Gibbs Measures for the Dimer Model

Gibbs Measures for the Dimer Model

Probability Theory and Mathematical Physics

8 October 2026

Valentin Ivanov

We study the classification of translation invariant Gibbs measures for the dimer model on planar, bipartite, periodic graphs. After developing the general theory of infinite-volume Gibbs measures, we construct a family of ergodic measures as limits of finite-volume Boltzmann measures on a sequence of tori, indexed by a magnetic field. Each such measure has a well-defined slope, given by the gradient of the free energy, and these slopes sweep out the interior of the Newton polygon of the characteristic polynomial of the model. We show that the measure of each slope in the interior of the Newton polygon is mixing, and compute its local statistics as determinants of a kernel given by a contour integral. Finally, we return to the general theory and establish the decomposition of an arbitrary translation invariant Gibbs measure over its ergodic components, which for the dimer model are indexed by slope. The classification is complete on the interior of the Newton polygon and only partial on its boundary, where we describe what obstructs it.

Mathematics, Mathematical Physics, Probability, Combinatorics