Range and critical generations of a random walk on Galton-Watson trees

2015/10/05 by Pierre Andreoletti, Xinxin Chen, Andreoletti, Pierre +1
Mathematics · Physics and Astronomy · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1510.01121

openalex publication_date 2015/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we consider a random walk in random environment on a tree and focus on the boundary case for the underlying branching potential. We study the range R_n of this walk up to time n and obtain its correct asymptotic in probability which is of order n/log n. Thisresult is a consequence of the asymptotical behavior of the number of visited sites at generations of order (log n)²,which turn out to be the most visited generations. Our proof which involves a quenched analysisgives a description of the typical environments responsible for the behavior of R_n.

Related

• Capacity of the range of tree-indexed random walk
• Spread of visited sites of a random walk along the generations of a branching process
• The number of generations entirely visited for recurrent random walks on random environment
• Heavy range of the randomly biased walk on Galton-Watson trees in the slow movement regime
• The heavy range of randomly biased walks on trees