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 ${\lambda }_0$ 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 $\beta \to \infty$. The theorem shows that the influence of dimension persists even for very strong branching.

arXiv comment12 pages, 2 figures