Free boundary dimers: random walk representation and scaling limit

TL;DR

This paper investigates the free boundary dimer model, establishing a random walk representation for the inverse Kasteleyn matrix and proving that the height function converges to the Gaussian free field with Neumann boundary conditions in the scaling limit.

Contribution

It introduces a new random walk representation for the inverse Kasteleyn matrix in free boundary dimer models and proves the Gaussian free field as the scaling limit of the height function.

Findings

Random walk representation for inverse Kasteleyn matrix established.

Height function converges to Gaussian free field with Neumann boundary conditions.

Results hold in the infinite volume limit in the upper half-plane.

Abstract

We study the dimer model on subgraphs of the square lattice in which vertices on a prescribed part of the boundary (the free boundary) are possibly unmatched. Each such unmatched vertex is called a monomer and contributes a fixed multiplicative weight $z>0$ to the total weight of the configuration. A bijection described by Giuliani, Jauslin and Lieb relates this model to a standard dimer model but on a non-bipartite graph. The Kasteleyn matrix of this dimer model describes a walk with transition weights that are negative along the free boundary. Yet under certain assumptions, which are in particular satisfied in the infinite volume limit in the upper half-plane, we prove an effective, true random walk representation for the inverse Kasteleyn matrix. In this case we further show that, independently of the value of $z>0$, the scaling limit of the height function is the Gaussian free field with Neumann (or free) boundary conditions, thereby answering a question of Giuliani et al.