The Annals of Probability

The bead model and limit behaviors of dimer models

Cédric Boutillier

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In this paper, we study the bead model: beads are threaded on a set of wires on the plane represented by parallel straight lines. We add the constraint that between two consecutive beads on a wire; there must be exactly one bead on each neighboring wire. We construct a one-parameter family of Gibbs measures on the bead configurations that are uniform in a certain sense. When endowed with one of these measures, this model is shown to be a determinantal point process, whose marginal on each wire is the sine process (given by eigenvalues of large hermitian random matrices). We prove then that this process appears as a limit of any dimer model on a planar bipartite graph when some weights degenerate.

Article information

Ann. Probab., Volume 37, Number 1 (2009), 107-142.

First available in Project Euclid: 17 February 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Dimers phase transition Harnack curves scaling limit


Boutillier, Cédric. The bead model and limit behaviors of dimer models. Ann. Probab. 37 (2009), no. 1, 107--142. doi:10.1214/08-AOP398.

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