First passage time of the frog model has a sublinear variance

Van Hao Can,Shuta Nakajima

doi:10.1214/19-ejp334

Abstract

In this paper, we show that the first passage time in the frog model on $\mathbb{Z} ^{d}$ with $d\geq 2$ has a sublinear variance. This implies that the central limit theorem does not hold at least with the standard diffusive scaling. The proof is based on the method introduced in [4, 11] combined with a control of the maximal weight of paths in a locally dependent site-percolation. We also apply this method to get the linearity of the lengths of optimal paths.

Highlights

Frog models are simple but well-known models in the study of the spread of infection
We are interested in the long time behavior of the infected individuals
One can show that Var(T(x)) = O(|x|1(1 + log |x|1)2A), for some constant A, see Corollary 2.3. This result is not enough to answer the question on the standard central limit theorem

Summary

Introduction

Frog models are simple but well-known models in the study of the spread of infection. One can show that Var(T(x)) = O(|x|1(1 + log |x|1)2A), for some constant A, see Corollary 2.3 This result is not enough to answer the question on the standard central limit theorem. The sublinearity of variance as in Theorem 1.1, which is called the superconcentration, was first discovered in the first passage percolation with Bernoulli edge weights by Benjamini, Kalai and Schramm [5]. We prove Lemma 2.6 to control the maximal weight of paths in locally dependent site-percolation by using a known result for independent site-percolation and tessellation arguments. Using a similar method, we prove the linearity of the length of optimal paths

The linearity of the lengths of optimal paths

Organization of this paper

Preliminaries