2026/07/29 by Gustavo Oshiro de Carvalho, Fábio Prates Machado, José Hermenegildo Ramírez-González
Mathematics · #math.PR #msc:60G50 #msc:60G52 #msc:60K35
38 pages, 2 figures
arxiv created 2026/07/29 · arxiv updated 2026/07/30
We study an interval-activation frog model on \(\mathbb Z\) with i.i.d. initial numbers of frogs \((ηx)x∈\mathbb Z\), satisfying \(0<𝔼[η0]<∞\). 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 \(π∈(0,1)\), with \(ℙ(L≥ k| π=p)=pk\). Every jump activates all sleeping frogs at the integer sites between its endpoints. Let \(D^→\) denote the maximal rightward displacement of a single frog before death. We derive survival and extinction criteria from the tail behavior of \(D^→\). If \(ℙ(|ξ1|≥ n)∼ n-αLξ(n)\), with \(Lξ\) 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 \(ℙ(|ξ1|>n)∼ cξn-α\); in the finite-variance case assume \(𝔼[ξ1]=0\) and \(Var(ξ1)=σ2∈(0,∞)\). Setting \(r=α\) in the stable case and \(r=2\) in the finite-variance case, if the law of \(π\) has density \(fπ(u)∼(1-u)β-1ℓ((1-u)-1)\) as \(u\uparrow1\), then, for \(0<β<1\), \(nℙ(D^→≥ n)∼ Cβn1-rβℓ(nr)\), with explicit \(Cβ\). Hence the sharp off-critical threshold is \(βc=1/r\): 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.