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作者：Hutchcroft, Tom
作者单位：University of Cambridge
摘要：We study self-avoiding walk on graphs whose automorphism group has a transitive nonunimodular subgroup. We prove that self-avoiding walk is ballistic, that the bubble diagram converges at criticality, and that the critical two-point function decays exponentially in the distance from the origin. This implies that the critical exponent governing the susceptibility takes its mean-field value, and hence that the number of self-avoiding walks of length n is comparable to the nth power of the connec...
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作者：Benaych-Georges, Florent; Bordenave, Charles; Knowles, Antti
作者单位：Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris Cite; Universite Federale Toulouse Midi-Pyrenees (ComUE); Universite de Toulouse; Institut National des Sciences Appliquees de Toulouse; Universite Toulouse III - Paul Sabatier; Centre National de la Recherche Scientifique (CNRS); University of Geneva
摘要：We consider inhomogeneous Erdos-Renyi graphs. We suppose that the maximal mean degree d satisfies d << log n. We characterise the asymptotic behaviour of the (1-omicron)(n)(1) largest eigenvalues of the adjacency matrix and its centred version. We prove that these extreme eigenvalues are governed at first order by the largest degrees and, for the adjacency matrix, by the nonzero eigenvalues of the expectation matrix. Our results show that the extreme eigenvalues exhibit a novel behaviour which...
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作者：Defosseux, Manon
作者单位：Universite Paris Cite
摘要：We consider the coadjoint action of a Loop group of a compact group on the dual of the corresponding centrally extended Loop algebra and prove that a Brownian motion in a Cartan subalgebra conditioned to remain in an affine Weyl chamber-which can be seen as a space time conditioned Brownian motion-is distributed as the radial part process of a Brownian sheet on the underlying Lie algebra.
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作者：Tikhomirov, Konstantin; Youssef, Pierre
作者单位：Princeton University; Universite Paris Cite
摘要：For any alpha is an element of (0,1) and any n(alpha) <= d <= n/2, we show that lambda(G) <= C-alpha root d with probability at least 1 - 1/n, where G is the uniform random undirected d-regular graph on n vertices, lambda(G) denotes its second largest eigenvalue (in absolute value) and C-alpha is a constant depending only on alpha. Combined with earlier results in this direction covering the case of sparse random graphs, this completely settles the problem of estimating the magnitude of lambda...
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作者：Ganguly, Shirshendu; Levine, Lionel; Sarkar, Sourav
作者单位：University of California System; University of California Berkeley; Cornell University
摘要：We study the following one-dimensional model of annihilating particles. Beginning with all sites of Z uncolored, a blue particle performs simple random walk from 0 until it reaches a nonzero red or uncolored site, and turns that site blue; then a red particle performs simple random walk from 0 until it reaches a nonzero blue or uncolored site, and turns that site red. We prove that after n blue and n red particles alternately perform such walks, the total number of colored sites is of order n(...
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作者：Chen, Zhen-Qing; Lou, Shuwen
作者单位：University of Washington; University of Washington Seattle; University of Toronto
摘要：In this paper, we introduce and study Brownian motion on a class of state spaces with varying dimension. Starting with a concrete case of such state spaces that models a big square with a flag pole, we construct a Brownian motion on it and study how heat propagates on such a space. We derive sharp two-sided global estimates on its transition density function (also called heat kernel). These two-sided estimates are of Gaussian type, but the measure on the underlying state space does not satisfy...
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作者：Funaki, Tadahisa; Yokoyama, Satoshi
作者单位：University of Tokyo; Waseda University
摘要：This paper studies the sharp interface limit for a mass conserving Allen-Cahn equation, added an external noise and derives a stochastically perturbed mass conserving mean curvature flow in the limit. The stochastic term destroys the precise conservation law, instead the total mass changes like a Brownian motion in time. For our equation, the comparison argument does not work, so that to study the limit we adopt the asymptotic expansion method, which extends that for deterministic equations us...
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作者：Baur, Erich; Miermont, Gregory; Ray, Gourab
作者单位：Ecole Normale Superieure de Lyon (ENS de LYON); University of Victoria
摘要：We study noncompact scaling limits of uniform random planar quadran-gulations with a boundary when their size tends to infinity. Depending on the asymptotic behavior of the boundary size and the choice of the scaling factor, we observe different limiting metric spaces. Among well-known objects like the Brownian plane or the self-similar continuum random tree, we construct two new one-parameter families of metric spaces that appear as scaling limits: the Brownian half-plane with skewness parame...
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作者：Hammond, Alan
作者单位：University of California System; University of California Berkeley
摘要：In last passage percolation models lying in the KPZ universality class, the energy of long energy-maximizing paths may be studied as a function of the paths' pair of endpoint locations. Scaled coordinates may be introduced, so that these maximizing paths, or polymers, now cross unit distances with unit-order fluctuations, and have scaled energy, or weight, of unit order. In this article, we consider Brownian last passage percolation in these scaled coordinates. In the narrow wedge case, one en...
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作者：Nam, Danny; Sly, Allan
作者单位：Princeton University
摘要：We study the Swendsen-Wang dynamics for the q-state Potts model on the lattice. Introduced as an alternative algorithm of the classical single-site Glauber dynamics, the Swendsen-Wang dynamics is a nonlocal Markov chain that recolors many vertices at once based on the random-cluster representation of the Potts model. In this work, we establish cutoff phenomenon for the Swendsen-Wang dynamics on the lattice at sufficiently high temperatures, proving that it exhibits a sharp transition from unmi...
