Gibbs Ensembles of Nonintersecting Paths

TL;DR

This paper introduces a family of determinantal point processes on a 2D lattice, interpreted as Gibbs ensembles of nonintersecting paths, with applications to tilings and connections to Toeplitz matrices.

Contribution

It establishes a new class of Gibbs ensembles for nonintersecting paths with correlation kernels extending the discrete sine kernel, linked to totally positive Toeplitz matrices.

Findings

Correlation kernels extend the discrete sine kernel.

Gibbs property follows from linear relations of kernels.

Includes non-translation-invariant tiling measures.

Abstract

We consider a family of determinantal random point processes on the two-dimensional lattice and prove that members of our family can be interpreted as a kind of Gibbs ensembles of nonintersecting paths. Examples include probability measures on lozenge and domino tilings of the plane, some of which are non-translation-invariant. The correlation kernels of our processes can be viewed as extensions of the discrete sine kernel, and we show that the Gibbs property is a consequence of simple linear relations satisfied by these kernels. The processes depend on infinitely many parameters, which are closely related to parametrization of totally positive Toeplitz matrices.