Persistent occupation and critical marked sums on branching random walks
Abstract
We study two path functionals of a branching random walk at an exponential normalization. For centered integer displacements bounded above by one, we construct on survival one ray whose local time at every large spatial level is bounded below by a positive multiple of the square of that level. For displacements in {−1,0,1}, overlapping forward profiles give a single ray with lower coefficient 4t/(σ2j1,12), where σ2 is the tilted variance and j1,1 the first positive zero of J1. The profile energy quotient is optimized exactly. For discounted marked path sums with real displacements, we prove critical-moment sufficiency without a drift assumption, and necessity under a finite ascending-ladder mean and a weighted sibling moment. At positive tilted drift, a stopping-line argument replaces the sibling condition by an Llog L hypothesis. A sparse mark construction identifies an exact universality obstruction in the constant-displacement subclass. At negative tilted drift, weighted powers give explicit sufficient logarithmic-tail conditions for every exponent t>0 and examples excluding a universal critical-moment converse. Under our sufficient conditions, path sums converge uniformly and attain their maxima. The exact optimal persistent local-time constant and a general negative-drift converse remain open.
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BibTeX
@misc{skiredj2026persistent,
title = {{Persistent occupation and critical marked sums on branching random walks}},
author = {Abderrahman Skiredj and Ismail Berrada},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33897},
eprint = {2610.33897},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33897},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Persistent occupation and critical marked sums on branching random walks
AU - Abderrahman Skiredj
AU - Ismail Berrada
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33897
UR - https://arxiv.org/abs/2610.33897
AB - We study two path functionals of a branching random walk at an exponential normalization. For centered integer displacements bounded above by one, we construct on survival one ray whose local time at every large spatial level is bounded below by a positive multiple of the square of that level. For displacements in {−1,0,1}, overlapping forward profiles give a single ray with lower coefficient 4t/(σ^2j_1,1^2), where σ^2 is the tilted variance and j_1,1 the first positive zero of J_1. The profile energy quotient is optimized exactly. For discounted marked path sums with real displacements, we prove critical-moment sufficiency without a drift assumption, and necessity under a finite ascending-ladder mean and a weighted sibling moment. At positive tilted drift, a stopping-line argument replaces the sibling condition by an Llog L hypothesis. A sparse mark construction identifies an exact universality obstruction in the constant-displacement subclass. At negative tilted drift, weighted powers give explicit sufficient logarithmic-tail conditions for every exponent t>0 and examples excluding a universal critical-moment converse. Under our sufficient conditions, path sums converge uniformly and attain their maxima. The exact optimal persistent local-time constant and a general negative-drift converse remain open.
ER -