重尾位移的杀死分支随机游走的尖锐相变
A Sharp Phase Transition for Killed Branching Random Walks with Heavy-Tailed Displacements
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中文总结 AI 辅助
本文研究重尾位移分支随机游走在超线性杀死屏障下的存活相变，证明临界值为4而非一阶矩启发式的2，并分析临界值处的非确定性。
中文摘要 AI 辅助
我们研究在超临界 Galton-Watson 树中，受确定性超线性杀死屏障且具有重尾位移的分支随机游走的存活性。对于固定的 $m>0$，我们考虑一族这样的屏障，并杀死祖先路径低于屏障的粒子。在一些标准假设下，我们建立了在 $m=4$ 处的尖锐相变：当 $m<4$ 时过程几乎必然灭绝，而当 $m>4$ 时以正概率存活。值得注意的是，自然的一阶矩启发式建议阈值为 $m=2$；真正的临界值 $4$ 源于沿无限存活射线的确定性约束。在临界值 $m=4$ 处，我们给出例子表明相同的标准假设不能确定存活行为。证明基于对无限存活射线和嵌入树的分析。
英文摘要
We study the survival of a branching random walk in a supercritical Galton-Watson tree subject to a deterministic, superlinear killing barrier with heavy-tailed displacements. For a fixed $m>0$, we consider a family of such barriers and kill particles whose ancestral paths fall below the barrier. Under some standing assumptions, we establish a sharp phase transition at $m=4$: the process becomes extinct almost surely for $m<4$, while survives with positive probability for $m>4$. Notably, a natural first-moment heuristic suggests the threshold $m=2$; the true critical value $4$ arises from a deterministic constraint along infinite surviving rays. At the critical value $m=4$, we provide examples showing that the same standing assumptions do not determine the survival behavior. The proofs are based on analysis of infinite surviving rays and embedded trees.
发表机构
- University College London（伦敦大学学院）
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