We formulate and prove a variational principle (in the sense of thermodynamics) for random
domino tilings, or equivalently for the dimer model on a square grid. This principle states that …

The determinant of the Laplacian on a graph arises in two related statistical mechanical
models, the uniform spanning tree model and the 2-dimensional lattice dimer model. For the …

A Gelfand–Tsetlin scheme of depth NN is a triangular array with mm integers at level mm,
m= 1, ..., N m= 1,…, N, subject to certain interlacing constraints. We study the ensemble of …

We study large-scale height fluctuations of random stepped surfaces corresponding to
uniformly random lozenge tilings of polygons on the triangular lattice. For a class of …

Plane partitions naturally appear in many problems of statistical physics and quantum field
theory, for instance, in the theory of faceted crystals and of topological strings on Calabi–Yau …

We introduce elliptic weights of boxed plane partitions and prove that they give rise to a
generalization of MacMahon's product formula for the number of plane partitions in a box …

We survey the generalized Baues problem of Billera and Sturmfels. The problem is one of
discrete geometry and topology, and asks about the topology of the set of subdivisions of a …

We introduce discrete time Markov chains that preserve uniform measures on boxed plane
partitions. Elementary Markov steps change the size of the box from a× b× c to (a− 1)×(b+ …

The tilings of rhombi in two dimensions and of rhomboedra in three dimensions are studied
when they are constrained by fixed boundary conditions. We establish a link between those …

2 4 YY 3 5 Page 1 THE PLANAR DIMER MODEL WITH BOUNDARY: A SURVEY. RICHARD
KENYON 1. Introduction A dimer covering of a nite graph is a perfect matching of the graph …