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作者：Berger, Noam; Conache, Diana; Johansson, Anders; Oberg, Anders
作者单位：Technical University of Munich; University of Gavle; Uppsala University
摘要：In this paper we solve two open problems in ergodic theory. We prove first that if a Doeblin function g (a g-function) satisfies lim sup(n ->infinity)var(n)log g / n(-1/2 )<2, then we have a unique Doeblin measure (g-measure). This result indicates a possible phase transition in analogy with the long-range Ising model. Secondly, we provide an example of a Doeblin function with a unique Doeblin measure that is not weakly mixing, which implies that the sequence of iterates of the transfer operat...
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作者：Amorino, Chiara; Belomestny, Denis; Pilipauskaite, Vytaute; Podolskij, Mark; Zhou, Shi-Yuan
作者单位：Pompeu Fabra University; Barcelona School of Economics; Aalborg University; University of Luxembourg
摘要：This paper investigates the estimation of the interaction function for a class of McKean-Vlasov stochastic differential equations. The estimation is based on observations of the associated particle system at time T, considering the scenario where both the time horizon T and the number of particles N tend to infinity. Our proposed method recovers polynomial rates of convergence for the resulting estimator. This is achieved under the assumption of exponentially decaying tails for the interaction...
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作者：Gorin, Vadim; Kenyon, Richard
作者单位：University of California System; University of California Berkeley; Yale University
摘要：We study limit shapes in two equivalent models: the six-vertex model in the c -> 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\rightarrow 0$$\end{document} limit and the random Mallows permutation with restricted permutation matrix. We give the Euler-Lagrange equation for the limit shape and show how to solve ...
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作者：Alvarado, Jose D.; de Oliveira, Leonardo Goncalves; Griffiths, Simon
作者单位：University of Ljubljana
摘要：We consider the question of determining the probability of triangle count deviations in the Erdos-Renyi random graphs G (n,m) and G (n,p)with densities larger than n(-1/2)(log n)(1/2). In particular, we determine the log probability log P(N-Delta(G) > (1+delta)p(3)n(3))up to a constant factor across essentially the entire range of possible deviations, in both the G (n,m) and G (n,p) model. For the G(n,p) model, we also prove a stronger result, up to a(1+o(1)) factor, in the non-localised regim...
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作者：Favaro, S.; Hanin, B.; Marinucci, D.; Nourdin, I.; Peccati, G.
作者单位：University of Turin; Princeton University; University of Rome Tor Vergata; University of Luxembourg
摘要：We study the distribution of a fully connected neural network with random Gaussian weights and biases in which the hidden layer widths are proportional to a large constant n. Under mild assumptions on the non-linearity, we obtain quantitative bounds on normal approximations valid at large but finite n and any fixed network depth. Our theorems show both for the finite-dimensional distributions and the entire process, that the distance between a random fully connected network (and its derivative...
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作者：Chen, Xin; Chen, Zhen-Qing; Kumagai, Takashi; Wang, Jian
作者单位：Shanghai Jiao Tong University; University of Washington; University of Washington Seattle; Waseda University; Fujian Normal University; Fujian Normal University; Fujian Normal University
摘要：We consider random conductance models with long range jumps on Z(d), where the one-step transition probability from x to y is proportional to w( x,y) | x - y | (-d-alpha )with alpha is an element of ( 0 , 2). Assume that {w x,y } (x,y)is an element of E are independent, identically distributed and uniformly bounded non-negative random variables with E w(x,y) = 1, where E is the set of all unordered pairs on Z(d). We obtain a quantitative version of stochastic homogenization for these random wa...
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作者：Gamarnik, David; Jagannath, Aukosh; Kizildag, Eren C.
作者单位：Massachusetts Institute of Technology (MIT); University of Waterloo; University of Illinois System; University of Illinois Urbana-Champaign
摘要：We study the Isingp-spin glass model for largep. We show that for any inversetemperature root ln 2infinity, there exists exponentially many well-separated clusters such that(a) each cluster has exponentially small Gibbs mass, and (b) the clusters collectivelycontain all but a vanishing fraction of Gibbs mass. Moreover, these clusters consist ofconfigurations with energy near beta. Range of temperatures for which shattering occursis within thereplica symmetricregion. To the best of our knowledg...
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作者：Guo, Xiaoqin; Tran, Hung V.
作者单位：University System of Ohio; University of Cincinnati; University of Wisconsin System; University of Wisconsin Madison
摘要：We consider random walks in a uniformly elliptic, balanced, i.i.d. random environment in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}<^>d$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepack...
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作者：Hutchcroft, Tom
作者单位：California Institute of Technology
摘要：In long-range percolation on Z(d), we connect each pair of distinct points x and y by an edge independently at random with probability 1-exp(-beta parallel to x-y parallel to(-d-alpha)), where alpha> 0 is fixed and beta >= 0 is a parameter. In a previous paper, we proved that if 0 < alpha < d then the critical two-point function satisfies the spatially averaged upper bound 1/ r(d) & sum;(d)( x is an element of[-r,r]) (P)beta(c)(0 <-> x) <= r(-d+alpha) for every r >= 1. This upper bound is beli...
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作者：Drewitz, Alexander; Prevost, Alexis; Rodriguez, Pierre-Francois
作者单位：University of Cologne; University of Bonn; Imperial College London; University of Cambridge
摘要：We investigate the bond percolation model on transient weighted graphs G induced by the excursion sets of the Gaussian free field on the corresponding metric graph. Under the sole assumption that its sign clusters do not percolate, we derive an extension of Lupu's formula for the two-point function at criticality. We then focus on the low-dimensional case 0 < nu < (alpha)/(2) , where alpha governs the polynomial volume growth of G and nu the decay rate of the Green's function on G. In particul...
