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

Abstract

We study the frog model on $\mathbb{Z}$ with particle wise discrete Weibull lifetimes. Each particle has an i.i.d. survival parameter $\ddot{I}€\in (0,1)$; conditionally on $\ddot{I}€=p$, its lifetime $\hat{I}ž$ satisfies \[ P(Ξ\ge k\mid π=p)=p^{k^γ},\qquad k\in\mathbb{N}_0,γ>0. \] The law of $\ddot{I}€$ has right edge density \[ f_π(u)\sim(1-u)^{β-1},L\big((1-u)^{-1}\big)\qquad (u\uparrow 1), \] with ${\hat{I}}^2>0$ and $L$ slowly varying; let $\hat{I}\cdot$ denote the common law of the i.i.d. initial occupation numbers $\{\hat{I}{\cdot }_x{\}}_{x\in \mathbb{Z}}$. 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 ${\hat{I}}^3-$dependent threshold \[ β_c:=\frac{1}{2γ}. \] If ${\hat{I}}^2>{\hat{I}}_c^2$ and $E(\hat{I}\cdot )<\infty$, the process becomes extinct almost surely; if ${\hat{I}}^2<{\hat{I}}_c^2$ and $P(\hat{I}\cdot =0)<1$, it survives with positive probability. At the boundary ${\hat{I}}^2={\hat{I}}_c^2$ we provide explicit criteria in terms of $\mathrm{limsup}/\mathrm{liminf}$ of $L(n^{2{\hat{I}}^3})$. The case ${\hat{I}}^3=1$ (geometric lifetimes) recovers the benchmark ${\hat{I}}_c^2=\frac{1}{2}$ and the critical refinements previously obtained for random geometric lifetimes.

23 pages, 2 figures