Frog model on $\mathbb{Z}$ with random survival parameter

TL;DR

This work analyzes a frog model on $\mathbb{Z}$ where each active particle has a lifetime distributed geometrically with a random parameter $p$ drawn from $\pi$. Focusing on $\pi \sim \text{Beta}(\alpha,\beta)$, the authors identify a phase transition at $\beta=0.5$: extinction almost surely for $\beta>0.5$ and positive survival probability for certain $\alpha,\beta$ when $\beta\le 0.5$, with a further threshold $\alpha_0$ at $\beta=0.5$ determining survival. A key methodological contribution is the coupling of the frog model with rumor/firework processes to translate survival into percolation events, enabling precise percolation criteria via maximal displacements $D^{\rightarrow}$, $D^{\leftarrow}$, and $D^*$. The paper also proves that, whenever survival occurs, the root is visited infinitely often (recurrence), highlighting how environmental randomness can create survival and recurrence in one dimension. Collectively, these results illustrate the power of percolation couplings in analyzing spatially structured stochastic processes with random environments.

Abstract

We study the frog model on \( \mathbb{Z} \) with geometric lifetimes, introducing a random survival parameter. Active and inactive particles are placed at the vertices of \( \mathbb{Z} \). The lifetime of each active particle follows a geometric random variable with parameter \( 1-p \), where \( p \) is randomly sampled from a distribution \( π\). Each active particle performs a simple random walk on \( \mathbb{Z} \) until it dies, activating any inactive particles it encounters along its path. In contrast to the usual case where \( p \) is fixed, we show that there exist non-trivial distributions \( π\) for which the model survives with positive probability. More specifically, for $π\sim Beta(α,β)$, we establish the existence of a critical value \( β=0.5 \), that separates almost sure extinction from survival with positive probability. Furthermore, we show that the model is recurrent whenever it survives with positive probability.

Figures (1)

• Figure 1: Phase diagram for $\text{FM}(\mathbb{Z},\mathcal{G}_\pi,\eta)$ where $\pi\sim Beta(\alpha,\beta)$ and $\eta$ is such that $P(\eta=0)<1$ and $E(\eta)<\infty$. The shaded part represents the region where $P(\text{FM}(\mathbb{Z},\mathcal{G}_\pi,\eta) \text{ survives})=0$ while the hatched part represents the region where $P(\text{FM}(\mathbb{Z},\mathcal{G}_\pi,\eta) \text{ survives})>0$. The black circle and the dashed line represent the unknown parts of the diagram.

Theorems & Definitions (19)

• Definition 1.1
• Proposition 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 1.6
• Theorem 1.7
• Remark 1.8
• Lemma 2.1
• proof