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作者：Forman, Noah; Pal, Soumik; Rizzolo, Douglas; Winkel, Matthias
作者单位：McMaster University; University of Washington; University of Washington Seattle; University of Delaware; University of Oxford
摘要：We introduce diffusions on a space of interval partitions of the unit interval that are stationary with the Poisson-Dirichlet laws with parameters (a, 0) and (a, a). The construction has two steps. The first is a general construction of interval partition processes obtained previously by decorating the jumps of a Levy process with independent excursions. Here, we focus on the second step which requires explicit transition kernels and, what we call, pseudostationarity. This allows us to study p...
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作者：Feng, Renjie; Wei, Dongyi
作者单位：Chinese Academy of Sciences; University of Science & Technology of China, CAS; Peking University
摘要：In this article, we study the smallest gaps of the circular beta-ensemble (C beta E) on the unit circle, where beta is any positive integer. The main result is that the smallest gaps, after being normalized by n(beta+2/beta+1), will converge in distribution to a Poisson point process with some explicit intensity. And thus one can derive the limiting density of the kth smallest gap, which is proportional to x(k(beta+1)-1)e(-x beta+1). In particular, the results apply to the classical COE, CUE a...
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作者：Ghoussoub, Nassif; Kim, Young-Heon; Palmer, Aaron Zeff
作者单位：University of British Columbia
摘要：We provide a solution to the problem of optimal transport by Brownian martingales in general dimensions whenever the transport cost satisfies certain subharmonic properties in the target variable as well as a stochastic version of the standard twist condition frequently used in deterministic Monge transport theory. This setting includes, in particular, the case of the distance cost c(x, y) = | x - y|. We prove existence and uniqueness of the solution and characterize it as the first time Brown...
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作者：Dario, Paul; Gu, Chenlin
作者单位：Tel Aviv University; Universite PSL; Ecole Normale Superieure (ENS)
摘要：We study the heat kernel and the Green's function on the infinite supercritical percolation cluster in dimension d >= 2 and prove a quantitative homogenization theorem for these functions with an almost optimal rate of convergence. These results are a quantitative version of the local central limit theorem proved by Barlow and Hambly in (Electron. J. Probab. 14 (2009) 127). The proof relies on a structure of renormalization for the infinite percolation cluster introduced in (Comm. Pure Appl. M...
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作者：Perkowski, Nicolas; Rosati, Tommaso
作者单位：Free University of Berlin
摘要：We study the scaling limit of a branching random walk in static random environment in dimension d = 1, 2 and show that it is given by a superBrownian motion in a white noise potential. In dimension 1 we characterize the limit as the unique weak solution to the stochastic PDE partial derivative t mu=(Delta+xi)mu+root 2 nu mu xi for independent space white noise xi and space-time white noise (xi) over tilde. In dimension 2 the study requires paracontrolled theory and the limit process is describ...
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作者：Bailleul, Ismael; Catellier, Remi; Delarue, Francois
作者单位：Centre National de la Recherche Scientifique (CNRS); Universite de Rennes; Centre National de la Recherche Scientifique (CNRS); Universite Cote d'Azur
摘要：We address propagation of chaos for large systems of rough differential equations associated with random rough differential equations of mean field type dX(t) = V(X-t, L(X-t)) dt + F(X-t, L(X-t)) dW(t), where W is a random rough path and L(X-t) is the law of X-t. We prove propagation of chaos, and provide also an explicit optimal convergence rate. The analysis is based upon the tools we developed in our companion paper (Electron. J. Probab. 25 (2020) 21) for solving mean field rough differenti...
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作者：Gheissari, Reza; Lubetzky, Eyal
作者单位：New York University
摘要：Consider the 3D Ising model on a box of side length n with minus boundary conditions above the xy-plane and plus boundary conditions below it. At low temperatures, Dobrushin (1972) showed that the interface separating the predominantly plus and predominantly minus regions is localized: its height above a fixed point has exponential tails. Recently, the authors proved a law of large numbers for the maximum height M-n of this interface: for every beta large, M-n/log n -> c beta in probability as...
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作者：Chen, Xinxing; Dagard, Victor; Derrida, Bernard; Hu, Yueyun; Lifshits, Mikhail; Shi, Zhan
作者单位：Shanghai Jiao Tong University; Universite PSL; Ecole Normale Superieure (ENS); Universite Paris Cite; Universite PSL; College de France; Saint Petersburg State University; Sorbonne Universite; Universite Paris Cite
摘要：We are interested in the nearly supercritical regime in a family of maxtype recursive models studied by Collet, Eckman, Glaser and Martin (Comm. Math. Phys. 94 (1984) 353-370) and by Derrida and Retaux (J. Stat. Phys. 156 (2014) 268-290) and prove that, under a suitable integrability assumption on the initial distribution, the free energy vanishes at the transition with an essential singularity with exponent 12. This gives a weaker answer to a conjecture of Derrida and Retaux (J. Stat. Phys. 1...
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作者：Lammers, Piet
作者单位：University of Cambridge
摘要：Linde, Moore and Nordahl introduced a generalisation of the honeycomb dimer model to higher dimensions. The purpose of this article is to describe a number of structural properties of this generalised model. First, it is shown that the samples of the model are in one-to-one correspondence with the perfect matchings of a hypergraph. This leads to a generalised Kasteleyn theory: the partition function of the model equals the Cayley hyperdeterminant of the adjacency hypermatrix of the hypergraph....
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作者：Hammersley, William R. P.; Siska, David; Szpruch, Lukasz
作者单位：University of Edinburgh
摘要：This paper concerns the McKean-Vlasov stochastic differential equation (SDE) with common noise. An appropriate definition of a weak solution to such an equation is developed. The importance of the notion of compatibility in this definition is highlighted by a demonstration of its role in connecting weak solutions to McKean-Vlasov SDEs with common noise and solutions to corresponding stochastic partial differential equations (SPDE5). By keeping track of the dependence structure between all comp...
