Rate of divergence of time constant for frog model with vanishing initial density

TL;DR

This paper investigates how the time constant for propagation in the frog model diverges as the initial density of sleeping particles approaches zero, revealing different divergence rates in two and higher dimensions.

Contribution

It provides a detailed analysis of the divergence rate of the time constant in the frog model with vanishing initial density, highlighting dimension-dependent differences.

Findings

Divergence of the time constant as initial density vanishes

Different divergence rates between 2D and higher dimensions

Quantitative characterization of divergence rates

Abstract

The frog model with a Bernoulli initial configuration is an interacting particle system on the $d$-dimensional lattice ($d \geq 2$) with two types of particles: active and sleeping. Active particles perform independent simple random walks. In contrast, although the sleeping particles do not move at first, they become active and start moving once touched by the active particles. Initially, only the origin has a single active particle, and the other sites have sleeping particles according to a Bernoulli distribution. After the original active particle starts moving, further active particles are gradually generated under the above rule and propagate across the lattice. The time required for the propagation of active frogs is expected to increase as the parameter of the Bernoulli distribution decreases, since fewer frogs are available. The aim of this paper is to investigate this increase in the vanishing density limit. In particular, we observe that it diverges and the rate of divergence differs significantly between $d=2$ and $d \geq 3$.