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作者：Stauffer, Alexandre; Taggi, Lorenzo
作者单位：University of Bath; Technical University of Darmstadt
摘要：We consider the activated random walk model on general vertex-transitive graphs. A central question in this model is whether the critical density mu(c) for sustained activity is strictly between 0 and 1. It was known that mu c > 0 on Z(d), d >= 1, and that mu(c) < 1 on Z for small enough sleeping rate. We show that mu(c) -> 0 as lambda -> 0 in all vertex-transitive transient graphs, implying that mu(c) < 1 for small enough sleeping rate. We also show that mu(c) < 1 for any sleeping rate in any...
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作者：Bassino, Frederique; Bouvel, Mathilde; Feray, Valentin; Gerin, Lucas; Pierrot, Adeline
作者单位：Centre National de la Recherche Scientifique (CNRS); CNRS - Institute for Information Sciences & Technologies (INS2I); Universite Paris 13; University of Zurich; Institut Polytechnique de Paris; Ecole Polytechnique; Centre National de la Recherche Scientifique (CNRS); Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS)
摘要：We study uniform random permutations in an important class of pattern-avoiding permutations: the separable permutations. We describe the asymptotics of the number of occurrences of any fixed given pattern in such a random permutation in terms of the Brownian excursion. In the recent terminology of permutons, our work can be interpreted as the convergence of uniform random separable permutations towards a Brownian separable permuton.
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作者：Tang, Wenpin; Tsai, Li-Cheng
作者单位：University of California System; University of California Berkeley; Columbia University
摘要：We study the Up the River problem formulated by Aldous (2002), where a unit drift is distributed among a finite collection of Brownian particles on R+, which are annihilated once they reach the origin. Starting K particles at x = 1, we prove Aldous' conjecture [Aldous (2002)] that the push-the-laggard strategy of distributing the drift asymptotically (as K -> infinity) maximizes the total number of surviving particles, with approximately 4/root pi root K surviving particles. We further establi...
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作者：Fribergh, Alexander; Kious, Daniel
作者单位：Universite de Montreal; New York University; NYU Shanghai
摘要：We consider biased random walks in positive random conductances on the d-dimensional lattice in the zero-speed regime and study their scaling limits. We obtain a functional law of large numbers for the position of the walker, properly rescaled. Moreover, we state a functional central limit theorem where an atypical process, related to the fractional kinetics, appears in the limit.
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作者：Garban, Christophe; Pete, Gabor; Schramm, Oded
作者单位：Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet; CNRS - National Institute for Mathematical Sciences (INSMI); Hungarian Academy of Sciences; HUN-REN; HUN-REN Alfred Renyi Institute of Mathematics; Budapest University of Technology & Economics; Microsoft
摘要：We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on a version of the triangular lattice in the complex plane have unique scaling limits, which are invariant under rotations, scalings, and, in the case of the MST, also under translations. However, they are not expected to be conformally invariant. We also prove some geometric properties of the limiting MST. The topology of convergence is the space of spanning trees introduced by Aizenman et al. [Random Structures Algori...
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作者：Grama, Ion; Lauvergnat, Ronan; Le Page, Emile
作者单位：Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
摘要：Consider a Markov chain (X-n)(n >= 0) with values in the state space X. Let f be a real function on X and set S-n = Sigma(n)(i=1) f(X-i), n >= 1. Let P-x be the probability measure generated by the Markov chain starting at X-0 = x. For a starting point y is an element of R, denote by tau(y) the first moment when the Markov walk (y + S-n)(n >= 1) becomes nonpositive. Under the condition that S-n has zero drift, we find the asymptotics of the probability P-x (tau(y) > n) and of the conditional l...
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作者：Seo, Insuk
作者单位：Seoul National University (SNU)
摘要：This paper presents our study of the asymptotic behavior of a two-component system of Brownian motions undergoing certain form of singular interactions. In particular, the system is a combination of two different types of particles and the mechanical properties and the interaction parameters depend on the corresponding type of particles. We prove that the hydrodynamic limit of the empirical densities of two types is the solution of a partial differential equation known as the Maxwell-Stefan eq...
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作者：Kolesnikov, Alexander, V; Milman, Emanuel
作者单位：HSE University (National Research University Higher School of Economics); Technion Israel Institute of Technology; Technion Israel Institute of Technology
摘要：What is the optimal way to cut a convex bounded domain K in Euclidean space (R-n, vertical bar . vertical bar) into two halves of equal volume, so that the interface between the two halves has least surface area? A conjecture of Kannan, Lovasz and Simonovits asserts that, if one does not mind gaining a universal numerical factor (independent of n) in the surface area, one might as well dissect K using a hyperplane. This conjectured essential equivalence between the former nonlinear isoperimetr...
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作者：Beffara, Vincent; Chhita, Sunil; Johansson, Kurt
作者单位：Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA); Centre National de la Recherche Scientifique (CNRS); Durham University; Royal Institute of Technology
摘要：Domino tilings of the two-periodic Aztec diamond feature all of the three possible types of phases of random tiling models. These phases are determined by the decay of correlations between dominoes and are generally known as solid, liquid and gas. The liquid-solid boundary is easy to define microscopically and is known in many models to be described by the Airy process in the limit of a large random tiling. The liquid-gas boundary has no obvious microscopic description. Using the height functi...
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作者：Do, Yen; Oanh Nguyen; Vu, Van
作者单位：University of Virginia; Yale University; Yale University; National University of Singapore
摘要：In this paper, we prove optimal local universality for roots of random polynomials with arbitrary coefficients of polynomial growth. As an application, we derive, for the first time, sharp estimates for the number of real roots of these polynomials, even when the coefficients are not explicit. Our results also hold for series; in particular, we prove local universality for random hyperbolic series.
