Conditions for Equivalence of Random Interlacements and Random Walk Reflected off of Infinity

TL;DR

This paper establishes the precise conditions under which random interlacements and reflected random walk models are equivalent on transient graphs, linking this equivalence to harmonic functions and spanning forests.

Contribution

It characterizes the equivalence of the two models in terms of harmonic functions with finite Dirichlet energy and spanning forest properties.

Findings

Models are equivalent iff all finite energy harmonic functions are constant.

Equivalence holds for $\\mathbb{Z}^d$, Cartesian products, Cayley graphs.

Fails for all transient trees.

Abstract

On a transient weighted graph, there are two models of random walk which continue after reaching infinity: random interlacements, and random walk reflected off of infinity, recently introduced in arXiv:2506.18827 [math.PR]. We prove these two models are equivalent if and only if all harmonic functions of the underlying graph with finite Dirichlet energy are constant functions, or equivalently, the free and wired spanning forests coincide. In particular, examples where the models are equivalent include $\mathbb{Z}^d$, cartesian products, and many Cayley graphs, while examples that fail the condition include all transient trees.