A Sharp Phase Transition for Killed Branching Random Walks with Heavy-Tailed Displacements
Abstract
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.
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BibTeX
@misc{samanta2026sharp,
title = {{A Sharp Phase Transition for Killed Branching Random Walks with Heavy-Tailed Displacements}},
author = {Ramkrishna Jyoti Samanta},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32480},
eprint = {2609.32480},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32480},
}
RIS
TY - UNPB
TI - A Sharp Phase Transition for Killed Branching Random Walks with Heavy-Tailed Displacements
AU - Ramkrishna Jyoti Samanta
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32480
UR - https://arxiv.org/abs/2609.32480
AB - 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.
ER -