-
作者：Haggstrom, Olle
作者单位：Chalmers University of Technology
摘要：Oded Schramm (1961-2008) influenced greatly the development of percolation theory beyond the usual Z(d) setting; in particular, the case of nonamenable lattices. Here, we review some of his work in this field.
-
作者：Saloff-Coste, L.; Zuniga, J.
作者单位：Cornell University; Stanford University
摘要：We study time-inhomogeneous Markov chains with finite state spaces using Nash and logarithmic-Sobolev inequalities, and the notion of c-stability. We develop the basic theory of such functional inequalities in the time-inhomogeneous context and provide illustrating examples.
-
作者：Bobkov, Sergey; Madiman, Mokshay
作者单位：University of Minnesota System; University of Minnesota Twin Cities; Yale University
摘要：A concentration property of the functional - log f (X) is demonstrated, when a random vector X has a log-concave density f on R-n. This concentration property implies in particular an extension of the Shannon-McMillan-Breiman strong ergodic theorem to the class of discrete-time stochastic processes with log-concave marginals.
-
作者：Coulibaly-Pasquier, Kolehe A.
作者单位：University of Luxembourg
摘要：Using Huisken's results about the mean curvature flow on a strictly convex hypersurface and Kendall-Cranston's coupling, we will build a stochastic process without birth and show that there exists a unique law of such a process. This process has many similarities with the circular Brownian motion studied by Emery and Schachermayer, and Arnaudon. In general this process is not a stationary process; it is linked to some differential equation without initial condition. We will show that this diff...
-
作者：Gloria, Antoine; Otto, Felix
作者单位：Max Planck Society
摘要：We consider a discrete elliptic equation on the d-dimensional lattice Z(d) with random coefficients A of the simplest type: they are identically distributed and independent from edge to edge. On scales large w.r.t. the lattice spacing (i.e., unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric homogenized matrix A(hom) = a(hom) Id is characterized by xi . A(hom)xi = <(xi + del ph...
-
作者：Beffara, Vincent; Nolin, Pierre
作者单位：Ecole Normale Superieure de Lyon (ENS de LYON); New York University
摘要：We investigate the so-called monochromatic arm exponents for critical percolation in two dimensions. These exponents, describing the probability of observing j disjoint macroscopic paths, are shown to exist and to form a different family from the (now well understood) polychromatic exponents. More specifically, our main result is that the monochromatic j-arm exponent is strictly between the polychromatic j-arm and (j + 1)-arm exponents.
-
作者：Schramm, Oded; Smirnov, Stanislav; Garban, Christophe
作者单位：University of Geneva; Ecole Normale Superieure de Lyon (ENS de LYON); Centre National de la Recherche Scientifique (CNRS)
摘要：We prove Tsirelson's conjecture that any scaling limit of the critical planar percolation is a black noise. Our theorems apply to a number of percolation models, including site percolation on the triangular grid and any subsequential scaling limit of bond percolation on the square grid. We also suggest a natural construction for the scaling limit of planar percolation, and more generally of any discrete planar model describing connectivity properties.
-
作者：Vershynin, Roman
作者单位：University of Michigan System; University of Michigan
摘要：For probability distributions on R-n, we study the optimal sample size N = N(n, p) that suffices to uniformly approximate the pth moments of all one-dimensional marginals. Under the assumption that the marginals have bounded 4p moments, we obtain the optimal bound N = O(n(p/2)) for p > 2. This bound goes in the direction of bridging the two recent results: a theorem of Guedon and Rudelson [Adv. Math. 208 (2007) 798-823] which has an extra logarithmic factor in the sample size, and a result of ...
-
作者：Sznitman, Alain-Sol
作者单位：Swiss Federal Institutes of Technology Domain; ETH Zurich
摘要：The vacant set of random interlacements on Z(d), d >= 3, has nontrivial percolative properties. It is known from Sznitman [Ann. Math. 171 (2010) 2039-2087], Sidoravicius and Sznitman [Comm. Pure Appl. Math. 62 (2009) 831-858] that there is a nondegenerate critical value u(*) such that the vacant set at level u percolates when u < u(*) and does not percolate when u > u(*). We derive here an asymptotic upper bound on u(*), as d goes to infinity, which complements the lower bound from Sznitman [P...
-
作者：Amir, Gideon; Angel, Omer; Valko, Benedek
作者单位：Bar Ilan University; University of British Columbia; University of Wisconsin System; University of Wisconsin Madison
摘要：In the multi-type totally asymmetric simple exclusion process (TASEP) on the line, each site of Z is occupied by a particle labeled with some number, and two neighboring particles are interchanged at rate one if their labels are in increasing order. Consider the process with the initial configuration where each particle is labeled by its position. It is known that in this case a.s. each particle has an asymptotic speed which is distributed uniformly on [-1, 1]. We study the joint distribution ...
