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May 2016 From loop clusters and random interlacements to the free field
Titus Lupu
Ann. Probab. 44(3): 2117-2146 (May 2016). DOI: 10.1214/15-AOP1019


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.


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Titus Lupu. "From loop clusters and random interlacements to the free field." Ann. Probab. 44 (3) 2117 - 2146, May 2016.


Received: 1 May 2014; Revised: 1 March 2015; Published: May 2016
First available in Project Euclid: 16 May 2016

zbMATH: 1348.60141
MathSciNet: MR3502602
Digital Object Identifier: 10.1214/15-AOP1019

Primary: 60G15, 60G60, 60J25, 60K35

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 3 • May 2016
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