The Frog Model on $\mathbb{Z}$ with Discrete Weibull Lifetimes and Random Parameter $p$

TL;DR

This paper analyzes a variant of the frog model on the integer lattice with particles having discrete Weibull lifetimes and random survival parameters, establishing a sharp phase transition between extinction and survival depending on the parameters.

Contribution

It introduces a new model with Weibull-distributed lifetimes and random parameters, extending previous geometric lifetime models, and derives explicit phase transition criteria.

Findings

Sharp extinction and survival thresholds depending on parameters

Extension of the Beta family for survival parameters

Recovery of known results for geometric lifetimes when b3=1

Abstract

We study the frog model on $\mathbb{Z}$ with particle wise discrete Weibull lifetimes. Each particle has an i.i.d. survival parameter $\pi\in(0,1)$; conditionally on $\pi=p$, its lifetime $\Xi$ satisfies \[ P(\Xi\ge k\mid \pi=p)=p^{k^{\gamma}},\qquad k\in\mathbb{N}_0,\gamma>0. \] The law of $\pi$ has right edge density \[ f_\pi(u)\sim(1-u)^{\beta-1},L\big((1-u)^{-1}\big)\qquad (u\uparrow 1), \] with $\beta>0$ and $L$ slowly varying; let $\eta$ denote the common law of the i.i.d. initial occupation numbers $\{\eta_x\}_{x\in\mathbb{Z}}$. The survival parameter distribution strictly extends the Beta family, while the lifetime distribution extends the geometric case. We prove a sharp extinction and survival dichotomy with the $\gamma-$dependent threshold \[ \beta_c:=\frac{1}{2\gamma}. \] If $\beta>\beta_c$ and $E(\eta)<\infty$, the process becomes extinct almost surely; if $\beta<\beta_c$ and $P(\eta=0)<1$, it survives with positive probability. At the boundary $\beta=\beta_c$ we provide explicit criteria in terms of $\limsup/\liminf$ of $L(n^{2\gamma})$. The case $\gamma=1$ (geometric lifetimes) recovers the benchmark $\beta_c=\frac{1}{2}$ and the critical refinements previously obtained for random geometric lifetimes.