Skip to Main content Skip to Navigation
Journal articles

Height representation of XOR-Ising loops via bipartite dimers

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 1 √ π a Gaussian free field [dT07b]. 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 [Wil11], stating that the scaling limit of XOR-Ising loops are " contour lines " of the Gaussian free field.
Document type :
Journal articles
Complete list of metadatas

Cited literature [55 references]  Display  Hide  Download
Contributor : Gestionnaire Hal-Upmc <>
Submitted on : Tuesday, May 17, 2016 - 2:17:55 PM
Last modification on : Friday, March 27, 2020 - 4:00:16 AM
Long-term archiving on: : Friday, August 19, 2016 - 4:51:59 PM


Publication funded by an institution


Distributed under a Creative Commons Attribution 4.0 International License



Cédric Boutillier, Béatrice de Tilière. Height representation of XOR-Ising loops via bipartite dimers. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2014, 19, pp.80. ⟨10.1214/EJP.v19-2449⟩. ⟨hal-01316626⟩



Record views


Files downloads