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作者：Bethencourt, Loic
作者单位：Centre National de la Recherche Scientifique (CNRS); Universite Paris Cite; Sorbonne Universite
摘要：We consider a particle with position (X-t)(t >= 0) living in R+, whose velocity (V-t)(t >= 0) is a positive recurrent diffusion with heavy-tailed invariant distribution when the particle lives in (0, infinity). When it hits the boundary x = 0, the particle restarts with a random strictly positive velocity. We show that the properly rescaled position process converges weakly to a stable process reflected on its infimum. From a P.D.E. point of view, the time-marginals of (X-t, V-t)(t >= 0) solve...
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作者：Gorin, Vadim; Huang, Jiaoyang
作者单位：University of California System; University of California Berkeley; University of California System; University of California Berkeley; University of Pennsylvania
摘要：We introduce dynamical versions of loop (or Dyson-Schwinger) equations for large families of two-dimensional interacting particle systems, including Dyson Brownian motion, Nonintersecting Bernoulli/Poisson random walks, beta -corners processes, uniform and Jack-deformed measures on Gelfand-Tsetlin patterns, Macdonald processes, and (q, kappa)-distributions on lozenge tilings. Under technical assumptions we show that the dynamical loop equations lead to Gaussian field type fluctuations. As an a...
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作者：Basu, Riddhipratim; Bhatia, Manan; Ganguly, Shirshendu
作者单位：Tata Institute of Fundamental Research (TIFR); University of California System; University of California Berkeley
摘要：First passage percolation (FPP) on Z d or R d is a canonical model of a random metric space where the standard Euclidean geometry is distorted by random noise. Of central interest is the length and the geometry of the geodesic, the shortest path between points. Since the latter, owing to its length minimization, traverses through atypically low values of the underlying noise variables, it is an important problem to quantify the disparity between the environment rooted at a point on the geodesi...
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作者：Narayanan, Hariharan; Sheffield, Scott
作者单位：Tata Institute of Fundamental Research (TIFR); Massachusetts Institute of Technology (MIT)
摘要：Suppose a, beta are Lipschitz, strongly concave functions from [0, 1] to R and y is a concave function from [0, 1] to R such that a(0) = y(0) = 0, a(1)=8(0) = 0 and 8(1)=y(1) = 0. For an nxn Hermitian matrix W, let spec(W) denote the vector in R whose coordinates are the eigenvalues of W listed in nonincreasing order. Let lambda=aa, mu=-beta on (0, 1) and v=ay, at all points of (0, 1], where a is the left derivative. Let lambda(i);= n2(a)-a()), for i = [n], and similarly, () == n2(B()-B(2)) an...
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作者：Biskup, Marek; Louidor, Oren
作者单位：University of California System; University of California Los Angeles; Technion Israel Institute of Technology
摘要：We consider a continuous-time random walk on a regular tree of finite depth and study its favorite points among the leaf vertices. For the walk started from a leaf vertex and stopped upon hitting the root, we prove that, in the limit as the depth of the tree tends to infinity, the suitably scaled and centered maximal time spent at any leaf converges to a randomly-shifted Gumbel law. The random shift is characterized using a derivative-martingale like object associated with square-root local-ti...
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作者：Muirhead, Stephen
作者单位：University of Melbourne
摘要：We establish the sharpness of the phase transition for a wide class of Gaussian percolation models, on Z(d) or R-d, d >= 2, with correlations decaying at least algebraically with exponent alpha > 0, including the discrete Gaussian free field (d >= 3, alpha = d - 2), the discrete Gaussian membrane model (d >= 5, alpha = d - 4), and many other examples both discrete and continuous. In particular, we do not assume positive correlations. This result is new for all strongly correlated models (i.e.,...
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作者：Xu, Changji; Yang, Fan; Yau, Horng-Tzer; Yin, Jun
作者单位：Harvard University; Tsinghua University; Harvard University; University of California System; University of California Los Angeles
摘要：We consider Hermitian random band matrices H = (h(xy)) on the d-dimensional lattice (Z/LZ)(d), where the entries h(xy) = (h) over bar (yx) are independent centered complex Gaussian random variables with variances s(xy) = E|h(xy)|(2). The variance matrix S = (s(xy)) has a banded profile so that sxy is negligible if |x - y| exceeds the band width W. For dimensions d >= 7, we prove the bulk eigenvalue universality of H under the condition W >> L95/(d+95). Assuming that W >= L-epsilon for a small ...
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作者：Hen, Yu-Ting
作者单位：University of Victoria
摘要：In order to give a dual, annealed description of the two-dimensional stochastic heat equation (SHE) from regularizing the noise, we consider the Schrodinger semigroup of the many -body delta -Bose gas from mollifying the delta potentials. The main theorem proves the convergences of the corresponding approximate semigroups when they act on bounded functions. For the proof we introduce a mean -field Poisson system to expand the Feynman- Kac formula of the approximate semigroups. This expansion y...
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作者：Gess, Benjamin; Tsatsoulis, Pavlos
作者单位：University of Bielefeld
摘要：Quantitative estimates for the top Lyapunov exponents for systems of stochastic reaction-diffusion equations are proven. The treatment includes reaction potentials with degenerate minima. The proof relies on an asymptotic expansion of the invariant measure, with careful control on the resulting error terms. As a consequence of these estimates, synchronisation by noise is deduced for systems of stochastic reaction-diffusion equations for the first time.
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作者：Dominguez, Tomas; Mourrat, Jean-christophe
作者单位：University of Toronto; Ecole Normale Superieure de Lyon (ENS de LYON)
摘要：We consider the problem of recovering the community structure in the stochastic block model with two communities. We aim to describe the mutual information between the observed network and the actual community structure in the sparse regime, where the total number of nodes diverges while the average degree of a given node remains bounded. Our main contributions are a conjecture for the limit of this quantity, which we express in terms of a Hamilton-Jacobi equation posed over a space of probabi...
