arXiv subjects

Clement Rau

Publications and source records attributed to Clement Rau.

3 recordsLinked to original sources

arXiv2009

A stochastic min-driven coalescence process and its hydrodynamical limit

A stochastic system of particles is considered in which the sizes of the particles increase by successive binary mergers with the constraint that each coagulation event involves a particle with minimal size. Convergence of a suitably renormalised version of this process to a deterministic hydrodynamical limit is shown and the time evolution of the minimal size is studied for both deterministic and stochastic models.

math.PR↗

arXiv2007

Existence of graphs with sub exponential transitions probability decay and applications

In this paper, we present a complete proof of the construction of graphs with bounded valency such that the simple random walk has a return probability at time $n$ at the origin of order $exp(-n^α),$ for fixed $α\in [0,1[$ and with Folner function $exp(n^{\frac{2α}{1-α}})$. We begin by giving a more detailled proof of this result contained in (see \cite{ershdur}). In the second part, we give an application of the existence of such graphs. We obtain bounds of the correct order for some functional of the local time of a simple random walk on an infinite cluster on the percolation model.

math.PR↗

arXiv2006

Sur le nombre de points visités par une marche aléatoire sur un amas infini de percolation

In this article, we consider random walk on the infinite cluster of bond percolation on $\Z^d (d \geq 2)$. We show that the Laplace transformation of the number of visited points $N\_n$, has a behaviour as the random walk was on $\Z^d$. More precisely, for all $0<α<1$, we proved that there exist constants $C\_i$ and $C\_s$ such that for all infinite cluster that contains the origin, we have: $$ e^{-C\_i n^{\frac{d}{d+2}}} \leq \E\_0^ω (α^{N\_n}) \leq e^{-C\_sn^{\frac{d}{d+2}}}.$$ Our approach is based on finding an isoperimetric inequalities on the infinite cluster, lifted on a wreath product which give good behaviour. The problem of the isoperimetry on wreath product was already raised by A.Ershler.

math.PR↗