The Annals of Probability

From loop clusters and random interlacements to the free field

Titus Lupu

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It was shown by Le Jan that the occupation field of a Poisson ensemble of Markov loops (“loop soup”) of parameter $\frac{1}{2}$ associated to a transient symmetric Markov jump process on a network is half the square of the Gaussian free field. We construct a coupling between these loops and the free field such that an additional constraint holds: the sign of the free field is constant on each cluster of loops. As a consequence of our coupling we deduce that the loop clusters of parameter $\frac{1}{2}$ do not percolate on periodic lattices. We also construct a coupling between the random interlacement on $\mathbb{Z}^{d}$, $d\geq 3$, introduced by Sznitman, and the Gaussian free field, such that the set of vertices visited by the interlacement is contained in a one-sided level set of the free field. We deduce an inequality between the critical level for the percolation by level sets of the free field and the critical parameter for the percolation of the vacant set of the random interlacement.

Article information

Ann. Probab., Volume 44, Number 3 (2016), 2117-2146.

Received: May 2014
Revised: March 2015
First available in Project Euclid: 16 May 2016

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G15: Gaussian processes 60G60: Random fields 60J25: Continuous-time Markov processes on general state spaces

Gaussian free field loop soup Poisson ensemble of Markov loops percolation by loops random interlacements


Lupu, Titus. From loop clusters and random interlacements to the free field. Ann. Probab. 44 (2016), no. 3, 2117--2146. doi:10.1214/15-AOP1019.

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