-
作者：Asselah, Amine; Schapira, Bruno; Sousi, Perla
作者单位：Universite Gustave-Eiffel; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Aix-Marseille Universite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); University of Cambridge
摘要：We study the scaling limit of the capacity of the range of a random walk on the integer lattice in dimension four. We establish a strong law of large numbers and a central limit theorem with a non-Gaussian limit. The asymptotic behaviour is analogous to that found by Le Gall in '86 [Comm. Math. Phys. 104 (1986) 471-507] for the volume of the range in dimension two.
-
作者：Ding, Jian; Dunlap, Alexander
作者单位：University of Pennsylvania; Stanford University
摘要：Let {Y-B(x) : is an element of B} be a discrete Gaussian free field in a two-dimensional box of B side length S with Dirichlet boundary conditions. We study Liouville first-passage percolation: the shortest-path metric in which each vertex x is given a weight of e(gamma YB(x)) for some gamma > 0. We show that for sufficiently small but fixed gamma > 0, for any sequence of scales {S-k} there exists a subsequence along which the appropriately scaled and interpolated Liouville FPP metric converge...
-
作者：Bloom, Thomas; Dauvergne, Duncan
作者单位：University of Toronto
摘要：We consider random polynomials of the form H-n(z) = Sigma(n)(j=0)xi(j)q(j)(z) where the {xi(j)} are i.i.d. nondegenerate complex random variables, and the {q(j)(z)} are orthonormal polynomials with respect to a compactly supported measure tau satisfying the Bernstein-Markov property on a regular compact set K subset of C. We show that if P(|xi(0)| > e(vertical bar z vertical bar)) = o(vertical bar z vertical bar(-1)), then the normalized counting measure of the zeros of H-n converges weakly in...
-
作者：Jiao, Yong; Osekowski, Adam; Wu, Lian
作者单位：Central South University; University of Warsaw
摘要：We introduce a notion of strong differential subordination of noncommutative semimartingales, extending Burkholder's definition from the classical case (Ann. Probab. 22 (1994) 995-1025). Then we establish the maximal weak-type (1, 1) inequality under the additional assumption that the dominating process is a submartingale. The proof rests on a significant extension of the maximal weak-type estimate of Cuculescu and a Gundy-type decomposition of an arbitrary noncommutative submartingale. We als...
-
作者：Eberle, Andreas; Guillin, Arnaud; Zimmer, Raphael
作者单位：University of Bonn; Universite Clermont Auvergne (UCA); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
摘要：We introduce a new probabilistic approach to quantify convergence to equilibrium for (kinetic) Langevin processes. In contrast to previous analytic approaches that focus on the associated kinetic Fokker-Planck equation, our approach is based on a specific combination of reflection and synchronous coupling of two solutions of the Langevin equation. It yields contractions in a particular Wasserstein distance, and it provides rather precise bounds for convergence to equilibrium at the borderline ...
-
作者：Domelevo, Komla; Petermichl, Stefanie
作者单位：Universite de Toulouse; Universite Toulouse III - Paul Sabatier
摘要：We prove optimal L-2 bounds for a pair of Hilbert space valued differentially subordinate martingales under a change of law. The change of law is given by a process called a weight and sharpness, and in this context refers to the optimal growth with respect to the characteristic of the weight. The pair of martingales is adapted, uniformly integrable and cadlag. Differential subordination is in the sense of Burkholder, defined through the use of the square bracket. In the scalar dyadic setting ...
-
作者：Lawler, Gregory; Sun, Xin; Wu, Wei
作者单位：University of Chicago; Columbia University; University of Warwick
摘要：The loop-erased random walk (LERW) in Z(4) is the process obtained by erasing loops chronologically for a simple random walk. We prove that the escape probability of the LERW renormalized by (log n) 1/3 converges almost surely and in L-p for all p > 0. Along the way, we extend previous results by the first author building on slowly recurrent sets. We provide two applications for the escape probability. We construct the two-sided LERW, and we construct a +/- 1 spin model coupled with the wired ...
-
作者：Grothaus, Martin; Wang, Feng-Yu
作者单位：University of Kaiserslautern; Tianjin University
摘要：For a contraction C-0-semigroup on a separable Hilbert space, the decay rate is estimated by using the weak Poincare inequalities for the symmetric and antisymmetric part of the generator. As applications, nonexponential convergence rate is characterized for a class of degenerate diffusion processes, so that the study of hypocoercivity is extended. Concrete examples are presented.
-
作者：Ding, Jian; Xu, Changji
作者单位：University of Pennsylvania; University of Chicago
摘要：Place an obstacle with probability 1 - p independently at each vertex of Z(d), and run a simple random walk until hitting one of the obstacles. For d >= 2 and p strictly above the critical threshold for site percolation, we condition on the environment where the origin is contained in an infinite connected component free of obstacles, and we show that the following path localization holds for environments with probability tending to 1 as n -> infinity: conditioned on survival up to time n we h...
-
作者：Hulley, Hardy; Ruf, Johannes
作者单位：University of Technology Sydney; University of London; London School Economics & Political Science
摘要：The following conditions are necessary and jointly sufficient for an arbitrary cadlag local martingale to be a uniformly integrable martingale: (A) The weak tail of the supremum of its modulus is zero; (B) its jumps at the first-exit times from compact intervals converge to zero in L-1 on the events that those times are finite; and (C) its almost sure limit is an integrable random variable.
