Approximating critical parameters of branching random walks

TL;DR

This paper studies how truncations in branching random walks affect critical parameters, providing convergence results and applying them to percolation clusters.

Contribution

It introduces new methods to approximate critical parameters of branching random walks under truncation and analyzes their convergence to the original parameters.

Findings

Weak and strong critical parameters converge under truncation.

Results apply to branching random walks on percolation clusters.

Provides a framework for approximating critical thresholds in complex networks.

Abstract

Given a branching random walk on a graph, we consider two kinds of truncations: by inhibiting the reproduction outside a subset of vertices and by allowing at most $m$ particles per site. We investigate the convergence of weak and strong critical parameters of these truncated branching random walks to the analogous parameters of the original branching random walk. As a corollary, we apply our results to the study of the strong critical parameter of a branching random walk restricted to the cluster of a Bernoulli bond percolation.