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

TL;DR

This work analyzes the frog model on $\mathbb{Z}^d$ with Bernoulli initial occupancy and investigates how the time constant $\mu_r$ diverges as the initial density $r$ vanishes. The authors prove that $\mu_r(x) \asymp \delta_d(r)\|x\|_1$ with $\delta_d(r)=\sqrt{\frac{|\,\log r\,|}{r}}$ for $d=2$ and $\delta_d(r)=1/\sqrt{r}$ for $d\ge3$, providing matching lower and upper bounds and a corollary on the shrinking asymptotic shape. The main methods combine chaining arguments for active frog trajectories, a percolation framework over $r$-good boxes, and a recursive seed-activation scheme aided by large-deviation controls. The results reveal that the rate of divergence of propagation speed depends critically on dimension, offering insights into scaling limits and asymptotic geometry under vanishing initial density.

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 $d \geq 3$.

Figures (3)

• Figure 1: Schematic picture of the $r$-good event. Starting from any frog in the box of size $2r^{-(1/2+\epsilon)}$ at the center, we can reach any of the boxes in the coordinate directions in time comparable to $r^{-d/2} \delta_d(r)$.
• Figure 2: Schematic picture of the event $\mathcal{S}_r(v)$. Every frog in $\Theta_r^\text{in}(v)$ hits at least one frog in each of $\Theta_r^\text{out}(v)$ and $\Theta_r^\text{in}(u)$.
• Figure 3: Schematic picture of $\mathcal{A}_r$. Smaller boxes are $\Theta_r^\text{out}(v)$ and the frogs reach many of them in a relatively short time. Then from one of those boxes, we can find a frog that reaches $y$.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['thm:divergence']}
• Proposition 3.1
• proof : Proof of Theorem \ref{['thm:upper']}