Dimers on Rail Yard Graphs

TL;DR

This paper introduces a versatile model of dimer coverings on rail yard graphs, linking it to algebraic structures and providing explicit formulas for correlation functions, with applications to well-known tiling problems.

Contribution

It develops a general framework for RYG dimer models, connecting transfer matrices to the boson-fermion correspondence and expressing the model as a Schur process.

Findings

Explicit correlation functions and inverse Kasteleyn matrix derived.

Unified approach to various tiling models including Aztec diamond.

New derivations of key probabilistic and combinatorial results for these models.

Abstract

We introduce a general model of dimer coverings of certain plane bipartite graphs, which we call rail yard graphs (RYG). The transfer matrices used to compute the partition function are shown to be isomorphic to certain operators arising in the so-called boson-fermion correspondence. This allows to reformulate the RYG dimer model as a Schur process, i.e. as a random sequence of integer partitions subject to some interlacing conditions. Beyond the computation of the partition function, we provide an explicit expression for all correlation functions or, equivalently, for the inverse Kasteleyn matrix of the RYG dimer model. This expression, which is amenable to asymptotic analysis, follows from an exact combinatorial description of the operators localizing dimers in the transfer-matrix formalism, and then a suitable application of Wick's theorem. Plane partitions, domino tilings of the Aztec diamond, pyramid partitions, and steep tilings arise as particular cases of the RYG dimer model. For the Aztec diamond, we provide new derivations of the edge-probability generating function, of the biased creation rate, of the inverse Kasteleyn matrix and of the arctic circle theorem.