Electronic Journal of Probability

Height representation of XOR-Ising loops via bipartite dimers

Cédric Boutillier and Béatrice de Tilière

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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.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 80, 33 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52C20: Tilings in $2$ dimensions [See also 05B45, 51M20]
Secondary: 55N33: Intersection homology and cohomology 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 82B23: Exactly solvable models; Bethe ansatz

Ising model bipartite dimer model height function homology theory

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Boutillier, Cédric; de Tilière, Béatrice. Height representation of XOR-Ising loops via bipartite dimers. Electron. J. Probab. 19 (2014), paper no. 80, 33 pp. doi:10.1214/EJP.v19-2449. https://projecteuclid.org/euclid.ejp/1465065722

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