Height representation of XOR-Ising loops via bipartite dimers

TL;DR

This paper establishes a connection between XOR-Ising loops and bipartite dimer models, showing that XOR-Ising loops can be viewed as level lines of a height function that converges to a Gaussian free field at criticality.

Contribution

It explicitly relates XOR-Ising loop configurations to bipartite dimer models and proves the scaling limit of XOR-Ising loops as contour lines of a Gaussian free field.

Findings

XOR-Ising loops are equivalent to level lines of a bipartite dimer height function.

At criticality, the height function converges to a Gaussian free field.

The results provide a discrete analogue of Wilson's conjecture for XOR-Ising loops.

Abstract

The XOR-Ising model on a graph consists of random spin configurations on vertices of the graph obtained by taking the product at each vertex of the spins of two independent Ising models. In this paper, we explicitly relate loop configurations of the XOR-Ising model and those of a dimer model living on a decorated, bipartite version of the Ising graph. This result is proved for graphs embedded in compact surfaces of genus g. Using this fact, we then prove that XOR-Ising loops have the same law as level lines of the height function of this bipartite dimer model. At criticality, the height function is known to converge weakly in distribution to a Gaussian free field. As a consequence, results of this paper shed a light on the occurrence of the Gaussian free field in the XOR-Ising model. In particular, they prove a discrete analogue of Wilson's conjecture, stating that the scaling limit of XOR-Ising loops are "contour lines" of the Gaussian free field.