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作者：Caputo, Pietro; Labbe, Cyril; Lacoin, Hubert
作者单位：Roma Tre University; Universite PSL; Universite Paris-Dauphine; Centre National de la Recherche Scientifique (CNRS); Instituto Nacional de Matematica Pura e Aplicada (IMPA)
摘要：By viewing the N-simplex as the set of positions of N - 1 ordered particles on the unit interval, the adjacent walk is the continuous-time Markov chain obtained by updating independently at rate 1 the position of each particle with a sample from the uniform distribution over the interval given by the two particles adjacent to it. We determine its spectral gap and prove that both the total variation distance and the separation distance to the uniform distribution exhibit a cutoff phenomenon, wi...
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作者：Duminil-Copin, Hugo; Kesten, Harry; Nazarov, Fedor; Peres, Yuval; Sidoravicius, Vladas
作者单位：University of Geneva; Cornell University; University System of Ohio; Kent State University; Kent State University Salem; Kent State University Kent; Microsoft; New York University; NYU Shanghai
摘要：We show that the number of maximal paths in directed last-passage percolation on the hypercubic lattice Z(d) (d >= 2) in which weights take finitely many values is typically exponentially large.
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作者：Barbu, Viorel; Roeckner, Michael
作者单位：Romanian Academy; University of Bielefeld
摘要：We construct weak solutions to the McKean-Vlasov SDE dX(t) = b(X(t), dL(X(t))/dx (X(t)))dt + sigma(X(t), dL(X(t))/dt (X(t)))dW(t) on R-d for possibly degenerate diffusion matrices sigma with X(0) having a given law, which has a density with respect to Lebesgue measure, dx. Here, L-X(t) denotes the law of X (t). Our approach is to first solve the corresponding nonlinear Fokker-Planck equations and then use the well-known superposition principle to obtain weak solutions of the above SDE.
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作者：Blanquicett, Daniel
作者单位：University of California System; University of California Davis
摘要：Consider a p-random subset A of initially infected vertices in the discrete cube [L](3), and assume that the neighborhood of each vertex consists of the a(i) nearest neighbors in the +/- e(i)-directions for each i is an element of {1, 2, 3}, where a(1) <= a(2) <= a(3). Suppose we infect any healthy vertex x is an element of [L](3) already having a(3) + 1 infected neighbors, and that infected sites remain infected forever. In this paper, we determine the critical length for percolation up to a ...
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作者：Albeverio, Sergio; De Vecchi, Francesco C.; Gubinelli, Massimiliano
作者单位：University of Bonn
摘要：We prove an explicit formula for the law in zero of the solution of a class of elliptic SPDE in R-2. This formula is the simplest instance of dimensional reduction, discovered in the physics literature by Parisi and Sourlas (Phys. Rev. Lett. 43 (1979) 744-745), which links the law of an elliptic SPDE in d + 2 dimension with a Gibbs measure in d dimensions. This phenomenon is similar to the relation between a Rd+1 dimensional parabolic SPDE and its R-d dimensional invariant measure. As such, di...
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作者：Bates, Erik; Chatterjee, Sourav
作者单位：Stanford University; Stanford University
摘要：Probabilistic models of directed polymers in random environment have received considerable attention in recent years. Much of this attention has focused on integrable models. In this paper, we introduce some new computational tools that do not require integrability. We begin by defining a new kind of abstract limit object, called partitioned subprobability measure, to describe the limits of endpoint distributions of directed polymers. Inspired by a recent work of Mukherjee and Varadhan on larg...
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作者：Chatterjee, Sourav; Harel, Matan
作者单位：Stanford University; Tel Aviv University
摘要：We consider a random geometric graph G(chi(n), r(n)), given by connecting two vertices of a Poisson point process chi(n) of intensity n on the d-dimensional unit torus whenever their distance is smaller than the parameter r(n). The model is conditioned on the rare event that the number of edges observed, vertical bar E vertical bar, is greater than (1 + delta)E(vertical bar E vertical bar), for some fixed delta > 0. This article proves that upon conditioning, with high probability there exists...
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作者：Dette, Holger; Tomecki, Dominik; Venker, Martin
作者单位：Ruhr University Bochum
摘要：We investigate moment sequences of probability measures on subsets of the real line under constraints of certain moments being fixed. This corresponds to studying sections of nth moment spaces, that is, the spaces of moment sequences of order n. By equipping these sections with the uniform or more general probability distributions, we manage to give for large n precise results on the (probabilistic) barycenters of moment space sections and the fluctuations of random moments around these baryce...
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作者：Remy, Guillaume; Zhu, Tunan
作者单位：Columbia University; Universite PSL; Ecole Normale Superieure (ENS)
摘要：We consider a subcritical Gaussian multiplicative chaos (GMC) measure defined on the unit interval [0, 1] and prove an exact formula for the fractional moments of the total mass of this measure. Our formula includes the case where log-singularities (also called insertion points) are added in 0 and 1, the most general case predicted by the Selberg integral. The idea to perform this computation is to introduce certain auxiliary functions resembling holomorphic observables of conformal field theo...
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作者：Schweiger, Florian
作者单位：University of Bonn
摘要：We show that the centred maximum of the four-dimensional membrane model on a box of sidelength N converges in distribution. To do so, we use a criterion of Ding, Roy and Zeitouni (Ann. Probab. 45 (2017) 3886-3928) and prove sharp estimates for the Green's function of the discrete Bilaplacian. These estimates are the main contribution of this work and might also be of independent interest. To derive them, we use estimates for the approximation quality of finite difference schemes as well as res...
