## Electronic Journal of Probability

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

#### Abstract

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

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

Dates
Accepted: 23 May 2019
First available in Project Euclid: 28 June 2019

https://projecteuclid.org/euclid.ejp/1561687603

Digital Object Identifier
doi:10.1214/19-EJP326

Mathematical Reviews number (MathSciNet)
MR3978220

Zentralblatt MATH identifier
07089008

#### Citation

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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