Supercritical Branching Random Walks on Nested Lattices
Abstract
In classical models of supercritical continuous-time branching random walks (BRWs) on a d-dimensional lattice, the dimension d is typically assumed to be fixed. When comparing the exponential growth rate λ0 of a particle population in a BRW with a single branching source across lattices of different dimensions, a question arises whether it is possible to construct a family of random walks on d-dimensional lattices such that the model for a (d+1)-dimensional lattice ``inherits'' the properties of the model for a d-dimensional lattice in a meaningful sense. To address this question, the paper introduces the concept of a ``random walk on nested lattices'' and investigates the effect of the dimension d on particle population growth using the ``inheritance'' condition. For an arbitrary symmetric random walk, a limit theorem is proved on the representation of the positive eigenvalue of the evolution operator for the mean particle numbers in terms of the process parameters and the dimension d, as the branching source intensity β→ ∞. The theorem shows that the influence of dimension persists even for very strong branching.
Related
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BibTeX
@misc{yarovaya2026supercritical,
title = {{Supercritical Branching Random Walks on Nested Lattices}},
author = {Elena Yarovaya},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27129},
eprint = {2610.27129},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27129},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Supercritical Branching Random Walks on Nested Lattices
AU - Elena Yarovaya
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27129
UR - https://arxiv.org/abs/2610.27129
AB - In classical models of supercritical continuous-time branching random walks (BRWs) on a d-dimensional lattice, the dimension d is typically assumed to be fixed. When comparing the exponential growth rate λ_0 of a particle population in a BRW with a single branching source across lattices of different dimensions, a question arises whether it is possible to construct a family of random walks on d-dimensional lattices such that the model for a (d+1)-dimensional lattice ``inherits'' the properties of the model for a d-dimensional lattice in a meaningful sense. To address this question, the paper introduces the concept of a ``random walk on nested lattices'' and investigates the effect of the dimension d on particle population growth using the ``inheritance'' condition. For an arbitrary symmetric random walk, a limit theorem is proved on the representation of the positive eigenvalue of the evolution operator for the mean particle numbers in terms of the process parameters and the dimension d, as the branching source intensity β→ ∞. The theorem shows that the influence of dimension persists even for very strong branching.
ER -