Meeting time of independent random walks in random environment

Instituto de Matemática e Estatística, Universidade de São Paulo, rua do Matão 1010, CEP 05508–090, São Paulo, SP, Brazil
gallesco@ime.usp.br

Received: 11 September 2010
Revised: 10 August 2011

Abstract

We consider, in the continuous time version, γ independent random walks on Z₊ in random environment in Sinai’s regime. Let T_γ be the first meeting time of one pair of the γ random walks starting at different positions. We first show that the tail of the quenched distribution of T_γ, after a suitable rescaling, converges in probability, to some functional of the Brownian motion. Then we compute the law of this functional. Eventually, we obtain results about the moments of this meeting time. Being E_ω the quenched expectation, we show that, for almost all environments ω, E_ω[T_γ^c] is finite for c < γ(γ − 1) / 2 and infinite for c > γ(γ − 1) / 2.