带随机存活参数与对称随机游走的整数集\\(\mathbb{Z}\\)上区间激活青蛙模型的灭绝与存活
Extinction and Survival in an Interval-Activation Frog Model on \mathbb{Z} with Random Survival Parameters and Symmetric Random Walks
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中文总结 AI 辅助
该研究针对整数集上带随机存活参数的区间激活青蛙模型，通过单蛙最大右位移的尾行为推导存活与灭绝判据，明确不同游走尾指数下的阈值及临界线间隙。
中文摘要 AI 辅助
我们研究整数集\\(\mathbb{Z}\\)上的区间激活青蛙模型，其初始青蛙数量\\((η_x)_{x\in\mathbb Z}\\)为独立同分布，满足\\(0<\mathbb{E}[η_0]<\infty\\)。原点处的青蛙初始为激活状态，其余所有青蛙处于休眠状态。每只青蛙进行对称整数值随机游走，且具有由独立同分布的存活参数\\(π\in(0,1)\\)决定的随机寿命\\(L\\)，满足\\(\mathbb{P}(L\ge k\mid π=p)=p^k\\)。每次跳跃会激活其端点之间所有整数位点上的休眠青蛙。令\\(D^\to\\)表示单只青蛙死亡前的最大向右位移。我们通过\\(D^\to\\)的尾行为推导出存活和灭绝判据。若\\(\mathbb{P}(|ξ_1|\ge n)\sim n^{-α}L_ξ(n)\\)（其中\\(L_ξ\\)为慢变函数），则当\\(0<α<1\\)时，模型以正概率存活；而当\\(α=1\\)时，存活和几乎必然灭绝均可能发生。对于\\(1<α<2\\)的情形，假设\\(\mathbb{P}(|ξ_1|>n)\sim c_ξn^{-α}\\)；在有限方差情形下，假设\\(\mathbb{E}[ξ_1]=0\\)且\\(\operatorname{Var}(ξ_1)=σ^2\in(0,\infty)\\)。在稳定情形下取\\(r=α\\)，在有限方差情形下取\\(r=2\\)，若当\\(u\uparrow1\\)时，\\(π\\)的分布具有密度\\(f_π(u)\sim(1-u)^{β-1}\ell((1-u)^{-1})\\)，则当\\(0<β<1\\)时，\\(n\mathbb{P}(D^\to\ge n)\sim C_βn^{1-rβ}\ell(n^r)\\)，其中\\(C_β\\)有显式表达式。由此得到精确的非临界阈值为\\(β_c=1/r\\)：当\\(β<1/r\\)时模型存活，当\\(β>1/r\\)时几乎必然灭绝，而临界线上的显式充分条件存在4倍因子的间隙。
英文摘要
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.
