Recurrence of the ℤ^d-valued infinite snake via unimodularity

2011/07/06 by Itai Benjamini, Itaï Benjamini, Nicolas Curien +2
Mathematics · Physics and Astronomy · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR

paper · pdf · doi:10.48550/arxiv.1107.1226

Published version available at http://ecp.ejpecp.org/article/view/1700/2003

openalex publication_date 2011/07/06 · arxiv created 2014/05/27 · arxiv updated 2014/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We use the concept of unimodular random graph to show that the branching simple random walk on ℤ^d indexed by a critical geometric Galton-Watson tree conditioned to survive is recurrent if and only if d ≤ 4.

Related

• Recurrence and transience of the critical random walk snake in random conductances
• The number of generations entirely visited for recurrent random walks on random environment
• On multidimensional branching random walks in random environment
• Scaling limit of the recurrent biased random walk on a Galton-Watson tree
• The looping constant of Z^d