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作者：Bourgade, P.; Dubach, G.
作者单位：New York University
摘要：We study the overlaps between eigenvectors of nonnormal matrices. They quantify the stability of the spectrum, and characterize the joint eigenvalues increments under Dyson-type dynamics. Well known work by Chalker and Mehlig calculated the expectation of these overlaps for complex Ginibre matrices. For the same model, we extend their results by deriving the distribution of diagonal overlaps (the condition numbers), and their correlations. We prove: (i) convergence of condition numbers for bul...
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作者：Menz, Georg; Tassy, Martin
作者单位：University of California System; University of California Los Angeles; Dartmouth College
摘要：We show the existence of a variational principle for graph homomorphisms from Z(m) to a d-regular tree. The technique is based on a discrete Kirszbraun theorem and a concentration inequality obtained through the dynamics of the model. As another consequence of the concentration inequality we also obtain the existence of a continuum of translation-invariant ergodic gradient Gibbs measures for graph homomorphisms from Z(m) to a regular tree. The method is sufficiently robust such that it could b...
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作者：Beliaev, Dmitry; McAuley, Michael; Muirhead, Stephen
作者单位：University of Oxford; University of London; King's College London; University of London; Queen Mary University London
摘要：The Nazarov-Sodin constant describes the average number of nodal set components of smooth Gaussian fields on large scales. We generalise this to a functional describing the corresponding number of level set components for arbitrary levels. Using results from Morse theory, we express this functional as an integral over the level densities of different types of critical points, and as a result deduce the absolute continuity of the functional as the level varies. We further give upper and lower b...
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作者：He, Yukun; Knowles, Antti
作者单位：University of Zurich; University of Geneva
摘要：We investigate to what extent the microscopic Wigner-Gaudin-Mehta-Dyson (WGMD) (or sine kernel) statistics of random matrix theory remain valid on mesoscopic scales. To that end, we compute the connected two-point spectral correlation function of a Wigner matrix at two mesoscopically separated points. In the mesoscopic regime, density correlations are much weaker than in the microscopic regime. Our result is an explicit formula for the two-point function. This formula implies that the WGMD sta...
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作者：Lupu, Titus; Sabot, Christophe; Tarres, Pierre
作者单位：Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); Universite Paris Cite; Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet; New York University; NYU Shanghai; Universite PSL; Centre National de la Recherche Scientifique (CNRS); Universite PSL; Universite Paris-Dauphine
摘要：We introduce a continuous space limit of the vertex reinforced jump process (VRJP) in dimension one, which we call linearly reinforced motion (LRM) on R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}$$\end{document}. It is constructed out of a convergent Bass-Burdzy flow. The proof goes through the repr...
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作者：Hazla, Jan; Mossel, Elchanan; Ross, Nathan; Zheng, Guangqu
作者单位：Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Massachusetts Institute of Technology (MIT); University of Melbourne; University of Kansas
摘要：We study the phenomenon of intransitivity in models of dice and voting. First, we follow a recent thread of research for n-sided dice with pairwise ordering induced by the probability, relative to 1/2, that a throw from one die is higher than the other. We build on a recent result of Polymath showing that three dice with i.i.d. faces drawn from the uniform distribution on {1, ... , n} and conditioned on the average of faces equal to (n+1)/2 are intransitive with asymptotic probability 1/4. We ...
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作者：Basak, Anirban; Zeitouni, Ofer
作者单位：Tata Institute of Fundamental Research (TIFR); International Centre for Theoretical Sciences, Bengaluru; Weizmann Institute of Science; New York University
摘要：Consider an N x N Toeplitz matrix T-N with symbol a(lambda) := Sigma(d1)(l=-d2) a(e)lambda(e), perturbed by an additive noise matrix N-gamma E-N, where the entries of E-N are centered i.i.d. random variables of unit variance and gamma > 1/2. It is known that the empirical measure of eigenvalues of the perturbed matrix converges weakly, as N -> infinity, to the law of a(U), where U is distributed uniformly on S-1. In this paper, we consider the outliers, i.e. eigenvalues that are at a positive ...
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作者：Helmuth, Tyler; Perkins, Will; Regts, Guus
作者单位：University of Bristol; University of Illinois System; University of Illinois Chicago; University of Illinois Chicago Hospital; University of Amsterdam
摘要：We develop an efficient algorithmic approach for approximate counting and sampling in the low-temperature regime of a broad class of statistical physics models on finite subsets of the lattice Zd and on the torus (Z/nZ)d. Our approach is based on combining contour representations from Pirogov-Sinai theory with Barvinok's approach to approximate counting using truncated Taylor series. Some consequences of our main results include an FPTAS for approximating the partition function of the hard-cor...
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作者：Adhikari, Arka; Huang, Jiaoyang
作者单位：Harvard University
摘要：In this paper, we compare the solutions of the Dyson Brownian motion for general beta and potential V and the associated McKean-Vlasov equation near the edge. Under suitable conditions on the initial data and the potential V, we obtain optimal rigidity estimates of particle locations near the edge after a short time t = o(1). Our argument uses the method of characteristics along with a careful estimate involving an equation of the edge. With the rigidity estimates as an input, we prove a centr...
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作者：Backhoff, Julio; Conforti, Giovani; Gentil, Ivan; Leonard, Christian
作者单位：University of Vienna; Institut Polytechnique de Paris; Ecole Polytechnique; Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet; University of Twente
摘要：We study the mean field Schrodinger problem (MFSP), that is the problem of finding the most likely evolution of a cloud ofinteractingBrownian particles conditionally on the observation of their initial and final configuration. Its rigorous formulation is in terms of an optimization problem with marginal constraints whose objective function is the large deviation rate function associated with a system of weakly dependent Brownian particles. We undertake a fine study of the dynamics of its solut...
