Conditional decoupling of random interlacements

TL;DR

This paper establishes a conditional decoupling inequality for random interlacements in dimensions three and higher, showing that the conditional distribution on a region given distant configurations closely resembles the unconditional distribution.

Contribution

It introduces a new decoupling inequality for random interlacements conditioned on configurations in distant regions, advancing understanding of spatial dependencies.

Findings

Conditional law on a box given distant configuration is similar to the unconditional law.

Decoupling inequality holds for dimensions d ≥ 3.

Enhances analysis of random interlacement models in high dimensions.

Abstract

We prove a conditional decoupling inequality for the model of random interlacements in dimension $d\geq 3$: the conditional law of random interlacements on a box (or a ball) $A_1$ given the (not very "bad") configuration on a "distant" set $A_2$ does not differ a lot from the unconditional law.