Extinction and Survival in an Interval-Activation Frog Model on $\mathbb{Z}$ with Random Survival Parameters and Symmetric Random Walks

We study an interval-activation frog model on $\mathbb{Z}$ with i.i.d.\ initial numbers of frogs $({\eta }_x{)}_{x\in \mathbb{Z}}$, satisfying $0<\mathbb{E}[{\eta }_0]<\mathrm{\infty }$. Frogs at the origin are initially active and all others are sleeping. Each frog performs a symmetric integer-valued random walk and has a random lifetime $L$ determined by an i.i.d.\ survival parameter $\pi \in (0,1)$, with $\mathbb{P}(L\ge k\mid \pi =p)=p^k$. Every jump activates all sleeping frogs at the integer sites between its endpoints. Let $D^{\to }$ denote the maximal rightward displacement of a single frog before death. We derive survival and extinction criteria from the tail behavior of $D^{\to }$. If $\mathbb{P}(\mathrm{\mid }{\xi }_1\mathrm{\mid }\ge n)\sim n^{-\alpha }{L}_{\xi }(n)$, with $L_{\xi }$ slowly varying, then survival holds with positive probability for $0<\alpha <1$, while for $\alpha =1$ both survival and almost sure extinction may occur. For $1<\alpha <2$, assume $\mathbb{P}(\mathrm{\mid }{\xi }_1\mathrm{\mid }>n)\sim c_{\xi }{n}^{-\alpha }$; in the finite-variance case assume $\mathbb{E}[{\xi }_1]=0$ and $\mathrm{Var}({\xi }_1)={\sigma }^2\in (0,\mathrm{\infty })$. Setting $r=\alpha$ in the stable case and $r=2$ in the finite-variance case, if the law of $\pi$ has density $f_{\pi }(u)\sim (1-u{)}^{\beta -1}\mathrm{\ell }((1-u{)}^{-1})$ as $u\uparrow 1$, then, for $0<\beta <1$, $n\mathbb{P}(D^{\to }\ge n)\sim C_{\beta }{n}^{1-r\beta }\mathrm{\ell }(n^r)$, with explicit $C_{\beta }$. Hence the sharp off-critical threshold is ${\beta }_c=1\mathrm{/}r$: survival holds for $\beta <1\mathrm{/}r$, extinction holds almost surely for $\beta >1\mathrm{/}r$, and explicit sufficient conditions on the critical line leave a factor-four gap.