Random interlacements and the Gaussian free field

Abstract

We consider continuous time random interlacements on ${\mathbb{Z}}^d$, $d\ge 3$, and characterize the distribution of the corresponding stationary random field of occupation times. When d = 3, we relate this random field to the two-dimensional Gaussian free field pinned at the origin by looking at scaled differences of occupation times of long rods by random interlacements at appropriately tuned levels. In the main asymptotic regime, a scaling factor appears in the limit, which is independent of the free field, and distributed as the time-marginal of a zero-dimensional Bessel process. For arbitrary $d\ge 3$, we also relate the field of occupation times at a level tending to infinity, to the d-dimensional Gaussian free field.

Published in at http://dx.doi.org/10.1214/11-AOP683 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Decoupling inequalities and interlacement percolation on G x Z
• On the fragmentation of a torus by random walk
• Connectivity Bounds for the Vacant Set of Random Interlacements

Cited by in corpus (11)

• Equality of critical parameters for percolation of Gaussian free field level-sets
• On scaling limits and Brownian interlacements
• Disconnection, random walks, and random interlacements
• The sign clusters of the massless Gaussian free field percolate on ${\mathbb{Z}}^d$, $d⩾3$ (and more)
• Exceptional points of two-dimensional random walks at multiples of the cover time
• On bulk deviations for the local behavior of random interlacements
• Dynamically accelerated cover times
• Large deviations for occupation time profiles of random interlacements
• Emergence of interlacements from the finite volume Bose soup
• Anatomy of a gaussian giant: supercritical level-sets of the free field on random regular graphs
• Entropic repulsion for the occupation-time field of random interlacements conditioned on disconnection