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The Green function of a transient symmetric Markov process can be interpreted as the covariance of a centered Gaussian process. This relation leads to several fruitful identities in law. Symmetric Markov processes and their associated Gaussian process both benefit from…

If $P_t$ is the semigroup associated with the Kawasaki dynamics on $Z^d$ and $f$ is a local function on the configuration space, then the variance with respect to the invariant measure $\mu$ of $P_t f$ goes to zero as $t\to…

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…