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

TL;DR

This paper explores the connections between Ising random currents, FK-percolation, loop-soups, and the Gaussian free field, revealing how these models relate through superimposition and probabilistic interpretations.

Contribution

It establishes a direct relation between the superimposed random current and Bernoulli percolation models, and interprets these connections via loop-soups and Gaussian free fields.

Findings

Superimposing random current with Bernoulli percolation yields FK-percolation.

Relations between Gaussian free field signs and loop-soups are clarified.

The study links multiple models in statistical physics through elementary observations.

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. We also 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.