We bound the probability that a continuous time simple random walk on the infinite percolation cluster on $Z^d$ returns to the origin at time $t$. We use this result to show that in dimensions 5 and higher the uniform spanning…

We are interested in the asymptotic analysis of the binary search tree (BST) under the random permutation model. Via an embedding in a continuous time model, we get new results, in particular the asymptotic behavior of the profile.

In this paper we prove that in the high temperature region of the Sherrington-Kirkpatrick model for a typical realization of the disorder the weighted average of spins $\sum_{i \leq N} t_i \sigma_i$ will be approximately Gaussian provided that $\max_{i \leq…

We prove that the logarithmic-Sobolev constant for Zero-Range Processes in a box of diameter $L$ may depend on $L$ but not on the number of particles. This is a first, but relevant and quite technical step, in the proof that…

A random incompressible velocity field in three dimensions composed by Poisson distributed Brownian vortex filaments is constructed. The filaments have a random thickness, length and intensity, governed by a measure $\gamma$. Under appropriate assumptions on $\gamma$ we compute the scaling…

Let $W$ be a one-dimensional Brownian motion starting from 0. Define $Y(t)= \int_0^t{ds \over W(s)}:= \lim_{\epsilon\to 0} \int_0^t 1_{(|W(s)|> \epsilon)} {ds\over W(s)}$ as Cauchy's principal value related to local time. We prove limsup and liminf results for the increments of…

Recently, Nicolas Bouleau has proposed an extension of the Donsker's invariance principle in the framework of Dirichlet forms. He proves that an erroneous random walk of i.i.d random variables converges in Dirichlet law toward the Ornstein-Uhlenbeck error structure on the…

In 1956, Dobrushin proved an important central limit theorem for non-homogeneous Markov chains. In this note, a shorter and different proof elucidating more the assumptions is given through martingale approximation.

Small-time asymptotic estimates of semigroups on a logarithmic scale are proved for all symmetric local Dirichlet forms on $\sigma$-finite measure spaces, which is an extension of the work by Hino and Ramírez [4].

We prove an estimate for the probability that a simple random walk in a simply connected subset $A$ of $Z^2$ starting on the boundary exits $A$ at another specified boundary point. The estimates are uniform over all domains of a…