Critical branching random walks on ℤ2: local survival probabilities and Yaglom limit theorems
Abstract
We consider a branching random walk on ℤ2 with critical offspring of mean $1$ and spatial motion governed by the jumps of a lazy simple random walk. For every site x∈ℤ2, we obtain a uniform asymptotical estimate for the local survival probability, i.e., the probability that there are particles at x at large time n. Using Stein's method, we establish a Yaglom-type theorem for the number of particles at x at time n when x is at distance of order √n from the origin. Moreover, at the position occupied by a typical particle at time n, the number of particles at that site, normalized by log n, converges in law to a Gamma distribution, thereby confirming a conjecture of Lalley and Zheng [Ann. Probab. 39 (2010), 327-368]. Finally, we prove that, conditional on local survival, the total number of particles at time n, divided by n, also converges weakly to a Gamma distribution.
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BibTeX
@misc{bai2026critical,
title = {{Critical branching random walks on $\mathbbZ^2$: local survival probabilities and Yaglom limit theorems}},
author = {Tianyi Bai and Xinxin Chen and Shen Lin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.15538},
eprint = {2609.15538},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15538},
}
RIS
TY - UNPB
TI - Critical branching random walks on ℤ^2: local survival probabilities and Yaglom limit theorems
AU - Tianyi Bai
AU - Xinxin Chen
AU - Shen Lin
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2609.15538
UR - https://arxiv.org/abs/2609.15538
AB - We consider a branching random walk on ℤ^2 with critical offspring of mean $1$ and spatial motion governed by the jumps of a lazy simple random walk. For every site x∈ℤ^2, we obtain a uniform asymptotical estimate for the local survival probability, i.e., the probability that there are particles at x at large time n. Using Stein's method, we establish a Yaglom-type theorem for the number of particles at x at time n when x is at distance of order √n from the origin. Moreover, at the position occupied by a typical particle at time n, the number of particles at that site, normalized by log n, converges in law to a Gamma distribution, thereby confirming a conjecture of Lalley and Zheng [Ann. Probab. 39 (2010), 327-368]. Finally, we prove that, conditional on local survival, the total number of particles at time n, divided by n, also converges weakly to a Gamma distribution.
ER -