Height fluctuations in the honeycomb dimer model

TL;DR

This paper proves that height fluctuations in the honeycomb dimer model, under certain boundary conditions, converge to a Gaussian free field and match the local ergodic Gibbs measure, revealing detailed fluctuation behavior.

Contribution

It establishes the convergence of surface fluctuations to a Gaussian free field and characterizes local statistics in the honeycomb dimer model with no facets.

Findings

Height fluctuations converge to a Gaussian free field.

Local statistics match the ergodic Gibbs measure with the same slope.

Results apply to boundary conditions approximating the wire frame.

Abstract

We study a model of random surfaces arising in the dimer model on the honeycomb lattice. For a fixed ``wire frame'' boundary condition, as the lattice spacing $\epsilon\to0$, Cohn, Kenyon and Propp [CKP] showed the almost sure convergence of a random surface to a non-random limit shape $\Sigma_0$. In [KO], Okounkov and the author showed how to parametrize the limit shapes in terms of analytic functions, in particular constructing a natural conformal structure on them. We show here that when $\Sigma_0$ has no facets, for a family of boundary conditions approximating the wire frame, the large-scale surface fluctuations (height fluctuations) about $\Sigma_0$ converge as $\epsilon\to0$ to a Gaussian free field for the above conformal structure. We also show that the local statistics of the fluctuations near a given point $x$ are, as conjectured in [CKP], given by the unique ergodic Gibbs measure (on plane configurations) whose slope is the slope of the tangent plane of $\Sigma_0$ at $x$.