An isomorphism theorem for random interlacements

TL;DR

This paper establishes a fundamental isomorphism between the occupation times of random interlacements and the Gaussian free field on weighted graphs, extending the second Ray-Knight theorem to this setting.

Contribution

It proves a new law identity linking random interlacement occupation times with the Gaussian free field, providing a unique characterization of their distribution.

Findings

Identity in law between occupation times and Gaussian free field

Extension of the second Ray-Knight theorem to weighted graphs

Unique determination of occupation time distribution

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.