Recurrence and transience of the critical random walk snake in random conductances

TL;DR

This paper investigates the recurrence and transience of a critical branching random walk in random environments on $\\mathbb{Z}^d$, showing recurrence in low dimensions and transience in higher ones under certain conditions.

Contribution

It provides new criteria for recurrence and transience of branching random walks in random conductance environments, using a truncated second moment method.

Findings

Recurrence holds in dimensions up to four.

Transience occurs in dimensions five and higher.

The method relies on estimates of the quenched Green's function.

Abstract

In this paper we study the recurrence and transience of the $\mathbb{Z}^d$-valued branching random walk in random environment indexed by a critical Bienaym\'e-Galton-Watson tree, conditioned to survive. The environment is made either of random conductances or of random traps on each vertex. We show that when the offspring distribution is non degenerate with a finite third moment and the environment satisfies some suitable technical assumptions, then the process is recurrent up to dimension four, and transient otherwise. The proof is based on a truncated second moment method, which only requires to have good estimates on the quenched Green's function.