An isomorphism theorem for random interlacements

Alain-Sol Sznitman

doi:10.1214/ecp.v17-1792

Abstract

We consider continuous-time random interlacements on a transient weighted graph. We prove an identity in law relating the field of occupation times of random interlacements at level u to the Gaussian free field on the weighted graph. This identity is closely linked to the generalized second Ray-Knight theorem, and uniquely determines the law of occupation times of random interlacements at level u.

Highlights

In this note we consider continuous-time random interlacements on a transient weighted graph E
We prove an identity in law relating the field of occupation times of random interlacements at level u to the Gaussian free field on the weighted graph
This identity is closely linked to the generalized second Ray-Knight theorem of [2], [4], and uniquely determines the law of occupation times of random interlacements at level u

Summary

Introduction

In this note we consider continuous-time random interlacements on a transient weighted graph E. The identity can be viewed as a kind of generalized second Ray-Knight theorem, see [2], [4], and characterizes the law of the field of occupation times of random interlacements at level u. An isomorphism theorem for random interlacements (informally, the durations of the successive steps of a trajectory are described by independent exponential variables of parameter 1, but occupation times at x get rescaled by a factor λ−x 1). The proof of Theorem 0.1 involves an approximation argument of the law of (Lx,u)x∈E stated in Theorem 2.1, which is of independent interest This approximation has a similar flavor to what appears at the end of Section 4.5 of [7], when giving a precise interpretation of random interlacements as “loops going through infinity”, see [3], p.

Notation and useful results

An approximation scheme for random interlacements

Proof of the isomorphism theorem

An application