Characterization of the critical values of branching random walks on weighted graphs through infinite-type branching processes

Abstract

We study the branching random walk on weighted graphs; site-breeding and edge-breeding branching random walks on graphs are seen as particular cases. We describe the strong critical value in terms of a geometrical parameter of the graph. We characterize the weak critical value and relate it to another geometrical parameter. We prove that, at the strong critical value, the process dies out locally almost surely; while, at the weak critical value, global survival and global extinction are both possible.

arXiv comment14 pages, corrected some typos and minor mistakes

References in corpus (1)

• Weak survival for branching random walks on graphs

Cited by in corpus (11)

• Survival, extinction and approximation of discrete-time branching random walks
• Survival of branching random walks in random environment
• Classification of catalytic branching processes and structure of the criticality set
• Extinction in lower Hessenberg branching processes with countably many types
• Rumor processes in random environment on N and on Galton-Watson trees
• Recent results on branching random walks
• Branching random walks and multi-type contact-processes on the percolation cluster of ${\mathbb{Z}}^d$
• Dynamical sensitivity of recurrence and transience of branching random walks
• Boolean percolation on digraphs and random exchange processes
• Random Walks on Directed Covers of Graphs
• Global survival of branching random walks and tree-like branching random walks