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Deviation bounds for the first passage time in the frog model

Naoki Kubota

TL;DR

This work analyzes the first passage time $T(0,x)$ in the frog model with random initial configuration on $\mathbb{Z}^d$. It establishes a time constant $\mu(\cdot)$ through the subadditive ergodic theorem and proves nontrivial large deviation bounds: a right-tail bound showing sublinear decay of $\mathbb{P}(T(0,x) \ge (1+\varepsilon)\mu(x))$ and a left-tail bound under finite mean initial occupancy, together with concentration results for the modified time $T^*(0,x)$. The central technique combines percolation-based renormalization, the construction of a high-probability set of 'good' sites, and concentration inequalities for $T^*(0,x)$, enabling control of $T(0,x)$ via a decomposition that mitigates infinite waiting times when sleeping frogs are sparse. The results illuminate how the propagation of activated frogs, the geometry of $\mathcal{C}_\infty$, and the initial distribution $\omega$ jointly govern deviation behavior of first passage times in high dimensions, with implications for information spreading in random media. Overall, the paper advances the understanding of deviation phenomena in frog-like infection processes by providing explicit tail bounds and a robust concentration framework grounded in subadditive and percolation techniques.

Abstract

We consider the so-called frog model with random initial configurations. The dynamics of this model is described as follows: Some particles are randomly assigned on any site of the multidimensional cubic lattice. Initially, only particles at the origin are active and these independently perform simple random walks. The other particles are sleeping and do not move at first. When sleeping particles are hit by an active particle, they become active and start moving in a similar fashion. The aim of this paper is to derive large deviation and concentration bounds for the first passage time at which an active particle reaches a target site.

Deviation bounds for the first passage time in the frog model

TL;DR

This work analyzes the first passage time in the frog model with random initial configuration on . It establishes a time constant through the subadditive ergodic theorem and proves nontrivial large deviation bounds: a right-tail bound showing sublinear decay of and a left-tail bound under finite mean initial occupancy, together with concentration results for the modified time . The central technique combines percolation-based renormalization, the construction of a high-probability set of 'good' sites, and concentration inequalities for , enabling control of via a decomposition that mitigates infinite waiting times when sleeping frogs are sparse. The results illuminate how the propagation of activated frogs, the geometry of , and the initial distribution jointly govern deviation behavior of first passage times in high dimensions, with implications for information spreading in random media. Overall, the paper advances the understanding of deviation phenomena in frog-like infection processes by providing explicit tail bounds and a robust concentration framework grounded in subadditive and percolation techniques.

Abstract

We consider the so-called frog model with random initial configurations. The dynamics of this model is described as follows: Some particles are randomly assigned on any site of the multidimensional cubic lattice. Initially, only particles at the origin are active and these independently perform simple random walks. The other particles are sleeping and do not move at first. When sleeping particles are hit by an active particle, they become active and start moving in a similar fashion. The aim of this paper is to derive large deviation and concentration bounds for the first passage time at which an active particle reaches a target site.

Paper Structure

This paper contains 16 sections, 17 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. The model
  3. Main results
  4. Earlier literature
  5. Organization of the paper
  6. Preliminaries
  7. Supercritical site percolation
  8. Upper tail estimate for the first passage time
  9. Right tail large deviation bound
  10. Step 1
  11. Step 2
  12. Step 3
  13. Left tail large deviation and concentration bounds
  14. Case (1)
  15. Case (2)
  16. ...and 1 more sections

Key Result

Proposition 1.1

There exists a norm $\mu(\cdot)$ (which is called the time constant) on $\mathbb{R}^d$ such that almost surely on the event $\{ \omega(0) \geq 1 \}$, Furthermore, $\mu(\cdot)$ is invariant under permutations of the coordinates and under reflections in the coordinate hyperplanes, and satisfies where $\xi_1$ is the first coordinate vector of $\mathbb{R}^d$.

Theorems & Definitions (29)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • ...and 19 more