Quenched invariance principle for random walks in random environments admitting a cycle decomposition

TL;DR

This paper establishes a quenched invariance principle for a class of non-reversible random walks in random environments with cycle decompositions, extending previous results beyond strong ellipticity and bounded cycles.

Contribution

It introduces a quenched invariance principle for non-reversible walks with cycle representations, broadening applicability beyond previous ellipticity constraints.

Findings

Proves a quenched invariance principle under moment conditions.

Extends previous results to unbounded cycle lengths.

Provides energy estimates for non-symmetric generators.

Abstract

We study a class of non-reversible, continuous-time random walks in random environments on $\mathbb{Z}^d$ that admit a cycle representation with finite cycle length. The law of the transition rates, taking values in $[0, \infty)$, is assumed to be stationary and ergodic with respect to space shifts. Moreover, the transition rate from $x$ to $y$, denoted by $c^\omega(x,y)$, is a superposition of non-negative random weights on oriented cycles that contain the edge $(x,y)$. We prove a quenched invariance principle under moment conditions that are comparable to the well-known p-q moment condition of Andres, Deuschel, and Slowik [2] for the random conductance model. A key ingredient in proving the sublinearity is an energy estimate for the non-symmetric generator. Our result extends that of Deuschel and K\"osters [12] beyond strong ellipticity and bounded cycle lengths.