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 \((η_x)_{x\in\mathbb Z}\), satisfying \(0<\mathbb{E}[η_0]<\infty\). 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 \(π\in(0,1)\), with \(\mathbb{P}(L\ge k\mid π=p)=p^k\). 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 \(\mathbb{P}(|ξ_1|\ge n)\sim n^{-α}L_ξ(n)\), with \(L_ξ\) slowly varying, then survival holds with positive probability for \(0<α<1\), while for \(α=1\) both survival and almost sure extinction may occur. For \(1<α<2\), assume \(\mathbb{P}(|ξ_1|>n)\sim c_ξn^{-α}\); in the finite-variance case assume \(\mathbb{E}[ξ_1]=0\) and \(\operatorname{Var}(ξ_1)=σ^2\in(0,\infty)\). Setting \(r=α\) in the stable case and \(r=2\) in the finite-variance case, if the law of \(π\) has density \(f_π(u)\sim(1-u)^{β-1}\ell((1-u)^{-1})\) as \(u\uparrow1\), then, for \(0<β<1\), \(n\mathbb{P}(D^\to\ge n)\sim C_βn^{1-rβ}\ell(n^r)\), with explicit \(C_β\). Hence the sharp off-critical threshold is \(β_c=1/r\): survival holds for \(β<1/r\), extinction holds almost surely for \(β>1/r\), and explicit sufficient conditions on the critical line leave a factor-four gap.