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2014 Height representation of XOR-Ising loops via bipartite dimers
Cédric Boutillier, Béatrice de Tilière
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Electron. J. Probab. 19: 1-33 (2014). DOI: 10.1214/EJP.v19-2449


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.


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Cédric Boutillier. Béatrice de Tilière. "Height representation of XOR-Ising loops via bipartite dimers." Electron. J. Probab. 19 1 - 33, 2014.


Accepted: 4 September 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1302.82014
MathSciNet: MR3256880
Digital Object Identifier: 10.1214/EJP.v19-2449

Primary: 52C20
Secondary: 55N33 , 60K35 , 82B20 , 82B23

Keywords: bipartite dimer model , height function , homology theory , Ising model


Vol.19 • 2014
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