The Frog Model on $\mathbb Z$ with Random Discrete Weibull Lifetimes and Biased Nearest-Neighbour Random Walks
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 $\pi\in(0,1)$. Conditionally on $\pi=p$, its lifetime $\Xi$ satisfies $$ \mathbb P(\Xi\ge k\mid \pi=p)=p^{k^\gamma}, \, k\in\mathbb{N}_0, $$ where $\gamma>0$. The distribution of $\pi$ is assumed to have right-edge density $$ f_\pi(u)\sim (1-u)^{\beta-1} L\left(\frac1{1-u}\right), \, u\uparrow1, $$ where $\beta>0$ 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 $\tau_n$ denotes the time needed by the underlying walk to reach distance $n$, then $$ \mathbb P(D^*\ge n)=\mathbb E[G(\tau_n)], \, G(k):=\mathbb P(\Xi\ge k). $$ Since $$ G(k)\sim \Gamma(\beta)k^{-\gamma\beta}L(k^\gamma), $$ and the biased nearest-neighbour random walk has linear hitting-time scale, the off-critical threshold is $\beta_c=1/\gamma$. If $\beta>\beta_c$ and the initial number of particles per site has finite mean, the model dies out almost surely. If $\beta<\beta_c$ and the initial configuration is not almost surely empty, the model survives with positive probability in the direction of the drift.