Upper tail estimates for the first passage time in the frog model

Naoki Kubota

Published 2016-12-27Version 1

We study the so-called frog model with random initial configurations. The dynamics of this model are described as follows: Some particles are randomly assigned on any site of the multidimensional cubic lattice $\mathbb{Z}^d$ ($d \geq 2$). Initially, only particles at the origin are active, and these independently perform simple random walks on $\mathbb{Z}^d$. Other particles are sleeping and do not move. When sleeping particles are attacked by an active particle, they become active and start moving in a similar fashion. In this context, the aim of this paper is to obtain some tail estimates for the first passage time at which an active particle reaches a target site. As a consequence, we derive order of intervals of initial positions of particles attaining the first passage time.