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

Abstract

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 \(0<\mathbb{E}[\eta_0]<\infty\). 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 \(\mathbb{P}(|\xi_1|\ge n)\sim n^{-\alpha}L_\xi(n)\), 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 \(\mathbb{P}(|\xi_1|>n)\sim c_\xi n^{-\alpha}\); in the finite-variance case assume \(\mathbb{E}[\xi_1]=0\) and \(\operatorname{Var}(\xi_1)=\sigma^2\in(0,\infty)\). Setting \(r=\alpha\) in the stable case and \(r=2\) in the finite-variance case, if the law of \(\pi\) has density \(f_\pi(u)\sim(1-u)^{\beta-1}\ell((1-u)^{-1})\) as \(u\uparrow1\), then, for \(0<\beta<1\), \(n\mathbb{P}(D^\to\ge n)\sim C_\beta n^{1-r\beta}\ell(n^r)\), with explicit \(C_\beta\). Hence the sharp off-critical threshold is \(\beta_c=1/r\): survival holds for \(\beta<1/r\), extinction holds almost surely for \(\beta>1/r\), and explicit sufficient conditions on the critical line leave a factor-four gap.