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 $\frac{1}{\sqrt{\pi}}$ 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 provide a step forward in the solution of Wilson's conjecture, stating that the scaling limit of XOR-Ising loops are level lines of the Gaussian free field.
Highlights
The double Ising model consists of two Ising models, living on the same graph
Using Nienhuis’ mapping [Nie84], the main contribution of this paper is to provide a coupling between the double Ising model and the bipartite dimer model, which keeps track of XOR loop configurations, and is valid for graphs embedded in surfaces of genus g
The second step consists of using Wu–Lin/Dubédat’s mapping [WL75, Dub11b] from the 6-vertex model of the medial graph to the bipartite dimer model on the decorated graph GQ
Summary
The double Ising model consists of two Ising models, living on the same graph. It is related [KW71, Wu71, Fan, Weg72] to other models of statistical mechanics, as the 8-vertex model [Sut, FW70] and the Ashkin–Teller model [AT43]. Fixing a monochromatic edge configuration, and applying low/high-temperature duality to the single Ising model corresponding to bichromatic edges, yields a rewriting of the double Ising partition function, as a sum over pairs of non-intersecting polygon configurations of the primal and dual graph, where primal polygon configurations exactly correspond to XOR loop configurations, see Proposition 4.2 and Corollary 4.3. XOR loop configurations of the double Ising model on G∗ have the same law as Poly configurations of the corresponding dimer model on the bipartite graph GQ:. Using Nienhuis’ mapping [Nie84], the main contribution of this paper is to provide a coupling between the double Ising model and the bipartite dimer model, which keeps track of XOR loop configurations, and is valid for graphs embedded in surfaces of genus g. It would yield a complete proof of the conjecture if we could overcome the same technical obstacles as those of the proof of the convergence of double dimer loops to CLE4
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