Electronic Journal of Probability

Inverting the coupling of the signed Gaussian free field with a loop-soup

Titus Lupu, Christophe Sabot, and Pierre Tarrès

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Lupu introduced a coupling between a random walk loop-soup and a Gaussian free field, where the sign of the field is constant on each cluster of loops. This coupling is a signed version of isomorphism theorems relating the square of the GFF to the occupation field of Markovian trajectories. His construction starts with a loop-soup, and by adding additional randomness samples a GFF out of it. In this article we provide the inverse construction: starting from a signed free field and using a self-interacting random walk related to this field, we construct a random walk loop-soup. Our construction relies on the previous work by Sabot and Tarrès, which inverts the coupling from the square of the GFF rather than the signed GFF itself.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 70, 28 pp.

Received: 29 November 2018
Accepted: 23 May 2019
First available in Project Euclid: 28 June 2019

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Zentralblatt MATH identifier

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 60J55: Local time and additive functionals
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 81T25: Quantum field theory on lattices 81T60: Supersymmetric field theories

Gaussian free field Ray-Knight identity self-interacting processes loop-soups random currents Ising model

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Lupu, Titus; Sabot, Christophe; Tarrès, Pierre. Inverting the coupling of the signed Gaussian free field with a loop-soup. Electron. J. Probab. 24 (2019), paper no. 70, 28 pp. doi:10.1214/19-EJP326. https://projecteuclid.org/euclid.ejp/1561687603

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