x_{i}) \\ \\ (i=1,2)\n$\nin Eq.(\\ref{Eq:1}) \nand then obtain\n\\begin{eqnarray}\n\\label{Eq:11}\nj_{i\\mu} (x_{1},x_{2})&=&j^{(\\rm convec)}_{i\\mu} (x_{1},x_{2})+j^{(\\rm spin)}_{i\\mu} (x_{1},x_{2}) \n\\end{eqnarray}\nwith\n\\begin{eqnarray}\nj^{(\\rm convec)}_{i\\mu} (x_{1},x_{2})\n&=&-\\frac{ideQ_{i}}{2m} \\langle \\bar{\\Psi}(x_{1},x_{2})\n\\left(\\frac{\\overleftrightarrow{\\partial}}{\\partial x^{}_{i\\mu}} \n\\right)\n\\Psi (x_{1},x_{2})\\rangle, \\\\\nj^{(\\rm spin)}_{i\\mu} (x_{1},x_{2})\n&=&-\\frac{ideQ_{i}}{2m} \\langle \\bar{\\Psi}(x_{1},x_{2})\n\\left(ig_{M}^{(i)}\\sigma_{\\mu\\nu}(\\frac{\\overrightarrow{\\partial}}{\\partial x^{}_{i\\nu}}+\\frac{\\overleftarrow{\\partial}}{\\partial x^{}_{i\\nu}}) \\right) \n\\Psi (x_{1},x_{2})\\rangle,\n\\end{eqnarray}\nwhere $\\langle \\cdots \\rangle$ means taking trace concerning \nthe Dirac indices, $\\bar{\\Psi}\\equiv -\\gamma_{4} \\Psi^{\\dagger} \\gamma_{4}$,\n$Q_{i}$ represent quark charges in units of $e$ \n($Q_{1}=Q_{u}=2\/3$ and $Q_{2}=Q_{d}=-1\/3$ for the present application) \nand $g_{M}^{(1)}=g_{M}^{(2)}\\equiv g_{M}$ are the parameters \nconcerning the anomalous magnetic moment of quarks. \nIt is worth mentioning that there exist two kinds of current, \n$j^{(\\rm convec)}$ and $j^{(\\rm spin)}$ \n(denoting the convection and spin currents respectively), \nwhich are conserved independently. \n\nSubstituting Eq.(\\ref{Eq:5}) and \n$A_{\\mu}(x_{i}) =(1\/\\sqrt{2q_{0}})\ne_{\\mu}^{*}(q) \ne^{-iq_{\\mu} x_{i\\mu}} \n$\ninto \n\\begin{eqnarray}\n\\label{Eq:12}\n-\\int d^4 x_{1} \\int d^{4}x_{2}~\\langle f|\n j_{1\\mu}(x_{1},x_{2}) A_{\\mu}(x_{1})|i \\rangle\n+(1\\leftrightarrow 2) ~~~~~~\n\\end{eqnarray}\nand equating it with the matrix element Eq.(\\ref{Eq:9}), \nwe obtain a formula to calculate \nthe decay amplitudes as \n\\begin{eqnarray}\n\\label{Eq:14}\n\\mathcal{M}_{fi}\n&=&-eQ_{1}\\int d^4 x \n\\langle\\bar{\\Phi}^{(-)}_{F}(v_{F}, x)\n\\left( P_{I\\mu}+P_{F\\mu} \\right. \\nonumber \\\\\n&&- \\frac{d}{2m}\ni\\frac{\\stackrel{\\leftrightarrow}{\\partial}}{\\partial x_{\\mu}}\n+\\frac{d}{2m}g_{M}\\sigma_{\\mu\\nu}iq_{\\nu} )\n\\Phi^{(+)}_{I}(v_{I},x) \\rangle e^{-i\\frac{2m}{d}q_{\\rho}x_{\\rho}}e^{*}_{\\mu}(q)\n+\\left(1\\leftrightarrow 2\\right), \n\\end{eqnarray}\nwhere $e^{}_{\\mu}(q)$ is the polarization vector \nof the photon. \nBy using this formula, we can derive \nthe covariant expressions of invariant amplitudes for the respective radiative \ntransitions summarized in Table \\ref{tab:1}. \n\\section{Numerical Predictions}\n\nTaking the following values of parameters, \n\\begin{itemize}\n\\item $M_{0}=0.75$ GeV and $\\Omega=1.11$ GeV$^2$ \\cite{Oda:1999},\nwhich give $M_{1}=1.29$ GeV and $M_{2}=1.67$ GeV \n\\item $g_{M}=0.82$, determined from the experimental width of\n$\\rho^{\\pm} \\to \\pi^{\\pm} \\gamma$ \n\\end{itemize} \nwe calculate numerical values of the radiative decay widths.\nThe results are shown in Table \\ref{tab:3} in comparison with \nexperiment and other quark-model predictions. \nFrom this table we can see that our results are in fairly \nagreement with experimental values. \n\\begin{table}[htbp]\n\\vspace{-1em}\n\\caption{Invariant amplitudes and formulas of decay width in the relevant transitions. \nHere $\\epsilon_{\\mu\\cdots}$ and $|\\boldsymbol{q}_{\\gamma}|$ denote the\npolarization tensors for initial mesons and physically emitted photon momentum \nat the rest frame of initial mesons, respectively, $\\omega=-v_{I\\mu} v_{F\\mu}=({M_{I}^2+M_{F}^2})\/(2M_{I}M_{F})$, and $F$ is \ndefined in the text. }\n\\begin{tabular}{lll}\n\\hline\n$j^{(\\rm spin)}_{\\mu}$ Process &\n$\\mathcal{M}_{fi}\n$\n&\\ $\\Gamma$\\\\\n\\hline\n${}^{3}S_{1}\\to {}^{1}S_{0} \\gamma$ \\ &\n$-e g_{\\rho} \\varepsilon_{\\mu\\nu\\rho\\alpha}e^{*}_{\\mu}(q)q_{\\nu}\n\\epsilon_{\\rho}(P_{I}) P_{I\\alpha}$\n&$\\frac{4\\alpha}{3} |\\boldsymbol{q}_{\\gamma}|^3 g_{\\rho}^2 $\n\\\\\n \n${}^{3}P_{J=1,2} \\to {}^{1}S_{0} \\gamma$ \\ &\n$ieg_{a}\\varepsilon_{\\mu\\nu\\rho\\alpha}e^{*}_{\\mu}(q)q_{\\nu}\n\\epsilon_{\\rho\\lambda}(P_{I})q_{\\lambda} P_{\\alpha} $\n&$\\frac{\\alpha}{2(2J+1)} \n|\\boldsymbol{q}_{\\gamma}|^5 g_{a}^2$\\\\\n\n${}^{3}D_{J} \\to {}^{1}S_{0} \\gamma$ \\ &\n$eg_{\\rho_{D}}\\varepsilon_{\\mu\\nu\\rho\\alpha}q_{\\nu}\\epsilon_{\\rho\\sigma\\lambda}(P_{I})q_{\\sigma}\nq_{\\lambda}P_{I\\alpha}e^{*}_{\\mu}(q)$\n&$C_{J}\\alpha |\\boldsymbol{q}_{\\gamma}|^{7}g^2_{\\rho_{D}}$ \\\\\n\n\\hline\n$j^{(\\rm convec.)}_{\\mu}$ Process &\n$\\mathcal{M}_{fi}$\n&\\ $\\Gamma$\\\\\n\\hline\n\n\n${}^{1}P_{1} \\to {}^{1}S_{0} \\gamma$ \\ &\n$i eg_{b_{1}} \\epsilon_{\\mu}(P_{I})e_{\\mu}^{*}(q)$\n&$\\frac{\\alpha}{3M^2_{I}} |\\boldsymbol{q}_{\\gamma}| g^2_{b_{1}}$\n\\\\\n\n${}^{1}D_{2} \\to {}^{1}S_{0} \\gamma$ \\ &\n$eg_{\\pi_{2}}\n(e^{*}_{\\mu}(q)\\epsilon_{\\mu\\kappa}(P_{I}) q_{\\kappa}\n+q_{\\kappa} \\epsilon_{\\kappa\\mu}(P_{I}) e^{*}_{\\mu}(q))$&\n$\\frac{\\alpha}{10M^2_{I}} |\\boldsymbol{q}_{\\gamma}|^3 g^2_{\\pi_{2}}$\\\\\n\\hline\n{Coupling parameters:}&\\\\\n\\multicolumn{3}{l}{$\ng_{\\rho}= g_{M} (Q_{u}+Q_{d})\\left(\\frac{1}{2M_{I}}+\\frac{1}{2M_{F}}\\right)F, \\ \\ \ng_{b_{1}}=\\frac{1}{\\sqrt{2\\beta^2}} \\frac{Q_{u}+Q_{d}}{2}\\left(M_I^2-M_F^2\\right) \\frac{1+\\omega}{2}F, \\ \\ \n$}\\\\\n\n\\multicolumn{3}{l}{$\ng_{a}=g_{M}(Q_{u}-Q_{d})\n(\\frac{1}{2M_{I}}+\\frac{1}{2M_{F}}) \\frac{1}{\\sqrt{2\\beta^2}}\\frac{M_{I}}{\\omega M_{F}}F,\\ \\ \\ \ng_{\\pi_{2}}=\\frac{1}{2\\beta^2}\n\\frac{Q_{u}-Q_{d}}{4}\\left(M_I^2-M_F^2\\right)\\frac{1+\\omega}{2\\omega}\\frac{{M_{I}}}{{M_{F}}}\nF, \\ \\ \\ \n$}\\\\\n\n\\multicolumn{3}{l}{$\ng_{\\rho_{D}}=g_{M}\\frac{Q_{u}+Q_{d}}{2}\n\\frac{1}{2\\beta^2}\\frac{M_{I}}{\\omega^2 M_{F}}\n\\left(\\frac{1}{2M_{I}}+\\frac{1}{2M_{F}}\\right) F, \\ \\ \\ \\ C_{J}=\\{ \\frac{4}{105},\\frac{1}{15}, \n\\frac{1}{45}\\} \\ {\\rm for} \\ \\{ {}^{3}D_{3}, {}^{3}D_{2}, {}^{3}D_{1} \\}\n$}\\\\\n\n\\hline\n\\end{tabular}\n\\label{tab:1}\n\\end{table}\n\\begin{table}[t]\n\\caption{Calculated widths in \ncomparison with experiment and other models. }\n\\begin{tabular}{llllll}\n\\hline\n&&\\multicolumn{4}{c}{$\\Gamma$\/keV}\\\\\n\\cline{3-6}\nProcess &$|\\boldsymbol{q}_{\\gamma}|$\/GeV&This work & Experiment \\cite{PDG:2012}&\nRef.~\\cite{Godfrey-Isgur:1985}&Ref.~\\cite{Rosner:1981}\n\\\\\n\\hline\n\\underline{$j^{(\\rm spin)}$ Process} &&&\\\\\n$\\rho^{\\pm}\\to \\pi^{\\pm} \\gamma$ &0.375&\n68 (input)& \n68 $\\pm$ 7&68 (input)\n&-\n\\\\\n\n$a_{1}(1260)^{\\pm} \\to \\pi^{\\pm} \\gamma$&0.608&\n278&640$\\pm$246~\\cite{Zielinski:1984}\n&314&(1.0-1.6)$\\cdot 10^3$\n\\\\\n$a_{2}(1320)^{\\pm} \\to \\pi^{\\pm} \\gamma$&0.652&\n237&287$\\pm$ 30\n&302(input)&375$\\pm$50\n\\\\\n\n$\\rho_{3}^{}(1690)^{\\pm} \\to \\pi^{\\pm} \\gamma$&0.839 &21&\n-&-&-\\\\\n\n$\\rho_{}(1700)^{\\pm} \\to \\pi^{\\pm} \\gamma$&0.854&\n14&\n-&-&-\\\\\n\n\\underline{$j^{(\\rm convec.)}$ Process} &&&\\\\\n$b_{1}^{}(1235)^{\\pm} \\to \\pi^{\\pm} \\gamma$ &0.607&57.8 (Model A) \n&230$\\pm$60~\\cite{Collick:1984}\n&397&184$\\pm$30\n\\\\\n&&\n116~ (Model B)&\n&&\n\\\\\n\n$\\pi_{2}(1670)^{\\pm} \\to \\pi^{\\pm} \\gamma$&0.829&\n335~ (Model A)&-\n&-&-\\\\\n&&\n521~ \n(Model B)\n&-&-&-\\\\\n\n\\hline\n\\end{tabular}\n\\label{tab:3}\n\\end{table}\n\\section{Discussion}\nLet us discuss in detail about obtained results. \nAt first we would like to remark that \na relativistic effect of the transition form factor $F$ \ncommonly contained in all decay amplitudes, \nplays a important role throughout this work. \nIt is given by the overlapping integral of 4-dimensional \nHO functions \n\\vspace{-1em}\n\\begin{eqnarray}\nF &=&\\int d^4 x~ f_{0}(v_{F}, x) f_{0}(v_{I}, x)e^{-i\\frac{q_{\\mu}x_{\\mu}}{2}}=\n\\frac{1}{\\omega}\\exp\\left(-\\frac{1}{16\\beta^2}\\frac{2v_{I\\mu}q_{\\mu}v_{F\\nu}q_{\\nu}}{\\omega}\\right)\n\\nonumber\\\\\n&\\stackrel{\\boldsymbol{P}_{I}= \\boldsymbol{0}}{=}& \\frac{M_{F}}{(P_{F})_{0}}\\exp\\left(-\\frac{|\\boldsymbol{q}|^2}{16\\beta^2} \n2(1+\\frac{|\\boldsymbol{P_{F}}|}{(P_{F})_{0}}) \\right). \n\\label{Eq:18}\n\\end{eqnarray}\nResulting ratio is $\\Gamma(a_{2}(1320)^{\\pm} \\to \\pi^{\\pm} \\gamma)\/\n\\Gamma(\\rho^{\\pm} \\to \\pi^{\\pm} \\gamma)=3.49$, \nwhich is independent from choice of the parameter $g_{M}$, \nin satisfactory agreement with data: $4.22\\pm 0.876$. Thus it turns out that \nour form factor yields a desirable damping effect. \n\nOn the other hand, concerning the $a_{1}(1260)$($b_{1}(1235)$) \n$\\to\\pi \\gamma$ process, \nit seems that our results are \npoorly-fitted to the experiments \\cite{Zielinski:1984,Collick:1984}, \nbeing both only measurements quoted in Ref. \\cite{PDG:2012}. \nHowever, it was pointed out that in Ref.~{\\cite{SELEX:2001}}, \nno evidence of the $a_{1}$ was found in the another \ncharge-exchange photo-production experiment \\cite{Condo:1993}, \nwhile a clear $a_{2}(1320)$ signal was observed. \nThis results suggest that total width of $a_{1}(1260)$ meson is \nextremely large or radiative $\\pi\\gamma$ width is rather small. \n\\footnote{Related works treating these \nmesons from another viewpoint \nhave been reported \\cite{Roca:2004, Nagahiro:2008, \nNagahiro:2009, Molona:2011}. }\nIn any case, a high-statistics confirmation of these process by \nthe COMPASS would be desirable. \n\nConcerning the $j^{\\rm (convec.)}$ process, \nthere are two possible ways for evaluating numerically \nthe factor $M_{I}^2-M_{F}^2$ in the couplings $g_{b_{1}}$ and $g_{\\pi_{2}}$: \nIn Table \\ref{tab:3}, we apply $M_{I}^2-M_{F}^2=N\\Omega $ in the Model A by using \nEq.(\\ref{Eq:4}), while $M_{I}^2-M_{F}^2=2M_{I} |\\boldsymbol{q}_{\\gamma}|$ in the Model B.\nIn both cases, it is shown that \nthe radiative decay widths of $\\pi_{2}(1670)$ into $\\pi \\gamma$\nhave large fraction, predicted\\footnote{\nHere we use $\\Gamma_{\\rm tot}(\\pi_{2}(1670))=260\\pm 9~ {\\rm MeV}$ \ntaken from Ref.~\\cite{PDG:2012}. \n} as \n${\\rm Br}(\\pi_{2}(1670)^{\\pm} \\to \\pi^{\\pm} \\gamma)\n=\\Gamma (\\pi_{2}(1670)^{\\pm} \\to \\pi^{\\pm} \\gamma)\/\n\\Gamma_{\\rm tot}(\\pi_{2}(1670))=(2.0 \\pm 0.069)\\times10^{-3} $ for the Model A, while \n$(1.3 \\pm 0.045) \\times10^{-3} $ for the Model B, respectively. \nThus it must have been detected at the COMPASS Primakoff \nmeasurements with \nenough statistics. \n\nIn summary we have investigated the radiative $\\pi \\gamma$ decays \nof the excited light-quark mesons in the COQM. \nRadiative decay widths of $D$-wave excited mesons are predicted. \nWe expect that forthcoming experiments at COMPASS will make these predictions to verify.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and results}\n\nIn this paper we investigate a new method to obtain overlap\nidentities for the spin glass models. The strategy we use is the\nexploitation of a bound on the fluctuations of a quantity that\ncompares a system with some Gaussian disorder with the system at a\nflipped ($J\\to -J$) disorder. While the disordered averages are\nsymmetric by interaction flip due to the symmetry of the\ndistribution, the difference among them is an interesting random\nvariable whose variance can be shown to grow at most like the volume\n(for extensive quantities).\n\nThe identities are then deduced using some form of integration by parts in the same\nperspective in which they appear from stochastic stability \\cite{AC} or the Ghirlanda Guerra\nmethod \\cite{GG} in the mean field case or, more recently, for short range finite\ndimensional models \\cite{CGi, CGi2} (see also \\cite{T,B}).\n\nThe interest of obtaining further information from the method of the\nidentities lies on the fact that they provide a constraint for the\noverlap moments (or their distribution) and have the potential to\nreduce its degrees of freedom toward, possibly, a {\\it mean field}\nstructure like it is expected to have the Sherrington Kirkpatrick\nmodel.\n\n\nMore specifically the results of this paper consist of overlap\nidentities for the quenched state which interpolate between a\nGaussian spin glass and the system where the couplings in a\nsubvolume (possibly coinciding with the whole volume) have been\nflipped. The interpolation is obtained by extending to the whole\ncircle the Guerra Toninelli interpolation \\cite{GT}. The bounds are\nderived from the concentration properties of the difference of the\nfree energy per particle in the two settings, original and flipped.\n\nAs an example, one may consider the result which is stated in\n\\cite{NS} (and quoted there as proved by Aizenman and Fisher) for\nthe difference $\\Delta F$ between the free energy of the\nEdwards-Anderson model on a $d$-dimensional lattice with linear size\n$L$ and a volume $L^d$ when going from periodic to antiperiodic\nboundary conditions on the hyperplane which is orthogonal to (say)\nthe $x$-direction. The mentioned property is a bound for the\nvariance of this quantity which grows no more than the volume of the\nhyperplane. Such an upper bound is equivalent to a bound for the\nstiffness exponent $\\theta \\le (d-1)\/2$ \\cite{SY,BM,FH} (See also the \ndiscussion of that exponent in \\cite{vE}). Although\nthat bound is not expected to be saturated we prove here that it\nimplies an identity for the equilibrium quantities. When expressed\nin terms of spin variables some of the overlap identities that we\nfind generalize the structure of truncated correlation function that\nappear in \\cite{Te} whose behaviour in the volume is related to the\nlow temperature phase properties of the model. Consequences of our\nbound can also be seen at the level of the difference of internal\nenergies. This second set of identities contains as a particular\ncase some of the Ghirlanda-Guerra identities.\n\nA quite interesting result, from the mathematical physics perspective, is\nprovided by the analysis of the identities for the random field model without interaction.\nWe show here that the new set of identities that we derive (and explicitly test) when\nconsidered together with the Ghirlanda Guerra ones provide a simple\nproof of triviality of the model i.e. the proof that the overlap is a non fluctuating\nquantity. We plan to apply the same method to the investigation of the random\nfield model with ferromagnetic interactions.\n\nThe plan of the paper is the following. In the next section\nwe define the setting of Gaussian spin glasses that we consider.\nThen in section 3 we prove a lemma for the first two moments\nof the difference of free energies. This is obtained by studying\na suitable interpolation on the circle for the linear combination\nof two independent Hamiltonians. Section 4 contains the proof\nof the concentration of measure results. The main results are\ngiven in section 5 and 6, where the new overlap identities\nare stated. Finally in section 7 we study the case of the\nrandom field model and shows how to derive the triviality of the model\nwithout making use of the explicit solution.\n\n\\section{Definitions}\n\\label{def}\n\nWe consider a disordered model of Ising configurations\n$\\sigma_n=\\pm 1$, $n\\in \\Lambda\\subset {\\cal L}$ for some subset\n$\\Lambda$ (volume $|\\Lambda|$) of a lattice ${\\cal L}$. We denote\nby $\\Sigma_\\Lambda$ the set of all $\\sigma=\\{\\sigma_n\\}_{n\\in \\Lambda}$, and\n$|\\Sigma_\\Lambda|=2^{|\\Lambda|}$. In the sequel the\nfollowing definitions will be used.\n\n\\begin{enumerate}\n\n\\item {\\it Hamiltonian}.\\\\ For every $\\Lambda\\subset {\\cal L}$ let\n$\\{H_\\Lambda(\\sigma)\\}_{\\sigma\\in\\Sigma_N}$\nbe a family of\n$2^{|\\Lambda|}$ {\\em translation invariant (in distribution)\nGaussian} random variables defined, in analogy with \\cite{RU}, according to\nthe general representation\n\\begin{equation}\nH_{\\Lambda}(\\sigma) \\; = \\; - \\sum_{X\\subset \\Lambda} J_X\\sigma_X\n\\label{hami}\n\\end{equation}\nwhere\n\\begin{equation}\n\\sigma_X=\\prod_{i\\, \\in X}\\sigma_i \\; ,\n\\end{equation}\n($\\sigma_\\emptyset=0$) and the $J$'s are independent Gaussian variables with\nmean\n\\begin{equation}\\label{mean_disorder}\n{\\rm Av}(J_X) = 0 \\; ,\n\\end{equation}\nand variance\n\\begin{equation}\\label{var_disorder}\n{\\rm Av}(J_X^2) = \\Delta^2_X \\; .\n\\end{equation}\nGiven any subset $\\Lambda^\\prime\\subseteq \\Lambda$, we also write\n\\begin{equation}\nH_\\Lambda(\\sigma)=H_{{\\Lambda}^\\prime}(\\sigma)+H_{{\\Lambda \\setminus {\\Lambda}^\\prime}}(\\sigma)\n\\end{equation}\nwhere\n\\begin{equation}\\label{hamiloc}\nH_{{\\Lambda}^\\prime}(\\sigma)= - \\sum_{X\\subset \\Lambda^\\prime} J_X\\sigma_X,\\quad H_{{\\Lambda \\setminus {\\Lambda}^\\prime}}(\\sigma)= - \\sum_{X\\subset \\Lambda \\atop X \\subset \\hspace{-0.18cm} \\setminus \\hspace{0.1cm} \\Lambda^\\prime} J_X\\sigma_X\\;,\n\\end{equation}\nand\n\\begin{equation}\nH_{{\\Lambda, {\\Lambda}^\\prime}}(\\sigma)=-H_{{\\Lambda}^\\prime}(\\sigma)+H_{{\\Lambda \\setminus {\\Lambda}^\\prime}}(\\sigma)\n\\end{equation}\nwill denote the Hamiltonian with the $J$ couplings inside the region $\\Lambda^\\prime$ that have been flipped.\n\\item {\\it Average and Covariance matrix}.\\\\\nThe Hamiltonian $H_{\\Lambda}(\\sigma)$ has covariance matrix\n\\begin{eqnarray}\\label{cov-matr}\n\\label{cc}\n{\\cal C}_\\Lambda (\\sigma,\\tau) \\; &:= &\\;\n\\mbox{Av}{H_\\Lambda(\\sigma)H_\\Lambda (\\tau)}\n\\nonumber\\\\\n& = & \\; \\sum_{X\\subset\\Lambda}\\Delta^2_X\\sigma_X\\tau_X\\, .\n\\end{eqnarray}\nBy the Schwarz inequality\n\\begin{equation}\\label{sw}\n|{\\cal C}_\\Lambda (\\sigma,\\tau)| \\; \\le \\; \\sqrt{{\\cal C}_\\Lambda\n(\\sigma,\\sigma)}\\sqrt{{\\cal C}_\\Lambda (\\tau,\\tau)} \\; = \\;\n\\sum_{X\\subset\\Lambda}\\Delta^2_X\n\\end{equation}\nfor all $\\sigma$ and\n$\\tau$.\n\\item {\\it Thermodynamic Stability}.\\\\\nThe Hamiltonian (\\ref{hami}) is thermodynamically stable if there exists\na constant $\\bar{c}$ such that\n\\begin{eqnarray}\n\\label{thst}\n\\sup_{\\Lambda\\subset {\\cal L}}\n\\frac{1}{|\\Lambda|}\\sum_{X\\subset\\Lambda}\\Delta^2_X\n\\; & \\le & \\; \\bar{c} \\; < \\; \\infty\\;.\n\\end{eqnarray}\nThanks to the relation (\\ref{sw}) a thermodynamically stable model fulfills the bound\n\\begin{eqnarray}\n\\label{pippo}\n{\\cal C}_\\Lambda (\\sigma,\\tau) \\; & \\le & \\; \\bar{c} \\, |\\Lambda|\n\\end{eqnarray}\nand has an order $1$ normalized covariance\n\\begin{eqnarray}\\label{norm_covar_matrix}\nc_{\\Lambda}(\\sigma,\\tau) \\; & : = & \\; \\frac{1}{|\\Lambda|}{\\cal C}_\\Lambda (\\sigma,\\tau)\\;.\n\\end{eqnarray}\n\\item {\\it Random partition function}.\n\\begin{equation}\\label{rpf}\n{\\cal Z}_\\Lambda(\\beta) \\; := \\; \\sum_{\\sigma \\in \\,\\Sigma_\\Lambda}\ne^{-\\beta{H}_\\Lambda(\\sigma)}\\equiv \\sum_{\\sigma \\in \\,\\Sigma_\\Lambda}e^{-\\bH_{{\\Lambda}^\\prime}(\\sigma)-\\bH_{{\\Lambda \\setminus {\\Lambda}^\\prime}}(\\sigma)}\n\\; ,\n\\end{equation}\n\\begin{equation}\\label{rpfflip}\n{\\cal Z}_{\\Lambda,\\Lambda^\\prime}(\\beta) \\; := \\; \\sum_{\\sigma \\in \\,\\Sigma_\\Lambda}\ne^{-\\betaH_{{\\Lambda, {\\Lambda}^\\prime}}(\\sigma)}\\equiv\\sum_{\\sigma \\in \\,\\Sigma_\\Lambda}\ne^{\\bH_{{\\Lambda}^\\prime}(\\sigma)-\\bH_{{\\Lambda \\setminus {\\Lambda}^\\prime}}(\\sigma)}\n\\; .\n\\end{equation}\n\\item {\\it Random free energy\/pressure}.\n\\begin{equation}\\label{rfe}\n-\\beta {\\cal F}_\\Lambda(\\beta) \\; := \\; {\\cal P}_\\Lambda(\\beta) \\; := \\; \\ln {\\cal Z}_\\Lambda(\\beta)\n\\; ,\n\\end{equation}\n\\begin{equation}\\label{rfeflip}\n-\\beta {\\cal F}_{\\Lambda,\\Lambda^\\prime}(\\beta) \\; := \\; {\\cal P}_{\\Lambda,\\Lambda^\\prime}(\\beta) \\; := \\; \\ln {\\cal Z}_{\\Lambda,\\Lambda^\\prime}(\\beta)\n\\; .\n\\end{equation}\n\n\\item {\\it Random internal energy}.\n\\begin{equation}\\label{rie}\n{\\cal U}_\\Lambda(\\beta) \\; := \\; \\frac{\\sum_{\\sigma \\in \\,\\Sigma_\\Lambda}\nH_{\\Lambda}(\\sigma)e^{-\\beta{H}_\\Lambda(\\sigma)}}{\\sum_{\\sigma \\in \\,\\Sigma_\\Lambda}\ne^{-\\beta{H}_\\Lambda(\\sigma)}}\n\\; ,\n\\end{equation}\n\\begin{equation}\\label{rie_flip}\n{\\cal U}_{\\Lambda,\\Lambda^\\prime}(\\beta) \\; := \\; \\frac{\\sum_{\\sigma \\in \\,\\Sigma_\\Lambda}\nH_{{\\Lambda, {\\Lambda}^\\prime}}(\\sigma)e^{-\\betaH_{{\\Lambda, {\\Lambda}^\\prime}}(\\sigma)}}{\\sum_{\\sigma \\in \\,\\Sigma_\\Lambda}\ne^{-\\betaH_{{\\Lambda, {\\Lambda}^\\prime}} (\\sigma)}}\n\\; .\n\\end{equation}\n\\item {\\it Quenched free energy\/pressure}.\n\\begin{equation}\n-\\beta F_{\\Lambda}(\\beta) \\; := \\; P_{\\Lambda}(\\beta) \\; := \\; \\mbox{Av}{ {\\cal P}_{\\Lambda}(\\beta) }\\; .\n\\end{equation}\n\\begin{equation}\n-\\beta F_{\\Lambda,\\Lambda^\\prime}(\\beta) \\; := \\; P_{\\Lambda,\\Lambda^\\prime}(\\beta) \\; := \\; \\mbox{Av}{ {\\cal P}_{\\Lambda,\\Lambda^\\prime}(\\beta) }\\; .\n\\end{equation}\n\\item $R$-{\\it product random Boltzmann-Gibbs state}.\n\\begin{equation}\n\\Omega_{\\Lambda} (-) \\; := \\;\n\\sum_{\\sigma^{(1)},...,\\sigma^{(R)}}(-)\\,\n\\frac{\ne^{-\\beta[H_\\Lambda(\\sigma^{(1)})+\\cdots\n+H_\\Lambda(\\sigma^{(R)})]}}{[{\\cal Z}_{\\Lambda}(\\beta)]^R}\n\\; .\n\\label{omega}\n\\end{equation}\n\\item {\\it Quenched equilibrium state}.\n\\begin{equation}\n\\<-\\>_{\\Lambda} \\, := \\mbox{Av}{\\Omega_{\\Lambda} (-)} \\; .\n\\end{equation}\n\\item\\label{obs} {\\it Observables}.\\\\\nFor any smooth bounded function $G(c_{\\Lambda})$\n(without loss of generality we consider $|G|\\le 1$ and no assumption of\npermutation invariance on $G$ is made) of the covariance matrix\nentries we introduce the random (with respect to $\\<-\\>$)\n$R\\times R$ matrix of elements $\\{q_{k,l}\\}$ (called {\\it generalized\noverlap}) by the formula\n\\begin{equation}\n\\ \\; := \\; \\mbox{Av}{\\Omega (G(c_{\\Lambda}))} \\; .\n\\end{equation}\nE.g.:\n$G(c_\\Lambda)= c_{\\Lambda}(\\sigma^{(1)},\\sigma^{(2)})c_{\\Lambda}(\\sigma^{(2)}\n,\\sigma^{(3)})$\n\\begin{equation}\n\\ \\; = \\;\n\\mbox{Av}{\\sum_{\\sigma^{(1)},\\sigma^{(2)},\\sigma^{(3)}}\nc_{\\Lambda}(\\sigma^{(1)},\\sigma^{(2)})c_{\\Lambda}(\\sigma^{(2)},\\sigma^{(3)})\n\\;\\frac{\ne^{-\\beta[\\sum_{i=1}^{3}H_\\Lambda(\\sigma^{(i)})]}}{[{\\cal Z}(\\beta)]^3}}\n\\end{equation}\n\\end{enumerate}\n{\\it Remark: In the following, whenever there is no risk of confusion,\nthe volume dependency in the quenched state or in the thermodynamic quantities\nwill be dropped.}\n\\section{Preliminary: interpolation on the circle}\\label{sec_interp}\nLet $\\xi = \\{\\xi_i\\}_{1\\leq i\\leq n}$ and $\\eta = \\{\\eta_i\\}_{1\\leq i\\leq n}$ be two independent\nfamilies of centered Gaussian random variables, each having covariance matrix $\\mathcal{C}$, i.e.\n\\begin{eqnarray}\n{\\rm Av}(\\xi_i\\xi_j) & = & \\mathcal{C}_{i,j} \\nonumber\\\\\n{\\rm Av}(\\eta_i\\eta_j) & = & \\mathcal{C}_{i,j} \\nonumber\\\\\n{\\rm Av}(\\xi_i\\eta_j) & = & 0.\n\\end{eqnarray}\nConsider the following linear combination of $\\xi$ and $\\eta$\n\\begin{eqnarray}\\label{interp_cerchio}\nx_i(t) & = & f(t) \\xi_i + g(t)\\eta_i \\nonumber\n\\end{eqnarray}\nwhere the parameter $t\\in [a,b]\\subset \\Bbb R$ and the two functions $f(t),g(t)$ take real values\nsubject to the constraint\n\\begin{equation}\n\\label{cerchio}\nf(t)^2+g(t)^2=1\\;.\n\\end{equation}\nChosing $f(t)=\\cos (t)$, $g(t)=\\sin (t)$ we obtain:\n\\begin{eqnarray}\nx_i(t) = \\cos(t)\\, \\xi_i +\\sin(t)\\, \\eta_i \\nonumber.\n\\end{eqnarray}\nBecause of the constraint (\\ref{cerchio}), for any given time $t\\in[a,b]$, the new centered Gaussian family\n$x(t)=\\{x_i(t)\\}_{1\\leq i\\leq n}$ has the same covariance structure of $\\xi$ and $\\eta$:\n\\begin{eqnarray} \\label{cov-tt}\n{\\rm Av}(x_i(t)x_j(t)) & = & \\mathcal{C}_{i,j},\\nonumber\n\\end{eqnarray}\nand hence the same distribution, independently of $t$ (i.e. $x(t)$ is a stationary Gaussian process).\n\nIn the abstract set-up described above, we regard $x(t)$ as an interpolating Hamiltonian which\nis a linear combination of the random Hamiltonians $\\xi$ and $\\eta$, with $t$-dependent weights\nthat are the coordinates of a point on the circle of unit radius.\\footnote{It is probably worth noting that\nany other parametrization of the unit circle would lead to the same expression as in (\\ref{varX1}).} We introduce the interpolating random pressure\n\\footnote{Here, in defining the interpolating (random) pressure, we absorb the temperature in the Hamiltonian.}:\n\\begin{equation}\n{\\cal P}(t) = \\ln Z(t)= \\ln\\sum_{i=1}^n e^{x_i(t)}\\;,\n\\end{equation}\nand the notation $\\_{t,s}$ to denote the expectation of the\ncovariance matrix in the deformed quenched state constructed from two\nindependent copies with Boltzmann weights $x(t)$, respectively $x(s)$.\nNamely:\n\\begin{eqnarray}\n\\_{t,s}={\\rm Av}\\sum_{i,j=1}^n\n\\mathcal{C}_{i,j} \\frac {e^{x_i(t)+x_j(s)}}{Z(t)Z(s)}\\;.\n\\end{eqnarray}\nThe definition is extended in the obvious way to more than two copies.\nWe will be interested in the random variable given by the difference of the pressures evaluated\nat the boundaries values\n\\begin{equation}\n{\\cal X}(a,b) = {\\cal P}(b)-{\\cal P}(a)\\;.\n\\end{equation}\nThe following lemma gives an explicit expression for the first two moments\nof this random variable.\n\\begin{lemma}\n\\label{lemma1}\nFor the random variable ${\\cal X}(a,b)$ defined above we have\n\\begin{equation}\\label{vaXzero}\n{\\rm Av}({\\cal X}(a,b))=0\n\\end{equation}\nand\n\\begin{eqnarray}\\label{varX1}\n{\\rm Av}[({\\cal X}(a,b))^2] & = &\n\\int_{a}^{b} \\int_{a}^{b} dt\\; d s\\; k_1(t,s)\\_{t,s}\n\\\\\n&-& \\int_{a}^{b} \\int_{a}^{b} dt\\; d s\\;\nk_2(t,s) \\left [ \\_{t,s} -\n2\\_{s,t,s} +\\_{t,s,s,t}\\right]\n\\nonumber \\end{eqnarray} with \\begin{eqnarray}\nk_1(t,s)=\\cos(t-s),\\qquad\\qquad\nk_2(t,s)=\\sin^2(t-s). \\end{eqnarray}\n\\end{lemma}\n{\\bf Proof.}\nThe stationarity of the Gaussian process $x(t)$ implies that ${\\rm Av} ({\\cal P}(t))$\nis independent of $t$, this proves (\\ref{vaXzero}).\nAs far as the computation of the second moment is concerned, starting from\n\\begin{eqnarray}\n{\\rm Av}({\\cal X}(a,b)) = \\int_a^b dt{\\rm Av}({\\cal P}'(t)) = \\int_a^b dt \\sum_{i=1}^n {\\rm Av}\\left(x'_i(t)\n\\frac{e^{x_i(t)}}{Z(t)}\\right)\n\\end{eqnarray}\nwe have\n\\begin{eqnarray}\n\\label{var}\n{\\rm Av}[({\\cal X}(a,b))^2] & = & \\int_a^b dt \\int_a^b ds {\\rm Av}({\\cal P}'(t){\\cal P}'(s)) \\nonumber \\\\\n& = & \\int_a^b dt \\int_a^b ds \\sum_{i,j=1}^n {\\rm Av}\\left(x'_i(t)x'_j(s) \\frac{e^{x_i(t)+x_j(s)}}{Z(t)Z(s)}\\right).\n\\end{eqnarray}\nThe computation of the average in the rightmost term of the previous formula, which is reported in Appendix 1,\ngives\n\\begin{eqnarray}\\label{brontolo}\n& &{\\rm Av}\\left(x'_i(t)x'_j(s) \\frac{e^{x_i(t)+x_j(s)}}{Z(t)Z(s)}\\right)=\n\\cos(t-s) \\langle C_{1,2}\\rangle_{t,s}\\nonumber\\\\\n&-&\\sin^2(s-t)\\left( _{t,s}-2_{t,s,t}+_{t,s,s,t}\\right)\n\\end{eqnarray}\nproving (\\ref{varX1}).\n\\hfill \\ensuremath{\\Box}\n\\section{Bound on the fluctuations of the free energy difference}\nIt is a well established fact that the random free energy per particles of Gaussian spin glasses\nsatisfies concentration inequalities, implying in particular self-averaging.\nHere we prove that the same result holds for the variation in the random free energy\n(or equivalently the random pressure)\n\\begin{equation}\n{\\cal X}_{\\Lambda,{\\Lambda}^\\prime}={\\cal P}_\\Lambda- {\\cal P}_{\\Lambda,\\Lambda^\\prime}\n\\end{equation}\ninduced by the change of the signs of the interaction in the subset $\\Lambda^\\prime\\subseteq \\Lambda$.\nIn general, the fact that the random free energy per particle concentrates around its mean\nas the system volume increases of the free energy can be obtained either by martingales\narguments \\cite{PS,CGi2} or by general Gaussian concentration of measure \\cite{T,GT2}.\nHere we follow the second approach. Our formulation applies to both\nmean field and finite dimensional models and, for instance, includes\nthe non summable interactions in finite dimensions \\cite{KS} and the\n$p$-spin mean field model as well as the REM and GREM models.\n\\par\\noindent\nBefore stating the result, it is useful to notice that, as a consequence of the symmetry\nof the Gaussian distribution, the variation of the random pressure has a zero average:\n\\begin{equation}\\label{av_dif_fre_en}\n{\\rm Av}({\\cal X}_{\\Lambda,{\\Lambda}^\\prime})=0\\; .\n\\end{equation}\n\\begin{lemma}\n\\label{martin}\nFor every subset $\\Lambda^\\prime\\subset \\Lambda$ the disorder fluctuation of the free energy variation ${\\cal X}_{\\Lambda,{\\Lambda}^\\prime}$ satisfies the following inequality: for all $x > 0$\n\\begin{equation}\n\\label{concentra}\n\\mathbb P\\,\\left(|{\\cal X}_{\\Lambda,{\\Lambda}^\\prime}| \\ge x\\right) \\;\\le\\; 2 \\exp{\\left(-\\frac{x^2}{8 \\pi \\beta^2 \\bar{c} |\\Lambda^\\prime|}\\right)}\n\\end{equation}\nwith $\\bar{c}$ the constant in the thermodynamic stability condition (cfr. Eq. (\\ref{thst})).\nThe variance of the free energy variation satisfies the bound\n\\begin{equation}\\label{sav}\nVar({{\\cal X}_{\\Lambda,{\\Lambda}^\\prime}}) \\; = \\; \\mbox{Av}{{\\cal X}_{\\Lambda,{\\Lambda}^\\prime}^{2}} \\;\\le\\; 16\\, \\pi\\, \\bar{c}\\, \\beta^2\\, {|\\Lambda^\\prime|}\n\\end{equation}\n\\end{lemma}\n{\\bf Proof.}\nConsider an $s > 0$ and let $x>0$. By Markov inequality, one has\n\\begin{eqnarray}\n\\label{markov}\n\\mathbb P\\,\\left\\{ {\\cal X}_{\\Lambda,{\\Lambda}^\\prime} \\ge x\\right\\}\n& = &\n\\mathbb P\\,\\left\\{\\exp [s {\\cal X}_{\\Lambda,{\\Lambda}^\\prime} ] \\ge \\exp (sx)\\right\\}\n\\nonumber\\\\\n&\\le & \\mbox{Av}{\\exp[s{\\cal X}_{\\Lambda,{\\Lambda}^\\prime}]} \\; \\exp(-sx)\n\\end{eqnarray}\nTo bound the generating function\n\\begin{equation}\n\\mbox{Av}{\\exp[s {\\cal X}_{\\Lambda,{\\Lambda}^\\prime}]}\n\\end{equation}\none introduces, for a parameter $t\\in [0,\\frac \\pi 2]$, the following interpolating partition functions:\n\\begin{eqnarray}\nZ^+(t)=\\sum_{\\sigma \\in \\,\\Sigma_\\Lambda} e^{-\\beta \\cos t\\, H^{(1)}_{\\Lambda^\\prime}(\\sigma)-\\beta H^{(3)}_{\\Lambda \\setminus \\Lambda^\\prime}(\\sigma)-\\beta \\sin t\\, H^{(2)}_{\\Lambda^\\prime}(\\sigma)},\\\\\nZ^-(t)=\\sum_{\\sigma \\in \\,\\Sigma_\\Lambda} e^{\\beta \\cos t\\, H^{(1)}_{\\Lambda^\\prime}(\\sigma)-\\beta H^{(3)}_{\\Lambda \\setminus \\Lambda^\\prime}(\\sigma)+\\beta \\sin t\\, H^{(2)}_{\\Lambda^\\prime}(\\sigma)}\\;.\n\\end{eqnarray}\nHere the hamiltonians $H^{(1)}_{\\Lambda^\\prime}(\\sigma)$, $H^{(2)}_{\\Lambda^\\prime}(\\sigma)$, $H^{(3)}_{\\Lambda \\setminus \\Lambda^\\prime}(\\sigma)$, defined according to (\\ref{hamiloc}),\ndepend on three independent copies $\\{J^{(1)}_X\\}_{X\\subset \\Lambda},\\; \\{J^{(2)}_X\\}_{X\\subset \\Lambda},\\; \\{J^{(3)}_X\\}_{X\\subset \\Lambda}$\nof the Gaussian disorder characterized by (\\ref{mean_disorder}),(\\ref{var_disorder}). Now we are ready to consider the\ninterpolating function\n\\begin{equation}\n\\phi(t) = \\ln Av_3 Av_1\\left\\{\\exp\\left(s \\; Av_2\\left\\{\\ln \\frac{{Z^+}(t)}{{Z^-}(t)} \\right\\}\\right)\\right\\}\\;,\n\\end{equation}\nwhere $Av_1\\{-\\}$, $Av_2\\{-\\}$ and $Av_3\\{-\\}$ denote expectation with\nrespect to the three independent families of Gaussian variables $J_X$.\nIt is immediate to verify that\n\\begin{equation}\n\\phi(0) = \\ln {\\rm Av} \\exp [s \\;{{\\cal X}_{\\Lambda,{\\Lambda}^\\prime}}] \\;,\n\\end{equation}\nand, using (\\ref{av_dif_fre_en}),\n\\begin{equation}\n\\phi\\left (\\frac \\pi 2\\right ) = 0 \\;.\n\\end{equation}\nThis implies that\n\\begin{equation}\n\\label{bush}\n\\mbox{Av}{\\exp[s{\\cal X}_{\\Lambda,{\\Lambda}^\\prime}]} = e^{\\phi(0)-\\phi(\\frac \\pi 2)} = e^{-\\int_{0}^{\\frac \\pi 2} \\phi'(t) dt}.\n\\end{equation}\nOn the other hand, the function $\\phi'(t)$ can be easily bounded. Defining\n\\begin{equation}\nK(t) = \\exp\\left (s \\; Av_2\\left\\{\\ln \\frac{{Z^+}(t)}{{Z^-}(t)} \\right\\}\\right )\n\\end{equation}\nthe derivative is given by\n\\begin{equation}\\\n\\phi'(t)=\\phi'_+(t)+\\phi'_-(t)\n\\end{equation}\nwhere\n\\begin{equation}\\label{der_phi}\n\\phi'_+(t)=\\frac{s Av_3 Av_1\\left\\{K(t)Av_2\\left\\{\\frac{{Z^+(t)}^\\prime}{Z^+(t)}\\right\\}\\right\\}}{Av_3 Av_1 \\left\\{K(t)\\right\\}},\\quad\n\\phi'_-(t)=-\\frac{s Av_3 Av_1\\left\\{K(t)Av_2\\left\\{\\frac{{Z^-(t)}^\\prime}{Z^-(t)}\\right\\}\\right\\}}{Av_3 Av_1 \\left\\{K(t)\\right\\}}.\n\\end{equation}\nThe first term in the derivative is\n\\begin{equation}\n\\phi'_+(t)=\\frac{sAv_3 Av_1\\left\\{K(t)Av_2\\left\\{\\sum_{\\sigma \\in \\,\\Sigma_\\Lambda} p^+_t(\\sigma)\\left[\\beta\\sin t\\,H^{(1)}_{\\Lambda^\\prime}(\\sigma)-\\beta\\cos t\\, H^{(2)}_{\\Lambda^\\prime}(\\sigma) \\right]\\right\\}\\right\\}}\n{Av_3 Av_1 \\left\\{K(t)\\right\\}}\n\\end{equation}\nwhere\n\\begin{equation}\np^+_t(\\sigma) =\\frac{e^{-\\beta \\cos t\\, H^{(1)}_{\\Lambda^\\prime}(\\sigma)-\\beta H^{(3)}_{\\Lambda \\setminus \\Lambda^\\prime}(\\sigma)-\\beta \\sin t\\, H^{(2)}_{\\Lambda^\\prime}(\\sigma)}}{Z^+(t)}\n\\end{equation}\nApplying the integration by parts formula, a simple computation gives\n\\begin{eqnarray}\\label{der_primo_addendo}\n& &\\beta\\sin t\\,Av_3Av_1\\left\\{K(t)\n\\; Av_2\\left\\{\\sum_{\\sigma} p^+_t(\\sigma)H^{(1)}_{\\Lambda^\\prime}(\\sigma) \\right \\} \\right\\}\\nonumber\\\\\n& = &\n-s \\beta^2 \\sin t\\, \\cos t\\, Av_3Av_1 \\left\\{K(t) \\sum_{X\\subset \\Lambda^\\prime} \\Delta_X^2[s^+_t(X)^2+\ns^+_t(X)s^-_t(X)]\\right \\}\n\\nonumber\\\\\n&-&\n\\beta^2 \\sin t\\, \\cos t\\, Av_3Av_1\\left\\{K(t)\n \\; Av_2\\left\\{\\sum_{\\sigma}{\\cal C}_{\\Lambda^\\prime}(\\sigma,\\sigma) p^+_t(\\sigma) \\right\\} \\right\\}\n\\nonumber\\\\\n& + &\n\\beta^2 \\sin t\\, \\cos t\\, Av_3Av_1\\left\\{K(t)\n \\; Av_2\\left\\{\\sum_{\\sigma,\\tau}{\\cal C}_{\\Lambda^\\prime}(\\sigma,\\tau) p^+_t(\\sigma)p^+_t(\\tau) ) \\right\\} \\right\\}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{der_secondo_addendo}\n& &-\\beta \\cos t\\, Av_3Av_1\\left\\{K(t)\n \\; Av_2\\left\\{\\sum_{\\sigma} p^t(\\sigma) \\;H^{(2)}_{\\Lambda^\\prime}(\\sigma) \\right\\} \\right\\}\\nonumber\\\\\n&= &\n\\beta^2 \\sin t\\, \\cos t\\, Av_3Av_1\\left\\{K(t)\n \\; Av_2\\left\\{\\sum_{\\sigma}{\\cal C}_{\\Lambda^\\prime} (\\sigma,\\sigma)p^+_t(\\sigma) \\right\\} \\right\\}\n\\nonumber\\\\\n& - &\n\\beta^2\\sin t\\, \\cos t\\, Av_1\\left\\{K(t)\n \\; Av_2\\left\\{\\sum_{\\sigma,\\tau}{\\cal C}_{\\Lambda^\\prime}(\\sigma,\\tau) p^+_t(\\sigma)p^+_t(\\tau) ) \\right\\} \\right\\}\n\\end{eqnarray}\nwhere\n\\begin{equation}\ns^+_t(X)=Av_2\\left\\{\\sum_{\\sigma}\\sigma_Xp^+_t(\\sigma)\\right\\},\\quad s^-_t(X)=Av_2\\left\\{\\sum_{\\sigma}\\sigma_Xp^-_t(\\sigma)\\right\\}\n\\end{equation}\nand\n\\begin{equation}\np^-_t(\\sigma) =\\frac{e^{\\beta \\cos t\\, H^{(1)}_{\\Lambda^\\prime}(\\sigma)-\\beta H^{(3)}_{\\Lambda \\setminus \\Lambda^\\prime}(\\sigma)+\\beta \\sin t\\, H^{(2)}_{\\Lambda^\\prime}(\\sigma)}}{Z^-(t)}\\;.\n\\end{equation}\nTaking the difference between (\\ref{der_primo_addendo}) and (\\ref{der_secondo_addendo}) one finds that\n\\begin{equation}\\label{der_phi_primo}\n\\phi'_+(t)=-s^2 \\beta^2 \\sin t\\, \\cos t\\,\\frac{ Av_3Av_1 \\left\\{K(t) \\sum_{X\\subset \\Lambda^\\prime} \\Delta_X^2[s^+_t(X)^2+\ns^+_t(X)s^-_t(X)]\\right \\}}\n{Av_3Av_1\\{K(t)\\}}\\;.\n\\end{equation}\nWith a similar computation one obtains also\n\\begin{equation}\\label{der_phi_primo}\n\\phi'_-(t)=-s^2 \\beta^2 \\sin t\\, \\cos t\\,\\frac{ Av_3Av_1 \\left\\{K(t) \\sum_{X\\subset \\Lambda^\\prime} \\Delta_X^2[s^-_t(X)^2+\ns^+_t(X)s^-_t(X)]\\right \\}}\n{Av_3Av_1\\{K(t)\\}}\\;,\n\\end{equation}\nthen we conclude that\n\\begin{equation}\n\\phi'(t)=-s^2 \\beta^2 \\sin t\\, \\cos t\\,\\frac{ Av_3Av_1 \\left\\{K(t) \\sum_{X\\subset \\Lambda^\\prime} \\Delta_X^2[s^+_t(X)+\ns^-_t(X)]^2\\right \\}}\n{Av_3Av_1\\{K(t)\\}}\\;.\n\\end{equation}\nUsing the thermodynamic stability condition (\\ref{pippo}), this yields\n\\begin{equation}\n|\\phi'(t)| \\le 4\\,\\beta^2\\,\\bar{c}\\,s^2\\,|\\Lambda^\\prime|\n\\end{equation}\nfrom which it follows, using (\\ref{bush})\n\\begin{equation}\n\\mbox{Av}{\\exp[s{\\cal X}_{\\Lambda,{\\Lambda}^\\prime}]} \\le \\exp\\left(2 \\pi \\beta^2\\,\\bar{c}\\, s^2\\,|\\Lambda^\\prime|\\right).\n\\end{equation}\nInserting this bound into the inequality (\\ref{markov}) and optimizing over $s$\none finally obtains\n\\begin{equation}\n\\mathbb P\\,\\left({\\cal X}_{\\Lambda,{\\Lambda}^\\prime} \\ge x\\right) \\;\\le\\;\\exp{\\left(-\\frac{x^2}{8\\pi\\,\\beta^2\\,\\bar{c}\\, |\\Lambda^\\prime|}\\right)}.\n\\end{equation}\nThe proof of inequality (\\ref{concentra}) is completed by observing\nthat one can repeat a similar computation for $\\mathbb P\\,\\left({\\cal X}_{\\Lambda,{\\Lambda}^\\prime} \\le -x\\right)$.\nThe result for the variance (\\ref{sav}) is then immediately proved, thanks to (\\ref{av_dif_fre_en}),\nusing the identity\n\\begin{equation}\n\\mbox{Av}{{\\cal X}_{\\Lambda,{\\Lambda}^\\prime}^2} = 2 \\int_{0}^{\\infty} x\\; \\mathbb P(|{\\cal X}_{\\Lambda,{\\Lambda}^\\prime}| \\ge x) \\;dx\\;.\n\\end{equation}\n\\hfill \\ensuremath{\\Box}\n\\section{Overlap identities from the difference of free energy}\nWe are now ready to state our first result.\n\\begin{theorem}\n\\label{cesare}\nGiven a volume $\\Lambda$, consider the Guassian spin glass\nwith Hamiltonian (\\ref{hami}). For a subvolume $\\Lambda^\\prime\\subseteq\\Lambda$ and\na parameter $t\\in [0,\\pi]$, let\n$$\n\\omega_t(-) = \\sum_{\\sigma} (-) e^{-H_\\sigma(t)}\/Z(t)\n$$\nwith\n$$\nH_{\\sigma}(t) = \\cos(t) H^{(1)}_{\\Lambda^\\prime}(\\sigma)+ \\sin(t) H^{(2)}_{\\Lambda^\\prime}(\\sigma)+ H_{{\\Lambda \\setminus {\\Lambda}^\\prime}}(\\sigma)\n$$\nbe the Boltzmann-Gibbs state which interpolates between the system with Gaussian\ndisorder and the system with a flipped disorder in the region $\\Lambda^\\prime$\n( $H^{(1)}_{\\Lambda^\\prime}$ and $H^{(2)}_{\\Lambda^\\prime}$ are two independent copies\nof the Hamiltonian in the subvolume $\\Lambda^\\prime$,\n$H_{{\\Lambda \\setminus {\\Lambda}^\\prime}}(\\sigma)$ is the Hamiltonian in the remaining part of the volume,\nthey are all independent). Then, the following identities hold\n\\begin{equation}\n\\label{id-freeenergy}\n\\lim_{\\Lambda,\\Lambda^\\prime \\nearrow \\Bbb Z^d}\n\\int_{0}^{\\pi}\\int_{0}^{\\pi} dt\\; ds\\; \\sin^2 (s-t) \\left [ \\<(c^{\\Lambda^\\prime}_{1,2})^2\\>_{t,s} - 2\\_{s,t,s}\n+\\_{t,s,s,t}\\right] = 0\n\\end{equation}\nwhere $\\<(c^{\\Lambda^\\prime}_{1,2})^2\\>_{t,s}$ (and analogously for the other terms)\nis the overlap of region $\\Lambda^\\prime\\subseteq\\Lambda$ in the quenched state\nconstructed form the interpolating Boltzmann-Gibbs state, e.g.\n$$\n\\<(c^{\\Lambda^\\prime}_{1,2})^2\\>_{t,s} = {\\rm Av}(\\omega_t\\omega_s (c^2_{\\Lambda^\\prime}(\\sigma,\\tau)))\\;.\n$$\n\\end{theorem}\n\n\\noindent\n{\\bf Proof:}\nThe proof is obtained from a suitable combination\nof the results in the previous sections. For a parameter\n$t\\in [0,\\pi]$ we consider the interpolating random pressure\n\\begin{equation}\n{\\cal P}(t) = \\ln \\sum_{\\sigma\\in\\Sigma_{\\Lambda}} e^{x_{\\sigma}(t)+ H_{{\\Lambda \\setminus {\\Lambda}^\\prime}}(\\sigma)}\n\\end{equation}\nwhere\n$$\nx_{\\sigma}(t) = \\cos(t) H^{(1)}_{\\Lambda^\\prime}(\\sigma)+ \\sin(t) H^{(2)}_{\\Lambda^\\prime}(\\sigma)\n$$\nwith $H^{(1)}_{\\Lambda^\\prime}(\\sigma), H^{(2)}_{\\Lambda^\\prime}(\\sigma)$ two independent copies of\nthe Hamiltonian for the subvolume $\\Lambda^\\prime\\subseteq\\Lambda$.\nThe boundaries values give the random pressure of the original system\nwhen $t=0$ and the random pressure of the system with the couplings $J$ flipped\non the subvolume $\\Lambda^\\prime$ when $t=\\pi$, i.e.\n$$\n{\\cal P}(0) = {\\cal P}_{\\Lambda}\\;,\n$$\n$$\n{\\cal P}(\\pi) = {\\cal P}_{\\Lambda,\\Lambda^\\prime}\\;.\n$$\nApplication of Lemma \\ref{lemma1} with $\\xi_{\\sigma} = H^{(1)}_{\\Lambda^\\prime}(\\sigma)$ and $\\eta_{\\sigma} = H^{(2)}_{\\Lambda^\\prime}(\\sigma)$\n(the presence of the additional term $H_{{\\Lambda \\setminus {\\Lambda}^\\prime}}(\\sigma)$ in the random interpolating pressure does not change the\nresult in the Lemma, as far as the quenched state is correctly interpreted) gives\n\\begin{eqnarray}\nVar({\\cal P}_{\\Lambda} - {\\cal P}_{\\Lambda,\\Lambda^\\prime})\n& = &\n\\Lambda^\\prime \\int_{0}^{\\pi}\\int_{0}^{\\pi} dt\\; ds\\; \\cos(s-t) \\_{t,s}\n\\\\\n&+& ({\\Lambda^\\prime})^{2}\\int_{0}^{\\pi}\\int_{0}^{\\pi} dt\\; ds\\; \\sin^2 (s-t) \\left [ \\<(c^{\\Lambda^\\prime}_{1,2})^2\\>_{t,s} - 2\\_{s,t,s}\n+\\_{t,s,s,t}\\right]\n\\nonumber\n\\end{eqnarray}\nOn the other hand, Lemma (\\ref{martin}) tell us that $Var({\\cal P}_{\\Lambda} - {\\cal P}_{\\Lambda,\\Lambda^\\prime})$\nis bounded above by a constant times the subvolume $\\Lambda^\\prime$. As a consequence,\nthe statement of the theorem follows.\n\\hfill \\ensuremath{\\Box}\n\n\n{\\it Remark:} When expressed in terms of the spin variables\nthe polynomial in the integral (\\ref{id-freeenergy})\ninvolves generalized truncated correlation functions.\nIndeed, for the model defined in Section 2,\nwe have the following expressions\n\\begin{eqnarray}\n\\omega_{t,s}((C^{\\Lambda^\\prime}_{1,2})^2)=\\sum_{X,Y\\subset \\Lambda^\\prime} \\Delta_X^2 \\Delta_Y^2\n\\omega_t(\\ensuremath{\\sigma}_X^{(1)}\\ensuremath{\\sigma}_Y^{(1)})\\omega_s(\\ensuremath{\\sigma}_X^{(2)}\\ensuremath{\\sigma}_Y^{(2)})\\nonumber\\\\\n\\omega_{s,t,s}(C^{\\Lambda^\\prime}_{1,2}C^{\\Lambda^\\prime}_{2,3})=\\sum_{X,Y\\subset \\Lambda^\\prime} \\Delta_X^2 \\Delta_Y^2\n\\omega_s(\\ensuremath{\\sigma}_X^{(1)})\\omega_t(\\ensuremath{\\sigma}_X^{(2)}\\ensuremath{\\sigma}_Y^{(2)})\\omega_s(\\ensuremath{\\sigma}_Y^{(3)})\\nonumber\\\\\n\\omega_{t,s,s,t}(C^{\\Lambda^\\prime}_{1,2}C^{\\Lambda^\\prime}_{3,4})=\\sum_{X,Y\\subset \\Lambda^\\prime} \\Delta_X^2 \\Delta_Y^2\n\\omega_t(\\ensuremath{\\sigma}_X^{(1)})\\omega_s(\\ensuremath{\\sigma}_X^{(2)})\\omega_s(\\ensuremath{\\sigma}_Y^{(3)})\\omega_t(\\ensuremath{\\sigma}_Y^{(4)})\n\\end{eqnarray}\nthus\n\\begin{eqnarray}\\label{comb_mom_av}\n& &\\omega_{t,s}((c^{\\Lambda^\\prime}_{1,2})^2)-2\\;\\omega_{s,t,s}(c^{\\Lambda^\\prime}_{1,2}c^{\\Lambda^\\prime}_{2,3})+\\omega_{t,s,s,t}(c^{\\Lambda^\\prime}_{1,2}c^{\\Lambda^\\prime}_{3,4})=\\\\\n& &\\frac{1}{|\\Lambda^\\prime|^2}\\sum_{X,Y\\subset \\Lambda^\\prime} \\Delta_X^2 \\Delta_Y^2 \\left [\\omega_{t}(\\ensuremath{\\sigma}_X\\ensuremath{\\sigma}_Y)- \\omega_{t}(\\ensuremath{\\sigma}_X)\\omega_{t}(\\ensuremath{\\sigma}_Y)\\right ]\n\\left [\\omega_{s}(\\ensuremath{\\sigma}_X\\ensuremath{\\sigma}_Y)- \\omega_{s}(\\ensuremath{\\sigma}_X)\\omega_{s}(\\ensuremath{\\sigma}_Y)\\right ]\\nonumber\n\\end{eqnarray}\nwhere replica indices have been dropped.\nFor the Edwards-Anderson model, which is obtained with\n$\\Delta_X^2 = 1$ if $X\\in B'=\\{(n,n^\\prime)\\in \\Lambda^\\prime\\times\\Lambda^\\prime,|n-n^\\prime|=1\\}$ and $\\Delta_X^2=0$\notherwise, the linear combination (\\ref{comb_mom_av}) of the moments of the link-overlap in the region $\\Lambda^\\prime$\nis written in terms of truncated correlation functions, that is\n\\begin{eqnarray}\n& &\\omega_{t,s}((c^{\\Lambda^\\prime}_{1,2})^2)-2\\;\\omega_{s,t,s}(c^{\\Lambda^\\prime}_{1,2}c^{\\Lambda^\\prime}_{2,3})+\\omega_{t,s,s,t}(c^{\\Lambda^\\prime}_{1,2}c^{\\Lambda^\\prime}_{3,4})=\\nonumber\\\\\n& &\\frac{1}{|\\Lambda^\\prime|^2}\\sum_{b,{b^\\prime}\\in B'}\\left [\\omega_{t}(\\ensuremath{\\sigma}_b\\ensuremath{\\sigma}_{b^\\prime})- \\omega_{t}(\\ensuremath{\\sigma}_b)\\omega_{t}(\\ensuremath{\\sigma}_{b^\\prime})\\right ]\n\\left [\\omega_{s}(\\ensuremath{\\sigma}_b\\ensuremath{\\sigma}_{b^\\prime})- \\omega_{s}(\\ensuremath{\\sigma}_b)\\omega_{s}(\\ensuremath{\\sigma}_{b^\\prime})\\right ],\\label{unodueunochi}\n\\end{eqnarray}\nwith $\\ensuremath{\\sigma}_b=\\ensuremath{\\sigma}_n\\ensuremath{\\sigma}_n^\\prime$, if $b=(n,n^\\prime)\\in B'$.\n\\section{Overlap identities from the difference of internal energy}\n\nIn this section we study the change in the internal energy after a\nflip of the couplings. We consider only the case of the flip of all\nthe couplings in the entire volume.\n\nLet us consider two centered gaussian families $\\xi=\\{\\xi_i\\}_{1\\le\ni\\le n}$, $\\eta=\\{\\eta_i\\}_{1\\le i\\le n}$ with covariance structure\ngiven by\n\\begin{eqnarray} \\mbox{Av}{\\xi_i\\xi_j}=\\mbox{Av}{\\eta_i\\eta_j}={\\cal C}_{i,j}\n\\end{eqnarray}\nwith ${\\cal C}_{i,i}=N$. We assume the thermodynamic stability\ncondition to hold. It follows that $N$ is proportional to the\nvolume. For example, in the case of the Edwards-Anderson model on a\n$d$-dimensional lattice we would have $N=d|\\Lambda|$. We introduce\nthe random free energies\n\\begin{eqnarray} {\\cal P}_\\xi(\\beta)=\\ln\nZ_\\xi(\\beta)=\\ln \\sum_i e^{-\\beta \\xi_i},\\quad {\\cal\nP}_\\eta(\\beta)=\\ln Z_\\eta(\\beta)=\\ln \\sum_i e^{-\\beta \\eta_i},\n\\end{eqnarray}\nwith the random Boltzmann-Gibbs state ${\\omega}_{\\xi}(-),{\\omega}_{\\eta}(-)$ and their\nquenched versions:\n\\begin{eqnarray} \\<-\\>_{\\xi}=\\mbox{Av}_\\xi {\\omega}_{\\xi}(-),\\qquad\n\\<-\\>_{\\eta}=\\mbox{Av}_\\eta {\\omega}_{\\eta}(-).\n\\end{eqnarray}\nWith a slight abuse of notation we\nwill use the previous symbols also to denote the product state\nacting on the replicated system. The free energy difference,\nobtained flipping the hamiltonian $\\eta$,\n\\begin{eqnarray} {\\cal X}(\\beta)={\\cal\nP}_\\xi(\\beta)-{\\cal P}_{-\\eta}(\\beta)\\equiv \\ln \\sum_i e^{-\\beta\n\\xi_i} - \\ln \\sum_i e^{\\beta \\eta_i},\n\\end{eqnarray}\nhas a $\\beta$-derivative\ngiven by the difference between the internal energies: \\begin{eqnarray}\n{\\cal X}^\\prime(\\beta)=-{\\omega}_{\\xi}(\\xi)-{\\omega}_{-\\eta}(\\eta). \\end{eqnarray} Using the symmetry of\nthe distribution of $\\eta$, we have the identities\\footnote{ Indeed,\nfrom the symmetry of the gaussian distribution, we have that for any\nfunction $f(\\eta)$ the following equalities hold:\n$\\mbox{Av}_\\eta f(\\eta)=\\mbox{Av}_\\eta f(-\\eta)=\\mbox{Av}_{-\\eta} f(-\\eta)$.\nIn particular if $g$ is a function of the configurations of the replicated system, applying the previous remark\nto $f(\\eta)={\\omega}_{\\eta}(g)$ we obtain:\n$\\_{\\eta}\\equiv \\mbox{Av}_\\eta {\\omega}_{\\eta} (g)= \\mbox{Av}_\\eta {\\omega}_{-\\eta} (g)= \\mbox{Av}_{-\\eta} {\\omega}_{-\\eta} (g)\\equiv \\_{-\\eta}$.\n\nThese properties will be tacitly used several time in this section.}\n\\begin{eqnarray}\n&&\\mbox{Av}_\\xi {\\omega}_{\\xi}(\\xi)=-\\beta(N-\\mbox{Av}_\\xi {\\omega}_{\\xi} (C_{1,2}))=-\\beta(N- \\_\\xi)\\label{en_int_h}\\\\\n&&\\mbox{Av}_\\eta {\\omega}_{-\\eta}(\\eta)=\\beta(N-\\mbox{Av}_\\eta {\\omega}_{-\\eta} (C_{1,2}))=\\beta(N-\n\\_\\eta).\\label{en_int_k}\n\\end{eqnarray}\nThe above formulae show that\nthe disorder average of ${\\cal X}^\\prime(\\beta)$ vanishes\n\\begin{eqnarray}\n\\mbox{Av}_{\\xi,\\eta}({\\cal X}^\\prime(\\beta)) =\\beta (\\_{\\eta}-\n\\_{\\xi})=0,\n\\end{eqnarray}\nsince, obviously, $\\_{\\eta} =\n\\_{\\xi}$. Here $C_{1,2}=\\{\\mathcal{C}_{i,j}\\}_{i,j}$\nrepresents the covariance matrix whose entries are regarded as\nconfigurations of two replicas labeled 1 and 2. Thus, using the\nidentity $\\mbox{Av}_\\eta({\\omega}_{-\\eta}(\\eta)^2)=\\mbox{Av}_\\xi({\\omega}_{\\xi}(\\xi)^2)$, we have that the\nvariance of ${\\cal X}^\\prime(\\beta)$ is given by:\n\\begin{eqnarray}\\label{avhkxprime}\n\\mbox{Av}_{\\xi,\\eta}({\\cal X}^\\prime(\\beta)^2)=2\\mbox{Av}_\\xi({\\omega}_{\\xi}(\\xi)^2)+2\\mbox{Av}_{\\xi,\\eta}({\\omega}_{\\xi}(\\xi){\\omega}_{-\\eta}(\\eta)).\n\\end{eqnarray}\nUsing the integration by parts formula, we obtain that\n\\begin{eqnarray}\\label{av_enintquad}\n\\mbox{Av}_\\xi({\\omega}_{\\xi}(\\xi)^2)=\n\\mbox{Av}_\\xi\\sum_{i,j} C_{i,j}\\frac{e^{-\\beta \\xi_i-\\beta \\xi_j}}{Z_\\xi(\\beta)^2}\n+\\mbox{Av}_\\xi\\sum_{i,j}\\sum_{k,\\ell} C_{i,k} C_{j,\\ell} \\frac{\\partial^2\n}{\\partial \\xi_\\ell \\partial \\xi_k}\\left [ \\frac{e^{-\\beta \\xi_i-\\beta \\xi_j}}{Z_\\xi(\\beta)^2} \\right ].\n\\end{eqnarray}\nThe second term in the right-hand side of the previous formula requires a repeated application of the integration\nby parts formula, which gives:\n\\begin{eqnarray}\n\\mbox{Av}_\\xi\\sum_{i,j}\\sum_{k,\\ell} C_{i,k} C_{j,\\ell} \\frac{\\partial^2\n}{\\partial \\xi_\\ell \\partial \\xi_k}\\left [ \\frac{e^{-\\beta \\xi_i-\\beta \\xi_j}}{Z_\\xi(\\beta)^2}\\right ]\n = \\beta^2N(N-2\\_{\\xi})+\\beta^2\\_{\\xi}\\nonumber\\\\\n-6\\beta^2\\_{\\xi}+6\\beta^2\\_{\\xi}.\\nonumber\n\\end{eqnarray}\nSince the first term in the right-side of (\\ref{av_enintquad}) is quenched average of\n$C_{1,2}$, we conclude that\n\\begin{eqnarray}\n\\mbox{Av}_\\xi({\\omega}_{\\xi}(\\xi)^2)&=&\\+\\beta^2N(N-2\\)+\\beta^2\\\\nonumber\\\\\n&-&6\\beta^2\\+6\\beta^2\\\\label{primo_addendo}\n\\end{eqnarray}\ndropping, here and in what follows, the unessential reference\nto $\\xi$ in the quenched averages. If the two families $\\xi$ and\n$\\eta$ were independent, then in (\\ref{avhkxprime}) the average of\nthe product would factorize $\\mbox{Av}_{\\xi,\\eta}({\\omega}_{\\xi}(\\xi){\\omega}_{-\\eta}(\\eta))=-\\beta^2(N-\n\\)^2$ giving:\n\\begin{eqnarray}\n\\mbox{Av}_{\\xi,\\eta}({\\cal X}^\\prime(\\beta)^2)&=&2\\+2\\beta^2\\left(\n\\-\\^2\\right)+12\\beta^2\n\\left(\\-\\ \\right).\\nonumber\n\\end{eqnarray}\nIn this case the self averaging of the normalized quantity\n${\\cal X}^\\prime(\\beta)^2\/N$ (see Theorem 2) would lead, in the large\nvolume limit $N\\rightarrow \\infty$, to the well known identity\n\\cite{G2}\n\\begin{eqnarray} \\langle c_{1,2}c_{2,3}\\rangle - \\langle\nc_{1,2}c_{3,4}\\rangle = \\frac 1 6 \\left( \\langle c_{1,2}^2\\rangle -\n\\langle c_{1,2} \\rangle^2 \\right).\n\\end{eqnarray}\nHowever, our concern here is the\ncomputation of the quadratic fluctuations of ${\\cal X}^\\prime(\\beta)$\nwhen the sign of a given hamiltonian $\\xi$ is flipped in the whole\nvolume. Therefore we have to set $\\xi$=$\\eta$ in (\\ref{avhkxprime}).\nThe computation requires, once again, the repeated use of the\nintegration by parts formula\n\\begin{eqnarray}\\label{avhxprimehmenoh}\n\\mbox{Av}_\\xi({\\omega}_{\\xi}(\\xi){\\omega}_{-\\xi}(\\xi))&=&\\mbox{Av}_\\xi\\left (\\sum_{i,j} \\xi_i \\xi_j\n\\frac{e^{-\\beta \\xi_i+\\beta\n\\xi_j}}{Z_\\xi(\\beta)Z_\\xi(-\\beta)}\\right)\n=\\mbox{Av}_\\xi\\sum_{i,j} C_{i,j}\\frac{e^{-\\beta \\xi_i+\\beta \\xi_j}}{Z_\\xi(\\beta)Z_\\xi(-\\beta)} \\nonumber \\\\\n&+&\\mbox{Av}_\\xi \\sum_{i,j} \\sum_{k,\\ell} C_{i,k} C_{j,\\ell}\\frac{\\partial^2\n}{\\partial \\xi_\\ell \\partial \\xi_k}\\left [ \\frac{e^{-\\beta \\xi_i+\\beta \\xi_j}}{Z_\\xi(\\beta)Z_\\xi(-\\beta)}\\right ].\n\\end{eqnarray}\nThe average in (\\ref{avhxprimehmenoh}) is expressed through a set of {\\it mixed} quenched state: for instance, the\nfirst term in right-hand side of the previous equation is\n\\begin{eqnarray}\n\\langle C_{1,2} \\rangle_{+,-}=\\mbox{Av}_\\xi \\sum_{i,j} \\mathcal{C}_{i,j} \\frac{e^{-\\beta \\xi_i+\\beta \\xi_j}}{Z_{\\xi}(\\beta)Z_{\\xi}(-\\beta)}.\n\\end{eqnarray}\nGeneralizing the previous definition we have, for instance, that $\\langle - \\rangle_{+,+,-,+}$ represents the thermal\naverage taken with the usual boltzmannfaktor\n(i.e. with the sign $-$ in the exponent) in the first, second and fourth copy, and with the opposite sign in the third one.\nMoreover, the symbol $\\langle -\\rangle_{+,+,+,\\ldots}$, with all the subscripts $+$ (or $-$, because of the symmetry of the\ngaussian distribution),\nis the usual quenched measure $\\langle\\ - \\rangle$. The explicit computation gives:\n\\begin{eqnarray}\n&&\\frac{\\partial^2}{\\partial \\xi_\\ell \\partial \\xi_k}\\left [ \\frac{e^{-\\beta \\xi_i+\\beta \\xi_j}}{Z_\\xi(\\beta)Z_\\xi(-\\beta)}\\right ]\n=-\\beta^2N^2+2\\beta^2N\\langle C_{1,2}\\rangle_{+,+}-\\beta^2\\langle C_{1,2}^2\\rangle_{+,-}\n+2\\beta^2\\langle C_{1,2}C_{2,3}\\rangle_{+,-,+}\\nonumber\\\\\n&&-4\\beta^2\\langle C_{1,2}C_{2,3} \\rangle_{+,+,-} +4\\beta^2\\langle\nC_{1,2}C_{3,4}\\rangle_{+,+,+,-} -\\beta^2\\langle\nC_{1,2}C_{3,4}\\rangle_{+,+,-,-}-\\beta^2\\langle\nC_{1,2}C_{3,4}\\rangle_{+,-,+,-}\\nonumber\n\\end{eqnarray}\nand finally:\n\\begin{eqnarray}\\label{varxprimo} &&\\mbox{Av}_\\xi ({\\cal X}^\\prime(\\beta)^2)\n=2(\\langle C_{1,2}\\rangle_{+,+} + \\langle C_{1,2}\\rangle_{+,-})+2\\beta^2 \\left(\\langle C_{1,2}^2\\rangle_{+,+}-\\langle C_{1,2}^2\\rangle_{+,-} \\right )\\\\\n&-&4\\beta^2\\left ( 3\\langle C_{1,2}C_{2,3}\\rangle_{+,+,+} - \\langle C_{1,2}C_{2,3}\\rangle_{+,-,+}+2\\langle C_{1,2}C_{2,3}\\rangle_{+,+,-}\\right)\\nonumber\\\\\n&+&2\\beta^2\\left(6\\langle C_{1,2}C_{3,4}\\rangle_{+,+,+,+}+4\\langle\nC_{1,2}C_{3,4}\\rangle_{+,+,+,-} -\\langle\nC_{1,2}C_{3,4}\\rangle_{+,+,-,-} -\\langle\nC_{1,2}C_{3,4}\\rangle_{+,-,+,-}\\right).\\nonumber\n\\end{eqnarray}\nIf we choose\nnow $\\xi$ to be the Hamiltonian family defined in section \\ref{def},\nwe obtain the following:\n\\begin{theorem}\nConsider the Guassian spin glass with Hamiltonian $\\xi$ given in (\\ref{hami}). In\nthe infinite volume limit\nand for almost all\nvalues of $\\beta$, we have\n\\begin{eqnarray} \\left [\\langle c_{1,2}^2\n\\rangle_{+,+} -\\langle c_{1,2}^2 \\rangle_{+,- } \\right ]\n-2 \\left [ 3\\langle c_{1,2}c_{2,3} \\rangle_{+,+,+}-\\langle c_{1,2}c_{2,3} \\rangle_{+,-,+}+2\\langle c_{1,2}c_{2,3} \\rangle_{+,+,-}\\right ]\\nonumber \\\\\n+\\left [ 6\\langle c_{1,2}c_{3,4} \\rangle_{+,+,+,+}+4\\langle\nc_{1,2}c_{3,4} \\rangle_{+,+,+,-}-\\langle c_{1,2}c_{3,4}\n\\rangle_{+,+,-,-} -\\langle c_{1,2}c_{3,4} \\rangle_{+,-,+,-}\\right\n]=0\n\\end{eqnarray}\nwhere $\\_{+,-}$ (and analogously for the\nother terms) is the overlap expectation in the quenched state\nconstructed form the mixed Boltzmann-Gibbs state with one copy given by\nthe original system and the other copy given by the flipped systems,\ne.g.\n$$\n\\_{+,-} = {\\rm Av}(\\omega_\\xi\\omega_{-\\xi}\n(c_{\\Lambda}^2(\\sigma,\\tau)))\\;.\n$$\n\\end{theorem}\n{\\bf Proof.}\\par\\noindent The proof is a simple consequence of well\nknown results. The sequence of convex functions ${\\cal\nP}_\\xi(\\beta)\/N$ converges almost everywhere in $J$ to the limiting\nvalue $a(\\beta)$ of its average and the convergence is self\naveraging (i.e. $\\mathrm{Var}({\\cal P}_\\xi(\\beta)\/N)\\rightarrow 0$).\nBy general convexity arguments \\cite{RU}\nit follows that the sequence of derivatives ${\\cal\nP}_\\xi^\\prime(\\beta)\/N$ converges to $u(\\beta)=a^\\prime(\\beta)$\nalmost everywhere in $\\beta$ and also that the convergence is self\naveraging ($\\mathrm{Var}({\\cal P}_\\xi^\\prime(\\beta)\/N)\\rightarrow\n0$, $\\beta$-a.e.) \\cite{S,OTW}. These remarks apply obviously also\nto ${\\cal P}_{-\\xi}(\\beta)\/N$ and to its derivative, with the same\nlimiting functions $a(\\beta)$ and $a^\\prime(\\beta)$. Thus we have\nthat ${\\cal X}(\\beta)\/N={\\cal P}_\\xi(\\beta)\/N-{\\cal P}_{-\\xi}(\\beta)\/N$\nand its derivative ${\\cal X}^\\prime(\\beta)\/N$ vanish a.e. in $J$ in the\nlarge volume limit. Moreover,\n$\\mathrm{Var}({\\cal X}^\\prime(\\beta)\/N)=\\mathrm{Var}({\\cal\nP}_\\xi^\\prime(\\beta)\/N)+\\mathrm{Var}({\\cal\nP}_{-\\xi}^\\prime(\\beta)\/N) -2\\mathrm{cov}\\left({\\cal\nP}_{\\xi}^\\prime(\\beta)\/N,{\\cal P}_{-\\xi}^\\prime(\\beta)\/N\\right)$,\nthus estimating the covariance with the Cauchy-Schwartz inequality\nwe have \\begin{eqnarray} \\mathrm{Var}({\\cal X}^\\prime(\\beta)\/N)\\le 4\\mathrm{Var}({\\cal\nP}_\\xi^\\prime(\\beta)\/N)\\rightarrow 0,\\quad \\beta -a.e. \\end{eqnarray} for\n$N\\rightarrow \\infty$. Therefore, dividing (\\ref{varxprimo}) by\n$N^2$ and taking the limit we obtain the result. \\hfill \\ensuremath{\\Box}\n\n\n\\section{Triviality of the Random Field model}\nIn this section we compute explicitly the expression appearing in\nTheorem \\ref{cesare} \\begin{eqnarray}\\label{unodueuno} \\_{t,s} -2\n\\_{s,t,s}+\\_{t,s,s,t} \\end{eqnarray} in the\nsimple case of the random field. We will show that this linear\ncombination of overlap moments vanishes pointwise for all values of\n$t$ and $s$. We will then deduce the triviality of the order\nparameter for the random field model.\n\nWe consider two families $J_i$ and $\\tilde{J}_i$ for $i=1,\\ldots, N$\nof independent normally distributed centered random variables with\nvariance 1:\n\\begin{eqnarray} {\\rm Av}(J_iJ_j)={\\rm Av}(\\tilde{J}_i\n\\tilde{J}_j)=\\delta_{i,j},\\quad {\\rm Av}(J_i \\tilde{J}_j)=0,\n\\end{eqnarray}\nand the\nrandom field hamiltonians\n\\begin{eqnarray} \\label{nerone}\\xi_\\sigma=\\sum_{i=1}^N\nJ_i\\sigma_i,\\quad \\eta_\\sigma= \\sum_{i=1}^N \\tilde{J}_i\\sigma_i.\n\\end{eqnarray}\nwhere $\\sigma_i=\\pm 1$. We have that $\\xi=\\{\\xi_\\sigma\\}_\\ensuremath{\\sigma}$ and\n$\\eta=\\{\\eta_\\sigma\\}_\\ensuremath{\\sigma}$ are two independent centered Gaussian families\n(each having $n=2^N$ elements indexed by the configurations\n$\\sigma$, $N$ being the volume) and covariance structure given by:\n\\begin{eqnarray}\\label{cov-randomf}\n&{\\rm Av}(\\xi_\\sigma \\xi_\\tau)\\equiv \\mathcal{C}_{\\ensuremath{\\sigma},\\tau}=Nq(\\sigma,\\tau),\\nonumber\\\\\n&{\\rm Av}(\\eta_\\sigma \\eta_\\tau)\\equiv \\mathcal{C}_{\\ensuremath{\\sigma},\\tau}= Nq(\\sigma,\\tau),\\nonumber\\\\\n&{\\rm Av}(\\xi_\\sigma \\eta_\\tau)=0.\n\\end{eqnarray}\nwhere $q(\\sigma,\\tau)$ is the {\\it site overlap} of the two configurations $\\ensuremath{\\sigma}$ and $\\tau$:\n\\begin{eqnarray}\nq(\\sigma,\\tau)=\\frac 1 N \\sum_{i,j=1}^N \\sigma_i\\tau_j\\; .\n\\end{eqnarray}\nThe interpolating Hamiltonian:\n\\begin{eqnarray}\nx_\\ensuremath{\\sigma}(t) = \\cos(t) \\xi_\\ensuremath{\\sigma} + \\sin(t)\\eta_\\ensuremath{\\sigma},\n\\end{eqnarray}\nwhich is a stationary Gaussian process with the same distribution of $\\xi$ and $\\eta$:\n\\begin{eqnarray}\n{\\rm Av}(x_\\sigma(t)x_\\tau(t)) = N q(\\sigma,\\tau)\n\\end{eqnarray}\ndefines the quenched deformed state on the replicated system, whose averages are\ndenoted with the usual notation, e.g. $\\langle - \\rangle_{t,s}$, $\\langle - \\rangle_{s,t,s}\\ \\ldots$ .\n\\begin{theorem}\n\\label{albano} Consider the random field spin glass with Hamiltonian\n(\\ref{nerone}). In the limit $N\\rightarrow \\infty$ and for all\nvalues of $t$ and $s$ we have \\begin{eqnarray}\n\\gamma_1\\_{t,s}+\\gamma_2\\_{t,s}^2+\\gamma_3\\_{s,t,s}+\\gamma_4\\_{t,s,s,t}=0\n\\end{eqnarray} for any choice of real $\\gamma_1,\\gamma_2,\\gamma_3,\\gamma_4$ with\n$\\gamma_1+\\gamma_2+\\gamma_3+\\gamma_4=0$.\\par\\noindent\n\\end{theorem}\n{\\bf Proof} The simple proof relies on\nthe following identities, derived in Appendix 2:\n\\begin{eqnarray}\n\\_{t,s}^2&=&\\sum_{i=1}^N ({\\rm Av}\\big(\\tanh(G_{i}(t))\\tanh(G_{i}(s))\\big))^2+\\mathcal{Q}_N(t,s),\\label{av_c12quad}\\\\\n\\_{t,s}&=&1+\\mathcal{Q}_N(t,s),\n\\label{av_c12q}\\\\\n\\_{s,t,s}&=&\\sum_{i=1}^N{\\rm Av}\\left(\\tanh^2(G_i(s))\\right)+\\mathcal{Q}_N(t,s),\n\\label{av_c12c23}\\\\\n\\_{t,s,s,t}&=&\\sum_{j=1}^N {\\rm Av}\\left(\\tanh^2(G_j(t))\\tanh^2(G_j(s))\\right)+\\mathcal{Q}_N(t,s),\n\\label{av_c12c_34}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\\label{defGi}\nG_i(t)=\\cos(t)J_i + \\sin(t) \\tilde{J}_i,\n\\end{eqnarray}\nand $\\mathcal{Q}_N(t,s)$ is a term of order $N^2$, see (\\ref{defQ}). Thus:\n\\begin{eqnarray}\n&&\\gamma_1\\_{t,s}+\\gamma_2\\_{t,s}^2+\\gamma_3\\_{s,t,s}+\\gamma_3\\_{t,s,s,t}=\\nonumber\\\\\n&&\\gamma_1+\\gamma_2\\sum_{i=1}^N ({\\rm Av}\\big(\\tanh(G_{i}(t))\\tanh(G_{i}(s))\\big))^2 +\\gamma_3\\sum_{i=1}^N{\\rm Av}\\left(\\tanh^2(G_i(s))\\right)\\nonumber\\\\\n&&+\\gamma_4\\sum_{j=1}^N {\\rm Av}\\left(\\tanh^2(G_j(t))\\tanh^2(G_j(s))\\right)\\nonumber\\\\\n&&+(\\gamma_1+\\gamma_2+\\gamma_3+\\gamma_4)\\mathcal{Q}_N(t,s),\\nonumber\n\\end{eqnarray}\ni.e. the linear combination of the covariance matrix moments is of order $N$. Thus, since $|\\tanh (x)|<1$, we have\n\\begin{eqnarray}\n&&\\left | \\gamma_1\\_{t,s}+\\gamma_2\\_{t,s}^2+\\gamma_3\\_{s,t,s}+\\gamma_4\\_{t,s,s,t} \\right |\\nonumber\\\\\n&&\\le |\\gamma_1|+(|\\gamma_2|+|\\gamma_3|+|\\gamma_4|)N,\\nonumber\n\\end{eqnarray}\nwhich can be rewritten, using the overlaps $q_{1,2},\\; q_{2,3},\\; q_{3,4}$ between replicas, as\n\\begin{eqnarray}\n&&\\left | \\gamma_1\\_{t,s}+\\gamma_2\\_{t,s}^2+\\gamma_3\\_{s,t,s}+\\gamma_4\\_{t,s,s,t} \\right |\\nonumber\\\\\n&&\\le \\frac{|\\gamma_2|+|\\gamma_3|+|\\gamma_4|}{N}+\\frac{|\\gamma_1|}{N^2}.\n\\end{eqnarray}\n\\hfill \\ensuremath{\\Box}\n\\par\\noindent\nAmong the relations of theorem \\ref{albano}, in the thermodynamic limit, we find the identity of\ntheorem \\ref{cesare} for the values $\\gamma_1=1, \\gamma_2=0, \\gamma_3=-2, \\gamma_4=1$:\n\\begin{eqnarray}\\label{unodueuno0}\n\\_{t,s}-2\\_{s,t,s}+\\_{t,s,s,t}=0\n\\end{eqnarray}\nand the Ghirlanda-Guerra identities: for $\\gamma_1=1, \\gamma_2=1, \\gamma_3=-2, \\gamma_4=0$ we find\n\\begin{eqnarray}\\label{gg1212}\n\\_{s,t,s}=\\frac 1 2\n\\_{t,s} + \\frac 1 2 \\_{t,s}^2\\; ;\n\\end{eqnarray}\nfor $\\gamma_1=1, \\gamma_2=2, \\gamma_3=0, \\gamma_4=-3$ we find\n\\begin{eqnarray}\\label{gg1323}\n\\_{s,t,s}=\\frac 1 3\n\\_{t,s} + \\frac 2 3 \\_{t,s}^2\\;. \\end{eqnarray}\nUsing\n(\\ref{gg1212}) and (\\ref{gg1323}) we can express (\\ref{unodueuno0})\nas:\n\\begin{eqnarray} \\label{roma}\n\\_{t,s}-2\\_{s,t,s}+\\_{t,s,s,t}=\\frac{1}{3}(\\_{t,s}\n- \\_{t,s}^2)\n\\end{eqnarray}\nThe identity derived from the flip of the\ncoupling thus imply a trivial order parameter distribution. Indeed,\nsince the identity (\\ref{unodueuno0}) is true for every $t$ and $s$\nwe can choose $t=s=0$ and then the interpolating states reduce to\nthe usual quenched Boltzmann-Gibbs state. From Eq. (\\ref{roma}) we\ndeduce a trivial overlap distribution.\n\n\\vspace{1.cm} \\noindent {\\bf Acknowledgements:} We thank S. Graffi\nfor many discussions over stiffness exponent and interpolation,\nG. Parisi for discussions on interface problems on spin glasses and for\nsuggesting to us the check of the new identity on the\nrandom field model. We thank C.Newman and D.Stein for showing\nto us the proof by martingale methods of the property mentioned in the\nintroduction and stated in \\cite{NS}. We also thank Francesco Guerra for \nsuggesting improvements in the presentation and Aernout van Enter for\nuseful comments. The authors acknowledge EURANDOM\nfor the warm ospitality during the \"Workshop on Statistical Mechanics and Applications\",\nthe grants Cultaptation (EU) and Funds for Strategic Research (University of Bologna).\n\n\\section{Appendix 1}\nIn this appendix we will use the Gaussian integration by parts formula for correlated Gaussian random\nvariables $z_1,\\ldots,z_n$:\n\\begin{eqnarray}\\label{gauss-per-parti}\n{\\rm Av}(z_j\\psi(z_1,\\ldots,z_n))=\\sum_{i=1}^n \\mbox{Av}{z_jz_i}\\mbox{Av}{\\frac{\\partial \\psi (z_1,\\ldots,z_n)}{\\partial z_i}},\n\\end{eqnarray}\nto compute the second moment of the pressure difference $\\mathcal{X}(a,b)$. We have to evaluate the\naverage inside the integral (\\ref{var})\n\\begin{eqnarray}\\label{gongolo}\n&&\\sum_{i,j=1}^n {\\rm Av}\\left(x'_i(t)x'_j(s)B(i,j;t,s) \\right)=\\\\\n&=&\\sin(t)\\sin(s) \\sum_{i,j=1}^n {\\rm Av} \\left( \\xi_i \\xi_j B(i, j;t,s) \\right)\n-\\sin(t)\\cos(s) \\sum_{i,j=1}^n {\\rm Av} \\left( \\xi_i \\eta_j B(i,j;t,s) \\right)\\nonumber\\\\\n&-&\\sin(s)\\cos(t) \\sum_{i,j=1}^n{\\rm Av} \\left( \\xi_j \\eta_i B(i,j;t,s) \\right)\\nonumber\n+ \\cos(t)\\cos(s) \\sum_{i,j=1}^n {\\rm Av} \\left( \\eta_i \\eta_j B(i,j;t,s) \\right).\\nonumber\n\\end{eqnarray}\nwhere, for the sake of notation, we have introduced the symbol\n$$\nB(i,j;t,s)=\\frac{e^{x_i(t)+x_j(s)}}{Z(t)Z(s)}.\n$$\nApplying (\\ref{gauss-per-parti}) twice, we obtain\n\\begin{eqnarray}\n\\label{t1pp}\n{\\rm Av} \\left ( \\xi_i \\xi_j B(i,j;t,s)\\right )=\n{\\cal C}_{i,j}{\\rm Av}\\left(B(i,j;t,s)\\right)+\\sum_{k,\\ell=1}^n {\\cal C}_{i,k}{\\cal C}_{j,\\ell}\n{\\rm Av}\\left ( \\frac{\\partial^2}{\\partial \\xi_k \\xi_\\ell} B(i,j;t,s)\\right)\n\\end{eqnarray}\n\\begin{eqnarray}\n\\label{t4pp}\n{\\rm Av} \\left ( \\eta_i \\eta_j B(i,j;t,s)\\right )=\n{\\cal C}_{i,j}{\\rm Av}\\left(B(i,j;t,s)\\right)+\\sum_{k,\\ell=1}^n {\\cal C}_{i,k}{\\cal C}_{j,\\ell}\n{\\rm Av}\\left ( \\frac{\\partial^2}{\\partial \\eta_\\ell \\eta_k}B(i,j;t,s)\\right)\n\\end{eqnarray}\n\\begin{eqnarray}\n\\label{t2pp}\n{\\rm Av} \\left ( \\xi_i \\eta_j B(i,j;t,s)\\right )=\n{\\rm Av} \\left ( \\xi_j \\eta_i B(i,j;t,s)\\right )=\n\\sum_{k,\\ell=1}^n {\\cal C}_{i,k}{\\cal C}_{j,\\ell}\n{\\rm Av}\\left ( \\frac{\\partial^2}{\\partial \\xi_k \\eta_\\ell} B(i,j;t,s)\\right)\n\\end{eqnarray}\nThe combination of the first two terms in the right hand sides of (\\ref{t1pp}) and (\\ref{t4pp}) with the trigonometric\ncoefficients given by (\\ref{gongolo}) produce the quenched expectation $\\cos(t-s) \\langle C_{1,2}\\rangle_{t,s}$.\\par\\noindent\nThe explicit computation of the derivatives is long but not difficult; the result is:\n\\begin{eqnarray}\n&\\frac{\\displaystyle\\partial^2}{\\displaystyle\\partial \\xi_k \\partial \\xi_\\ell}B(i,j;t,s)=B(i,j;t,s)\\left\\{\\cos^2(t) A_1+\\cos^2(s) A_2 +\\cos(t)\\cos(s)(A_3+A_4)\\right\\},\\nonumber\\\\%\\label{derxixiA}\\\\\n&\\frac{\\displaystyle\\partial^2}{\\displaystyle\\partial \\eta_k \\partial \\eta_\\ell}B(i,j;t,s)=B(i,j;t,s)\\left\\{\\sin^2(s) A_1+\\sin^2(s) A_2 +\\sin(t)\\sin(s)(A_3+A_4)\\right\\},\\nonumber\\\\%\\label{deretaetaA}\\\\\n&\\frac{\\displaystyle\\partial^2}{\\displaystyle\\partial \\xi_k \\eta_\\ell}B(i,j;t,s)=B(i,j;t,s)\\left\\{\\sin(t)\\cos(t) A_1 +\\sin(s)\\cos(s) A_2\\right.\\nonumber\\\\\n& \\left. + \\sin(t)\\cos(s) A_3 +\\sin(s)\\cos(t) A_4\\right\\},\\nonumber\n\\end{eqnarray}\nwhere $A_1,A_2,A_3,A_4$ are combinations of Kronecker delta functions depending on the indices $i, j,\\ell,k$ and Boltzmann\nweights for the hamiltonians $x(t)$ and $x(s)$.\\par\\noindent\nUsing the previous formulas for the second derivatives and formulas (\\ref{t1pp}),(\\ref{t4pp}) and (\\ref{t2pp}), we see that the right hand side of (\\ref{gongolo})\ncontains a linear combination of functions $A_j$ with trigonometric coefficients given by the product of four factors\ntaken from $\\{\\cos(t),\\sin(t),\\cos(s),\\sin(s)\\}$. It is not difficult to recognize that the coefficient of $A_3$ is $-\\sin^2(s-t)$\nwhile the other are zero. Thus,\n\\begin{eqnarray}\n\\sum_{i,j=1}^n {\\rm Av}\\left(x'_i(t)x'_j(s)B(i,j;t,s) \\right)=\\cos(t-s) \\langle C_{1,2}\\rangle_{t,s}\n-\\sin^2(s-t)\\mbox{Av}\\sum_{i,j=1}^n\\sum_{k,\\ell=1}^n\n{{\\cal C}_{i,k}{\\cal C}_{j,\\ell}}A_3B(i,j;t,s)\\nonumber\n\\end{eqnarray}\nand since\n$$\nA_3=\\dd{\\ell}{i}\\dd{k}{j}-\\dd{\\ell}{i}\\frac{e^{x_k(s)}}{Z(s)}-\\dd{k}{j}\\frac{e^{x_\\ell(t)}}{Z(t)}+\n\\frac{e^{x_\\ell(t)}}{Z(t)}\\frac{e^{x_k(s)}}{Z(s)}\n$$\nwe obtain\n$$\n\\mbox{Av}\\sum_{i,j=1}^n\\sum_{k,\\ell=1}^n\n{{\\cal C}_{i,k}{\\cal C}_{j,\\ell}}A_3B(i,j;t,s)=_{t,s}-2_{t,s,t}+_{t,s,s,t}\n$$\nwhich proves (\\ref{brontolo}).\n\\section{Appendix 2}\nIn this appendix we prove the identities (\\ref{av_c12quad}),(\\ref{av_c12q}),(\\ref{av_c12c23}),(\\ref{av_c12c_34}).\nRecalling the definition (\\ref{defGi}) of the Gaussian variables $G_i(t)$, we can define the interpolating partition function\n\\begin{eqnarray}\n&Z(t)=\\sum_\\sigma \\exp(x_\\ensuremath{\\sigma}(t))=\\sum_\\sigma \\exp(\\sum_{i=1}^N G_i(t)\\sigma_i).\n\\end{eqnarray}\nA simple computation shows that\n\\begin{eqnarray}\nZ(t)=2^N \\prod_{i=1}^N \\cosh G_i(t).\n\\end{eqnarray}\nFor any integer $M_{t,s}=\\sum_{i=1}^N {\\rm Av}\\big(\\tanh(G_{i}(t))\\tanh(G_{i}(s))\\big),\n\\end{eqnarray}\nthus\n$$\n\\_{t,s}^2=\\sum_{i=1}^N ({\\rm Av}\\big(\\tanh(G_{i}(t))\\tanh(G_{i}(s))\\big))^2+\\mathcal{Q}_N(t,s)\n$$\nwhere\n\\begin{eqnarray}\\label{defQ}\n\\mathcal{Q}_N(t,s)=2\\sum_{1\\le j<\\ell \\le N}{\\rm Av}\\left[ \\tanh(G_j(t))\\tanh(G_j(s))\\right]{\\rm Av} \\left [\\tanh(G_\\ell(t))\\tanh(G_\\ell(s))\\right]\n\\end{eqnarray}\nis a term of order $N^2$. This proves (\\ref{av_c12quad}).\\par\\noindent\nFor the squared overlap the following relation holds\n\\begin{eqnarray}\n\\omega_{t,s}(q_N^2)=\\frac{1}{N}+\\frac{2}{N^2}\\sum_{j=1}^{N-1}(N-j)\\tanh(G_j(t))\\tanh(G_j(s))\\omega_{t,s}^{N-j}(q_{N-1}).\n\\end{eqnarray}\nSince for $M\\le N$\n\\begin{eqnarray}\n\\omega^M_{t,s}(q_M)=\\frac 1 M \\sum_{j=N-M+1}^N\\tanh(G_j(t))\\tanh(G_j(s))\n\\label{av_qM}\n\\end{eqnarray}\nwe can write\n\\begin{eqnarray}\n\\omega_{t,s}(q_N^2)=\\frac{1}{N^2}+\\frac{2}{N^2}\\sum_{1\\le j<\\ell \\le N}\\tanh(G_j(t))\\tanh(G_j(s))\\tanh(G_\\ell(t))\\tanh(G_\\ell(s))\n\\label{boltzAv_c12q}\n\\end{eqnarray}\nand finally\n$$\n\\_{t,s}=1+2\\sum_{1\\le j<\\ell \\le N}{\\rm Av}\\left[ \\tanh(G_j(t))\\tanh(G_j(s))\\tanh(G_\\ell(t))\\tanh(G_\\ell(s))\\right].\n$$\nFrom the independence of the random variables $G_i(t)$ (see (\\ref{defGi})), we have that the average in the right hand side\nof the previous formula factorizes, thus we obtain (\\ref{av_c12q}):\n$$\n\\_{t,s}=1+\\mathcal{Q}_N(t,s).\n$$\n\\par\\noindent\nThe second term in (\\ref{unodueuno}) is computed considering the average of $q_N(\\sigma,\\gamma)q_N(\\gamma,\\tau)$ where $\\gamma,\\sigma,\\tau\\in S$.\nWe have\n\\begin{eqnarray}\n&\\omega^N _{s,t,s}(q_N(\\sigma,\\gamma)q_N(\\gamma,\\tau))=\\frac{\\displaystyle 1}{\\displaystyle N^2}\\tanh^2(G_1(s))\\nonumber\\\\\n&+\\frac{\\displaystyle 2}{\\displaystyle N^2}\n\\tanh(G_1(t))\\tanh(G_1(s))\\sum_{j=2}^N\\tanh(G_j(t))\\tanh(G_j(s))\\nonumber\\\\\n&+\\left(\\frac{\\displaystyle N-1}{\\displaystyle N}\\right)^2\\omega_{s,t,s}^{N-1}(q_{N-1}(\\sigma^\\prime,\\gamma^\\prime)q_{N-1}(\\gamma^\\prime,\\tau^\\prime)).\n\\end{eqnarray}\nwhere $\\sigma^\\prime,\\gamma^\\prime,\\tau^\\prime$ are the restriction of $\\sigma,\\gamma,\\tau$ to $S_{N-1}$. As in the previous cases, iterating\nthis formula and taking into account that $\\omega^1_{s,t,s}(q_1(\\sigma,\\gamma)q_1(\\gamma,\\tau))=\\tanh^2(G_N(s))$ we obtain\n\\begin{eqnarray}\n&\\omega^N _{s,t,s}(q_N(\\sigma,\\gamma)q_N(\\gamma,\\tau))=\\frac{\\displaystyle 1}{\\displaystyle N^2}\\sum_{i=1}^N\\tanh^2(G_i(s))\\nonumber\\\\\n&+\\frac{\\displaystyle 2}{\\displaystyle N^2}\\sum_{j=1}^{N-1}\n\\tanh(G_j(t))\\tanh(G_j(s))\\sum_{\\ell=j+1}^N\\tanh(G_\\ell(t))\\tanh(G_\\ell(s)),\n\\label{boltzAv_c12c23}\n\\end{eqnarray}\nthen\n$$\n\\_{s,t,s}=\\sum_{i=1}^N{\\rm Av}\\left(\\tanh^2(G_i(s))\\right)+\\mathcal{Q}_N(t,s).\n$$\nwhich proves (\\ref{av_c12c23}).\nThe computation of the last term in (\\ref{unodueuno}) is simple because in this case the random product\nmeasure factorizes:\n\\begin{eqnarray}\n\\omega_{t,s,s,t}(q(\\sigma,\\tau)q(\\gamma,\\kappa))=\\omega_{t,s}(q(\\sigma,\\tau))\\omega_{s,t}(q(\\gamma,\\kappa)).\n\\end{eqnarray}\nThen, using (\\ref{avn}) we have\n\\begin{eqnarray}\n\\omega_{t,s,s,t}(q(\\sigma,\\tau)q(\\gamma,\\kappa))=\\frac{1}{N^2}\\sum_{i,j=1}^N \\tanh(G_j(t))\\tanh(G_j(s))\\tanh(G_i(t))\\tanh(G_i(s))\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\_{t,s,s,t}=\\sum_{i,j=1}^N {\\rm Av}\\left(\\tanh(G_j(t))\\tanh(G_j(s))\\tanh(G_i(t))\\tanh(G_i(s))\\right)\n\\end{eqnarray}\nwhich, using the symmetry of $a_{i,j}={\\rm Av}\\left(\\tanh(G_j(t))\\tanh(G_j(s))\\tanh(G_i(t))\\tanh(G_i(s))\\right)$,\ngives (\\ref{av_c12c_34}):\n$$\n\\_{t,s,s,t}=\\sum_{j=1}^N {\\rm Av}\\left(\\tanh^2(G_j(t))\\tanh^2(G_j(s))\\right)+\\mathcal{Q}_N(t,s).\n$$\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nSuppose that a complex algebraic representation $\\rho:G\\to \\mathrm{GL}(V)$ of a complex algebraic Lie group $G$ is given and we want to understand the hierarchy of orbits: which orbit is in the closure of another one. The idea is simple: if $\\eta$ and $\\xi$ are two orbits then we calculate $[\\eta]$---the $G$-equivariant Poincar\\'e dual of $\\overline\\eta$ and restrict it to $\\xi$. Since $[\\eta]$ is supported on $\\overline\\eta$, if $\\xi$ is disjoint from $\\overline\\eta$, then the incidence class $[\\eta]|_\\xi$ is zero. If the $G$-action is rich enough then we have a chance for the opposite implication.\n\nIn this paper \n\\begin{itemize}\n\\item we give a sufficient condition (positivity) for the Incidence Property of an orbit $\\xi$: for any other $G$-invariant subvariety $X\\subset V$ we have that $[X]|_\\xi\\neq 0 \\iff \\xi\\subset [X]$.\n\\item We show that for many interesting cases, e.g. the quiver representations of Dynkin type positivity holds for all orbits.\n\\item We also study the case of singularities where positivity doesn't hold for all orbits.\n\\end{itemize}\n\nOur work was inspired by a conjecture of R. Rim\\'anyi in \\cite{rrthom}, that for singularities of holomorphic maps the incidence classes detect the hierarchy of contact singularity classes.\n\nTo study the conjecture we generalized the notion of incidence class to the general group representation setting. Let $\\rho:G\\to \\mathrm{GL}(V)$ be an algebraic representation of the complex algebraic Lie group $G$ on the vector space $V$. If $X\\subset V$ is a $G$-invariant subvariety of codimension $d$ then we can assign a $G$-equivariant cohomology class $[X]\\in H^{2d}_G(V)\\cong H^{2d}(BG) $. In different areas of mathematics this class has different names e.g. equivariant Poincar\\'e dual, multidegree, Joseph polynomial and---in the case of singularities---Thom polynomial.\n\nEquivariant cohomology shares some properties of ordinary cohomology. For example there is a restriction map: if $Y\\subset V$ is another $G$-invariant subset and $\\alpha\\in H^{*}_G(V)$ then $\\alpha|_Y\\in H^{*}_G(Y)$. So we can define the incidence class $[X]|_Y$ measuring the ``closeness\" of $X$ and $Y$.\n\nThe crucial observation is that if $\\xi$ is an orbit then $[X]|_\\xi$ is an equivariant Poincar\\'e dual itself: Let $x\\in \\xi$ and let us denote by $G_x$ the maximal compact subgroup of the stabilizer group of $x$. Then $[X]|_\\xi\\in H^{*}_G(\\xi)\\cong H^{*}_{G_x}(pt) $. Choosing a $G_x$-invariant normal space $N_x$ to $\\xi$ at $x$ we will show that \n\n{\\bf Proposition \\ref{inc=tp}.} \\ \\ $[X]|_\\xi=[X\\cap N_x]_{G_x}$.\n\nThis observation reduces the problem to finding a sufficient condition for a representation having the property that all non-empty $G$-invariant subvariety has a nonzero equivariant Poincar\\'e dual. This condition can be given in terms of the {\\em weights} of the representation. Suppose that $T\\subset G$ is a maximal torus. Then $V$ splits into 1-dimensional representations of $T$ with weights $w_i\\in \\check T=\\Hom(T,U(1))$. If $T$ has rank $r$ then $\\check T\\cong \\Z^r$.\n\n{\\bf Definition.} The representation $\\rho:G\\to \\mathrm{GL}(V)$ is {\\em positive} if the convex hull of its weights doesn't contain zero.\n\n{\\bf Theorem \\ref{main}.} {\\it If the representation $\\rho:G\\to \\mathrm{GL}(V)$ is positive then all non-empty $G$-invariant subvariety has a nonzero equivariant Poincar\\'e dual.}\n\nFrom this we immediately get:\n\n{\\bf Theorem \\ref{all}.} {\\it All orbits of the representation $\\rho:G\\to \\mathrm{GL}(V)$ have the Incidence Property if for all $x\\in V$ the normal representation of $G_x$ is positive (normal positivity).} \n\nIn section \\ref{sec:examples}. we show a simple condition which implies normal positivity and show that e.g. every quiver representation of Dynkin type satisfy this condition. \n\nAfter this generalization we faced the problem, familiar to many mathematicians, that our theory doesn't apply to the original conjecture. Most orbits of the representations corresponding to the case of singularities are not positive. Many times it has a trivial reason: the stabilizer is trivial. Even if we restrict ourselves to the orbits having at least a $U(1)$ symmetry, there are plenty of non positive examples. Nonetheless, calculations show that in many cases incidence still detects the hierarchy of orbits. We list some examples, counterexamples and conjectures.\n\nThe authors thank M. Kazarian and R. Rim\\'anyi for valuable discussions.\n\n\n\\section{The equivariant Poincar\\'e dual and the Incidence class} \\label{epd&inc}\n\nWe define equivariant cohomology of a $G$-space $X$ via the Borel construction and use cohomology with rational coefficients: $H^*_G(X):=H^*(EG\\times_G X;\\Q)$, where $EG\\to BG$ is the universal principal $G$-bundle over the classifying space $BG$ of $G$. \n\nWe will frequently use the following simple properties of equivariant cohomology:\n\\begin{enumerate}\n\\item $H^*_G(V)\\cong H^*_G(pt)$ if $V$ is contractible.\n\\item $H^*_G(X)\\cong H^*_K(X)$, where $K$ is a maximal compact subgroup of $G$. More generally it is true if $K$ is a deformation retract of $G$.\n\\item We can restrict to an invariant subspace, or to a subgroup. We can also combine them. If the subspace $Y\\subset X$ is invariant for a subgroup $S0$ for all $w\\in W_\\rho$. another possible name would be instability as positivity is equivalent to the condition that all points of $V$ are unstable in the GIT sense.\n\nWe need one more property of the equivariant Poincar\\'e dual (see \\cite[prop 4.1]{patak}) to prove Theorem \\ref{convex}.\n\n\\begin{theorem} \\label{tpposinweights} If \\,$Y\\subset V$ is a $\\T$-invariant subvariety then the class $[Y]\\in H^*_{\\T}(V)$ can be expressed as a non-zero polynomial of the weights of $\\rho$ with non-negative integer coefficients.\n\\end{theorem}\n\nTheorem \\ref{tpposinweights} immediately follows from the fact---well known and widely used by the specialists---that for complex torus actions the notions of equivariant Poincar\\'e dual and {\\em multidegree} coincide. The reason we refer to \\cite{patak} is that we haven't found an older proof in the literature. The proof is based on the identification of the multidegree to the equivariant intersection class \\cite{MR1614555}, and then the later to the equivariant Poincar\\'e dual. See \\cite{four}, page 255, for the first equivalence. As for the second one, there is a cycle map from the equivariant Chow group to the equivariant cohomology ring of $V$ (\\cite{MR1614555}, page 605). Scrutinizing this map, it turns out that it is in fact the ordinary cycle map of the product of some projective spaces, for which it is easy to prove that it is an isomorphism. A basic fact about multidegree is that the multidegree of a $\\T$-invariant subvariety is equal to the sum of multidegrees of $\\T$-invariant sub(vector)spaces (with possible positive integer multiplicities). Noticing that the equivariant Poincar\\'e dual of a subspace is the product of some weights of $\\rho$ we proved Theorem \\ref{tpposinweights}. Here we use the identification of a weight $w:\\T\\to GL(1)$ with the first Chern class $c_1(L_w)$ of the line bundle $L_w=E\\T\\times_w\\C$.\n\n \\begin{proof}[Proof of Theorem \\ref{convex}.] We can think of the weights as linear functionals on $\\R^r$---the real part of the Lie algebra of the torus $\\T$. If the convex hull of $W_\\rho$ doesn't contain 0 then there is a substitution $\\alpha_i\\mapsto z_i\\in\\R$ such that for all $w(\\alpha_1,\\dotsc,\\alpha_r)\\in W_\\rho$ we have that $w(z_1,\\dotsc,z_r)>0$. By Theorem \\ref{tpposinweights} the class $[Y]$ is a non-trivial linear combination of monomials of the weights of $\\rho$ with non-negative coefficients, so if we apply the same substitution we get a positive number, which implies that $[Y]=p(\\alpha_1,\\dotsc,\\alpha_r)\\neq 0$. \n\nOn the other hand, if 0 is in the convex hull, then there is a non-trivial linear combination $\\sum n_iw_i=0$ with $n_i\\geq 0$ integers. It implies that $p=\\prod x_i^{n_i}$ is a non-zero invariant polynomial (the coordinate $x_i$ corresponds to the one-dimensional eigenspace with weight $w_i$). Then $Y:=\\{x\\in V:p(x)=1\\}$ is a non-empty invariant subvariety. We have that $0\\not\\in Y$ so by the fact that $[Y]$ is localized to 0 we get that $[Y]=0$.\n \\end{proof}\n\n Theorem \\ref{convex} immediately implies\n\\begin{theorem}\\label{main} If the representation $\\rho:G\\to \\mathrm{GL}(V)$ is positive then all non-empty $G$-invariant subvariety has a nonzero equivariant Poincar\\'e dual. \n\\end{theorem}\n\nLet us recall now the definition of Incidence Property from the Introduction.\n\\begin{definition} \\label{def:ip} Given a representation $\\rho:G\\to \\mathrm{GL}(V)$ we say that an orbit $\\xi$ has the Incidence Property if for any other $G$-invariant subvariety $X\\subset V$ we have that $[X]|_\\xi\\neq 0 \\iff \\xi\\subset [X]$.\n\\end{definition} \nProposition \\ref{inc=tp} and Theorem \\ref{main} now implies that\n\\begin{theorem}\\label{all} All orbits of the representation $\\rho:G\\to \\mathrm{GL}(V)$ have the Incidence Property if for all $x\\in V$ the normal representation of $G_x$ is positive (normal positivity).\n\\end{theorem}\n\n\\section{Examples of representations with the Incidence Property} \\label{sec:examples}\nFirst we give a stronger condition than positivity, which is easier to check. The examples we have in mind are sub-representations of the left-right action of a subgroup of $\\mathrm{GL}(W)\\times \\mathrm{GL}(W)$ on the vector space $\\Hom(W,W)$. In these cases the weights are of the form $t-s$ where $t$ is a ``target\" weight and $s$ is a ``source\" weight.\n\n \\begin{proposition} \\label{i j\\}$, so by Proposition \\ref{in_s$, then $r \\succ s$. Suppose that $r \\not\\succ s$. Then $n_r \\leq n_s$, and consequently $m_{rs}=\\dim \\mathrm{Ext}_{\\mathbb{C}Q}(l_r, l_s) = \\dim \\mathrm{Ext}_{\\mathbb{C}Q}(\\tau^{n_r}l_r, \\tau^{n_r}l_s) = 0$, since $\\tau^{n_r} l_r$ is projective. This shows that indeed if $m_{rs}\\neq 0$, then $r \\succ s$, and we proved\n\n\\begin{theorem}\\label{quiver} For a representation corresponding to a Dynkin type quiver all orbits satisfy the Incidence Property. \\end{theorem}\n\n\\section{Singularities} \\label{sec:singularities}\n\nIn this section we study incidences of singularities of holomorphic map germs. On singularity we mean a contact orbit of map germs. The equivariant Poincar\\'e dual in this context is called {\\em Thom polynomial}. We use \\cite{rrthom} and \\cite{tpforgroup} as a general reference.\n\nThe space $\\mathcal{E}(n,p)$ of holomorphic map germs from $\\C^n$ to $\\C^p$ and the contact group $\\mathcal{K}(n,p)$ acting on it are infinite dimensional. To define $G$-equivariant cohomology for infinite dimensional groups some extra care is needed as it is explained in \\cite{rrthom}. Our main interest lies in studying {\\em finite codimensional} singularities where a finite dimensional reduction is available.\n\nOnly finite codimensional singularities have Thom polynomials so it is a natural choice. It is possible to restrict to infinite codimensional singularities (for example see \\cite{integer}), but to simplify the situation we stick with the finite codimensional case.\n\nEvery finite codimensional singularity $\\eta$ is {\\em finitely determined}, i.e. there is a $k\\in \\N$ such that for any two germs $f\\in \\eta$ and $g\\in \\mathcal{E}(n,p)$ we have that $g\\in \\eta$ if and only if their $k$\\textsuperscript{th} Taylor polynomials (or $k$-jets) $j^k(f)$ and $j^k(g)$ are contact equivalent. So we can reduce to the finite dimensional space of $k$-jets: $J^k(n,p)\\cong \\bigoplus_{i=1}^k\\Hom(\\sym^i\\C^n,\\C^p)$. We have the truncating map $t_k:\\mathcal{E}(n,p)\\to J^k(n,p)$. Similarly we can define $\\mathcal{K}^k(n,p)$, the group of $k$-jets of the contact group:\n\\[ \\mathcal{K}^k(n,p):=\\{(\\varphi,\\psi)\\in J^k(n,n)^\\times\\times J^k(n+p,n+p)^\\times:\\psi|_{\\C^n\\times0}=\\Id_{\\C^n},\\ \\pi_{\\C^n}\\psi=\\psi\\pi_{\\C^n}\\}. \\]\nThis group acts on $J^k(n,p)$ and for the truncating homomorphism $\\pi_k: \\mathcal{K}(n,p)\\to \\mathcal{K}^k(n,p)$ the map $t_k$ is $\\pi_k$-equivariant. The homomorphism $\\pi_k$ induces isomorphism on equivariant cohomology and for any two $k$-determined singularities $\\eta$ and $\\xi$:\n\\[ [\\eta]|_\\xi=[t_k\\eta]|_{t_k\\xi}. \\]\nWe don't prove this statement---as we avoided defining the left hand side---but use as a motivation to study the representation of the algebraic group $\\mathcal{K}^k(n,p)$ on the jet space $J^k(n,p)$. \n\nWe cannot expect that all orbits of this representation have the Incidence Property. There are many singularities $\\xi$ with small symmetry: the maximal torus of their stabilizer group is trivial. It implies that the restriction map is automatically trivial. To have a chance for the Incidence Property to be satisfied we have to require the existence of at least a $U(1)$ symmetry. Since the diagonal torus $U(1)^n\\times U(1)^p$ is maximal in $\\mathcal{K}^k(n,p)$ and all maximal tori are conjugate, this requirement implies that a representative $f=(f_1,\\dotsc,f_p)$ of the orbit $\\xi$ with diagonal symmetry can be chosen. Then $f$ is a {\\em weighted homogeneous} polynomial, i.e. there are weights $a_1,\\dotsc,a_n,b_1,\\dotsc,b_p\\in \\Z$ such that\n\\[ f_j(t^{a_1}x_1,\\dotsc,t^{a_n}x_n)=t^{b_j}f_j(x_1,\\dotsc,x_n) \\]\nfor $j=1,\\dotsc,p$. These polynomials are also called quasi-homogeneous.\n\nThe Incidence Property can fail even for weighted homogeneous polynomials, which can be detected by restricting them to themselves. According to the definition. Namely if $\\xi$ has the Incidence Property then $e(\\xi)=[\\xi]|_\\xi\\neq0$. \n\n\\begin{example}\\label{exa:e=0} For $\\xi=(x^2+3 yz, y^2+3 xz, z^2+3 xy)$ the normal Euler class $e(\\xi)=0$. \\end{example}\n\\begin{remark}\\label{family}\n This Euler class can be directly calculated using unfolding---see the second part of this section---but there is a deeper reason for the vanishing of $e(\\xi)$. This orbit is a member of a one-parameter family of orbits: $f_\\lambda=(x^2+\\lambda yz, y^2+\\lambda xz, z^2+\\lambda xy)$. (This is a famous example (\\cite{MR0432666}), the smallest codimensional occurence of a family in the equidimensional case.) The tangent direction of the family in the normal space $N_\\xi$ has 0 weight, which implies that $e(\\xi)=0$. In fact we don't know any example when $e(\\xi)=0$ but $\\xi$ is {\\em not in a family}.\n\\end{remark}\nThe following conjecture is a slightly modified version of Rim\\'anyi's from \\cite{rrthom}:\n\\begin{conjecture}The singularity $\\xi\\subset J^k(n,p)$ has the Incidence Property iff $e(\\xi)\\neq0$.\\end{conjecture}\n\n\\begin{remark} Defining the Incidence Property (Definition \\ref{def:ip}) in such a way that we require the condition $e(\\xi)=[\\xi]|_\\xi\\neq0$ to be satisfied seems formal, we know that $\\xi$ is a subset of $\\overline\\xi$ anyway. However the vanishing of $e(\\xi)$ sometimes can be related to the vanishing of a proper incidence $[\\eta]|_\\xi$ where $\\eta\\neq\\xi$ but $\\xi\\subset \\overline\\eta$. Consider the following abstract example:\n\\end{remark}\n\n\\begin{example} Let $GL(1)$ act on $\\C^2$ by $z(x,y)=(x,zy)$. The orbits of this representation are $f_\\lambda=\\{(\\lambda,0)\\}$ and $g_\\lambda=\\{(\\lambda,y):\\, y\\neq0\\}$. The fixed point $f_\\lambda$ has a $U(1)$ symmetry and $f_\\lambda\\subset\\overline{g_\\lambda}$ but the incidence $[g_\\lambda]|_{f_\\lambda}$ is zero. We have a geometric reason for this: $[g_\\mu]|_{f_\\lambda}=0$ if $\\mu\\neq\\lambda$ since $f_\\lambda\\not\\subset\\overline{g_\\mu}$. But $[g_\\mu]$ depends continuously on $\\mu$ so it is constant. We can also calculate directly: $\\overline{g_\\lambda}$ is a line so $[g_\\lambda]|_{f_\\lambda}$ is the Euler class of the complementary invariant line $\\{(\\lambda,0):\\lambda\\in \\C\\}$. But this is exactly the set of fixed points so the Euler class is zero.\n\nWe believe that such situation is not uncommon for singularities, i.e. there are contact orbits $f_\\lambda$ and $g_\\lambda$ depending on the parameter $\\lambda$ continuously such that $\\codim(f_\\lambda)=\\codim(g_\\lambda)+1$, $f_\\lambda\\subset\\overline{g_\\lambda}$ and $f_\\lambda$ has a $U(1)$ symmetry. The same reasoning shows that the incidence $[g_\\lambda]|_{f_\\lambda}$ is zero in this case. Unfortunately we were unable to verify the existence of such an example.\n\\end{example}\n\nTheorem \\ref{main} implies that\n\\begin{proposition} If a singularity $\\xi\\subset J^k(n,p)$ is positive i.e the normal action of the stabilizer group is positive, then $\\xi$ has the Incidence Property.\n\\end{proposition}\n \nComplicated singularities are usually not positive:\n\\begin{example} \\label{exa:xayb} The singularities $(x^a,y^b)$ for $2\\leq a\\leq b$ are positive exactly for $(x^2,y^2)$, $(x^2,y^3)$ and $(x^2,y^4)$. \\end{example}\n\nOn the other hand we know many positive cases:\n\\begin{example}\\label{exa:s2} All $\\Sigma^1$ and $\\Sigma^{2,0}$ singularities (type $A_n,\\ I_{a,b},\\ III_{a,b},\\ IV_a,\\ V_a$, for the notation see \\cite{mather}) are positive.\n\\end{example}\n\\begin{remark} In fact all the incidences of the singularities in Example \\ref{exa:s2} can be explicitly calculated since their adjacencies are well understood (see \\cite{rrthesis}). For example in the equidimensional case for $2\\leq a\\leq b$, $(a,b)\\neq(2,2)$, $2\\leq c\\leq d$ and $c+d2$ then $III_{2,2}$---the orbit of $(x^2,xy,y^2,0,\\dotsc,0)$---is open in $\\Sigma^2$. So for example for the latter case $[III_{2,2}]=[\\Sigma^2]$ (see \\cite{mather}).\n \\item By the Thom-Porteous-Giambelli formula \\cite{porteous} we have\n\\[ [\\Sigma^{2}(2,p)]=\\prod(\\beta_{i}-\\alpha_{1})(\\beta_{i}-\\alpha_{2})\\]\n in terms of Chern roots. In other words $\\alpha_1,\\alpha_2,\\beta_1,\\dotsc,\\beta_p$ denote the generators of $H^2_T(pt)$ where $T=U(1)^2\\times U(1)^p$ is the maximal torus acting on $J^1(2,p)$. (It can also be seen directly since $J^1(2,p)\\cap \\Sigma^{2}(2,p)=0$.)\n \\item Consequently for the weighted homogeneous polynomial $f\\in J^k(2,p)$ with one $U(1)$ symmetry, i.e. with stabilizer subgroup of rank one, and with weights $ a_1,a_2$ and $b_1,\\ldots,b_p$ we have \n\\begin{equation}[\\Sigma^2]|_f=\\prod(b_i-a_1)(b_i-a_2)g^{2p}\\label{resu} \\end{equation} \n where $ g $ is the generator of the cohomology ring $ H^2_{U(1)}(pt;\\Z)\\cong \\Z[g] $ of the symmetry group. In other words the incidence is zero if and only if there is a target weight equal to a source weight. \n\nIf $f$ has symmetry group $U(1) \\times U(1)$, then we can still restrict $[\\Sigma^2]|_f$ further to the subtorus $U$ of $U(1) \\times U(1)$ corresponding to the weighting of $f$. Then we get formula (\\ref{resu}) for $\\left( [\\Sigma^2] |_f \\right) |_U$. Hence if no target weight is equal to a source weight we still get that the incidence is not zero.\n \\item Notice that (\\ref{resu}) holds only if $f$ has no additional symmetry. For example if $f$ is contact equivalent to a monomial germ then the rank of the symmetry group is at least 2. If $f\\in \\Sigma^2(2,p)$ is monomial then the incidence $ [\\Sigma^2]|_f $ is not zero since we can restrict to the $U(1)$ symmetry corresponding to the source weights $ a_1=a_2=1 $ and see that $b_i\\geq 2$ for all $i\\leq p$ since $f\\in \\Sigma^2(2,p)$ implies that there are no linear terms. \n \\end{enumerate}\n\\end{remarks}\n\nThese remarks imply that to prove Theorem \\ref{sigma2} it is enough to prove:\n\\begin{proposition} Let $f\\in J^k(2,p)$ be a weighted homogeneous polynomial with weights $ a_1,a_2$ and $b_1,\\ldots,b_p$. If $a_1=b_1$ then $f$ is contact equivalent to a monomial germ.\n \\end{proposition}\n\\begin{proof}\nWe distinguish three cases and in all cases we will use the fact that if the ideals $ (f_1,\\dotsc,f_p) $ and $ (f^{'}_1,\\dotsc,f^{'}_p) $ agree then $f$ is contact equivalent to $f^{'}$\\negthinspace. In the first two cases we don't need the $a_1=b_1$ assumption.\n\n{\\bf Case 1:} If $a_1=0$ or $a_2=0$. If, say, $a_2=0$ then $f_i=x^{b_i\/a_1}g_i(y)$ for some polynomials $g_i(y)$. If the lowest power of $y$ in $g_i(y)$ is $k_i$ then $g_i(y)=y^{k_i}(h_i+y\\overline g_i(y)$ for some polynomials $\\overline g_i(y)$ and non-zero numbers $h_i$. The functions $h_i+y\\overline g_i(y)$ are units in the algebra $\\C[y]\/(y^{k+1})$ which implies that the ideal $ (f_1,\\dotsc,f_p) $ is equal to $ (x^{b_1\/a_1}y^{k_1},\\dotsc,x^{b_p\/a_1}y^{k_p}) $ i.e. $f$ is contact equivalent to a monomial germ.\n\n{\\bf Case 2:} If $a_1>0$ and $a_2<0$. Suppose that $\\deg x^uy^v=\\deg x^{u'}y^{v'}$ and $u\\geq u'$. Then $v\\geq v'$.It implies that $f_i(x,y)=x^{u_i}y^{v_i}(h_i+g_i(x,y))$ for some $u_i,\\ v_i$ non-negative integers and $h_i\\neq 0$ constant. Therefore $ (f_1,\\dotsc,f_p)=(x^{u_1}y^{v_1},\\dotsc,x^{u_p}y^{v_p})$.\n\n{\\bf Case 3:} If $a_1>0$ and $a_2>0$. Then $a_1=b_1$ implies that $f_1=y^{k}$ for $a_1=ka_2$. Without loss of generality we can assume that $a_1=k$ and $a_2=1$. From here we don't use the $a_1=b_1$ assumption. There is a unique decomposition $b_i=u_ik+d_i$ for $0 \\leq d_i < k$. Then $(f_i,y^k)=(x^{u_i}y^{d_i},y^k)$ for $2\\leq i\\leq p$, therefore $ (f_1,\\dotsc,f_p)=(y^k,x^{u_2}y^{d_2},\\dotsc,x^{u_p}y^{d_p})$.\n\\end{proof} \n\n\n\n\\subsection{Calculations of the Examples \\ref{exa:xayb} and \\ref{exa:s2}} \\label{sec:calc}\n\nLet us recall now that the normal space of the contact orbit of the singularity $\\xi=(f_1,\\dotsc,f_p)$ is isomorphic\nto $\\C^n\\oplus U_\\xi$ where $U_\\xi=J^k(n,p)\/(f_ie_j,\\partial f\/x_l:i,j\\leq p,l\\leq n)$ is the {\\em unfolding space} of $\\xi$. Here $e_j$ denotes the constant map $(0,\\dotsc,0,1,0,\\dotsc,0)$ with 1 in the $j$\\textsuperscript{th} coordinate. (see e.g. in \\cite{agv} or \\cite{tpforgroup}).\n\n{\\bf \\large $(x^a,y^b)$:} The unfolding space is spanned by the monomials $\\{(x^iy^j,0): i