The Annals of Probability

Random interlacements and the Gaussian free field

Alain-Sol Sznitman

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Abstract

We consider continuous time random interlacements on ${\mathbb{Z}}^{d}$, $d\ge3$, 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 $d\ge3$, we also relate the field of occupation times at a level tending to infinity, to the $d$-dimensional Gaussian free field.

Article information

Source
Ann. Probab., Volume 40, Number 6 (2012), 2400-2438.

Dates
First available in Project Euclid: 26 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1351258730

Digital Object Identifier
doi:10.1214/11-AOP683

Mathematical Reviews number (MathSciNet)
MR3050507

Zentralblatt MATH identifier
1261.60095

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J27: Continuous-time Markov processes on discrete state spaces 60F05: Central limit and other weak theorems

Keywords
Random interlacements Gaussian free field occupation times

Citation

Sznitman, Alain-Sol. Random interlacements and the Gaussian free field. Ann. Probab. 40 (2012), no. 6, 2400--2438. doi:10.1214/11-AOP683. https://projecteuclid.org/euclid.aop/1351258730


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