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

TL;DR

The paper analyzes the recurrence versus transience of a $\,\mathbb{Z}^d$-valued branching random walk in a random environment indexed by a critical Bienaymé–Galton–Watson tree conditioned to survive. The authors employ a truncated second moment method that hinges on precise quenched Green's function and heat-kernel estimates for random walks in random conductances or traps, leveraging a 0–1 law for recurrence and a particle-viewpoint ergodic framework. They prove that the critical snake is recurrent in dimensions $d\le 4$ and transient in $d\ge 5$ under suitable integrability assumptions on the environment, with conductance and trap settings treated (the trap case allowing bounded long-range correlations). These results extend understanding of critical BRWREs, provide a robust method relying on Green's function control, and raise questions about heat-kernel decay and optimal integrability thresholds in more general environments.

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 Bienaymé-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.

Figures (1)

• Figure 1: Illustration of the dynamical system. An (annealed) realization of the snake is a triplet $(\omega,{\mathcal{S}}_{\mathcal{T}_\infty} ,{\mathcal{T}_\infty} )$, where the tree ${\mathcal{T}_\infty}$ contains an infinite spine $(u_i)_{i\geq0}$ and is rooted in $u_0$, i.e. ${\mathcal{S}}_{\mathcal{T}_\infty} (u_0)=0$. The application $\Phi$ maps the realization into ${\mathbb A}$, by splitting ${\mathcal{T}_\infty}$ into a sequence of finite trees $({\mathcal{T}} _i)_{i\geq0}$, and defining ${\mathcal{S}}_{{\mathcal{T}} _i}={\mathcal{S}}_{{\mathcal{T}_\infty} }|_{{\mathcal{T}} _i}$ and $x_i={\mathcal{S}}_{{\mathcal{T}} _i}(u_i)$. Then, the transformation ${\mathcal{R}}:{\mathbb A}\to{\mathbb A}$ modifies this sequence by removing the first tree ${\mathcal{T}} _0$ and shifting the environment $\omega$ and the maps ${\mathcal{S}}_{{\mathcal{T}} _i}$, so that the new root $u_1$ is sent to $0$ in $\mathbb{Z}^d$: see \ref{['eq:ergodicity:R']} for the definitions of $\widetilde{\omega}$, $\widetilde{x}_{i+1}$ and $\widetilde{{\mathcal{S}}}_{{\mathcal{T}} _{i+1}}$, $i\geq0$.

Theorems & Definitions (38)

• Definition 1.1
• Proposition 1.3
• Theorem 1.5
• Remark 1.1
• Theorem 1.7
• Remark 1.2
• Proposition 2.1
• Lemma 2.2
• Remark 2.1
• proof