The Frog Model on $\mathbb Z$ with Random Discrete Weibull Lifetimes and Biased Nearest-Neighbour Random Walks

Abstract

We study the frog model on $\mathbb Z$ with particle-wise random discrete Weibull lifetimes and biased nearest-neighbour random walks. Each particle has an independent survival parameter $π\in(0,1)$. Conditionally on $π=p$, its lifetime $Ξ$ satisfies $$ \mathbb P(Ξ\ge k\mid π=p)=p^{k^γ}, \, k\in\mathbb{N}_0, $$ where $γ>0$. The distribution of $π$ is assumed to have right-edge density $$ f_π(u)\sim (1-u)^{β-1} L\left(\frac1{1-u}\right), \, u\uparrow1, $$ where $β>0$ and $L:(0,\infty)\to(0,\infty)$ is a slowly varying function at infinity. The main step is to estimate the tail of the maximal displacement of a single particle before death. If $τ_n$ denotes the time needed by the underlying walk to reach distance $n$, then $$ \mathbb P(D^*\ge n)=\mathbb E[G(τ_n)], \, G(k):=\mathbb P(Ξ\ge k). $$ Since $$ G(k)\sim Γ(β)k^{-γβ}L(k^γ), $$ and the biased nearest-neighbour random walk has linear hitting-time scale, the off-critical threshold is $β_c=1/γ$. If $β>β_c$ and the initial number of particles per site has finite mean, the model dies out almost surely. If $β<β_c$ and the initial configuration is not almost surely empty, the model survives with positive probability in the direction of the drift.