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 $\mathrm{\Xi }$ satisfies $$\mathbb{P}(\mathrm{\Xi }\ge k\mid \pi =p)=p^{k^{\gamma }},k\in {\mathbb{N}}_0,$$ where $\gamma >0$. The distribution of $\pi$ is assumed to have right-edge density $$f_{\pi }(u)\sim (1-u{)}^{\beta -1}L\left(\frac{1}{1-u}\right),u\uparrow 1,$$ where $\beta >0$ and $L:(0,\mathrm{\infty })\to (0,\mathrm{\infty })$ 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 $$\mathbb{P}(D^{\ast }\ge n)=\mathbb{E}[G({\tau }_n)],G(k):=\mathbb{P}(\mathrm{\Xi }\ge k)\mathrm{.}$$ Since $$G(k)\sim \mathrm{\Gamma }(\beta )k^{-\gamma \beta }L(k^{\gamma }),$$ and the biased nearest-neighbour random walk has linear hitting-time scale, the off-critical threshold is ${\beta }_c=1\mathrm{/}\gamma$. 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.