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作者：Candellero, Elisabetta; Ganguly, Shirshendu; Hoffman, Christopher; Levine, Lionel
作者单位：University of Warwick; University of California System; University of California Berkeley; University of Washington; University of Washington Seattle; Cornell University
摘要：We introduce a two-type internal DLA model which is an example of a nonunary abelian network. Starting with n oil and n water particles at the origin, the particles diffuse in Z according to the following rule: whenever some site x. Z has at least 1 oil and at least 1 water particle present, it fires by sending 1 oil particle and 1 water particle each to an independent random neighbor x +/- 1. Firing continues until every site has at most one type of particles. We establish the correct order f...
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作者：Chen, Wei-Kuo
作者单位：University of Minnesota System; University of Minnesota Twin Cities
摘要：The validity of the Parisi formula in the Sherrington-Kirkpatrick model (SK) was initially proved by Talagrand [Ann. of Math. (2) 163 (2006) 221263]. The central argument relied on a dedicated study of the coupled free energy via the two-dimensional Guerra-Talagrand (GT) replica symmetry breaking bound. It is believed that this bound and its higher dimensional generalization are highly related to the conjectures of temperature chaos and ultrametricity in the SK model, but a complete investigat...
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作者：Paquette, Elliot; Zeitouni, Ofer
作者单位：University System of Ohio; Ohio State University; New York University; Weizmann Institute of Science
摘要：Let lambda((N)) be the largest eigenvalue of the N x N GUE matrix which is the Nth element of the GUE minor process, rescaled to converge to the standard Tracy-Widom distribution. We consider the sequence {lambda((N))}(N >= 1) and prove a law of fractional logarithm for the lim sup: lim sup(N ->infinity) lambda((N)) / (logN)(2/3) = (1/4)(2/3) almost surely. For the lim inf, we prove the weaker result that there are constants c(1), c(2) > 0 so that -c(1) <= lim inf(N ->infinity) lambda((N)) / (...
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作者：Ding, Jian; Roy, Rishideep; Zeitouni, Ofer
作者单位：University of Chicago; Indian Institute of Management (IIM System); Indian Institute of Management Bangalore; Weizmann Institute of Science; New York University
摘要：We show that the centered maximum of a sequence of logarithmically correlated Gaussian fields in any dimension converges in distribution, under the assumption that the covariances of the fields converge in a suitable sense. We identify the limit as a randomly shifted Gumbel distribution, and characterize the random shift as the limit in distribution of a sequence of random variables, reminiscent of the derivative martingale in the theory of branching random walk and Gaussian chaos. We also dis...
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作者：Garet, Olivier; Gouere, Jean-Baptiste; Marchand, Regine
作者单位：Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Lorraine; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Tours; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Lorraine
摘要：We study the number N-n of open paths of length n in supercritical oriented percolation on Z(d) x N, with d >= 1, and we prove the existence of the connective constant for the supercritical oriented percolation cluster: on the percolation event {inf N-n > 0}, N-n(1/n) almost surely converges to a positive deterministic constant. The proof relies on the introduction of adapted sequences of regenerating times, on subadditive arguments and on the properties of the coupled zone in supercritical or...
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作者：Khoshnevisan, Davar; Kim, Kunwoo; Xiao, Yimin
作者单位：Utah System of Higher Education; University of Utah; Pohang University of Science & Technology (POSTECH); Michigan State University
摘要：Let xi denote space-time white noise, and consider the following stochastic partial differential equations on R+ x R: (i) (u) over dot = 1/2 u '' + u xi, started identically at one; and (ii) (Z) over dot = 1/2 Z '' + xi, started identically at zero. It is well known that the solution to (i) is intermittent, whereas the solution to (ii) is not. And the two equations are known to be in different universality classes. We prove that the tall peaks of both systems are multifractals in a natural lar...
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作者：Mueller, Carl; Mytnik, Leonid; Perkins, Edwin
作者单位：University of Rochester; Technion Israel Institute of Technology; University of British Columbia
摘要：We study the density X(t, x) of one-dimensional super-Brownian motion and find the asymptotic behaviour of P(0 < X( t, x) <= a) as a down arrow 0 as well as the Hausdorff dimension of the boundary of the support of X(t, .). The answers are in terms of the leading eigenvalue of the Ornstein-Uhlenbeck generator with a particular killing term. This work is motivated in part by questions of pathwise uniqueness for associated stochastic partial differential equations.
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作者：Balazs, Marton; Nagy, Attila Laszlo
作者单位：University of Bristol; Budapest University of Technology & Economics
摘要：We identify the ballistically and diffusively rescaled limit distribution of the second class particle position in a wide range of asymmetric and symmetric interacting particle systems with established hydrodynamic behavior, respectively (including zero-range, misanthrope and many other models). The initial condition is a step profile, which in some classical cases of asymmetric models, gives rise to a rarefaction fan scenario. We also point out a model with nonconcave, nonconvex hydrodynamics...
