## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 21 (2016), paper no. 13, 7 pp.

### A note on Ising random currents, Ising-FK, loop-soups and the Gaussian free field

Titus Lupu and Wendelin Werner

#### Abstract

We make a few elementary observations that relate directly the items mentioned in the title. In particular, we note that when one superimposes the random current model related to the Ising model with an independent Bernoulli percolation model with well-chosen weights, one obtains exactly the FK-percolation (or random cluster model) associated with the Ising model, and we point out that this relation can be interpreted via loop-soups, combining the description of the sign of a Gaussian free field on a discrete graph knowing its square (and the relation of this question with the FK-Ising model) with the loop-soup interpretation of the random current model.

#### Article information

**Source**

Electron. Commun. Probab., Volume 21 (2016), paper no. 13, 7 pp.

**Dates**

Received: 27 November 2015

Accepted: 12 January 2016

First available in Project Euclid: 17 February 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1455717135

**Digital Object Identifier**

doi:10.1214/16-ECP4733

**Mathematical Reviews number (MathSciNet)**

MR3485382

**Zentralblatt MATH identifier**

1338.60236

**Subjects**

Primary: 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

**Keywords**

Ising model random currents loop-soups Gaussian free field

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Lupu, Titus; Werner, Wendelin. A note on Ising random currents, Ising-FK, loop-soups and the Gaussian free field. Electron. Commun. Probab. 21 (2016), paper no. 13, 7 pp. doi:10.1214/16-ECP4733. https://projecteuclid.org/euclid.ecp/1455717135