Level set percolation for random interlacements and the Gaussian free field

TL;DR

This paper studies percolation phenomena in random interlacements and the Gaussian free field on high-dimensional integer lattices, revealing new insights into the structure of occupied and empty sites and their percolation properties.

Contribution

It establishes a connection between percolation in random interlacements and the Gaussian free field using an isomorphism theorem, providing new tools and results for level-set percolation analysis.

Findings

Percolation thresholds for occupied and empty sites identified.

New results on two-sided level-set percolation for the Gaussian free field.

Insights into the structure of percolation clusters in high dimensions.

Abstract

We consider continuous-time random interlacements on Z^d, d greater or equal to 3, and investigate the percolation model where a site x of Z^d is occupied if the total amount of time spent at x by all the trajectories of the interlacement at level u > 0 exceeds some given non-negative parameter, and empty otherwise. Thus, the set of occupied sites forms a subset of the interlacement at level u. We also investigate percolation properties of empty sites. A recent isomorphism theorem arXiv:1111.4818 of Sznitman enables us to "translate" some of the relevant questions into the language of level-set percolation for the Gaussian free field on Z^d, d greater or equal to 3, for which useful tools have been developed in arXiv:1202.5172. We also gain new insights of independent interest concerning "two-sided" level-set percolation, where a site x of Z^d is occupied if and only if the absolute value of the field variable at that site exceeds a given non-negative level.