Height fluctuations in interacting dimers

TL;DR

This paper proves that height fluctuations in a 2D interacting dimer model remain Gaussian like in the non-interacting case, with correlation decay depending on interaction strength, using lattice fermion representations and constructive field theory.

Contribution

It demonstrates that height fluctuations are Gaussian for small interactions and reveals non-universal decay of dimer correlations, extending known results to non-integrable models.

Findings

Height fluctuations converge to Gaussian Free Field for small interactions.

Dimer-dimer correlations decay with a critical exponent depending on interaction.

Path-independence of height difference operators is crucial for the proof.

Abstract

We consider a non-integrable model for interacting dimers on the two-dimensional square lattice. Configurations are perfect matchings of $\mathbb Z^2$, i.e. subsets of edges such that each vertex is covered exactly once ("close-packing" condition). Dimer configurations are in bijection with discrete height functions, defined on faces $\boldsymbol{\xi}$ of $\mathbb Z^2$. The non-interacting model is "integrable" and solvable via Kasteleyn theory; it is known that all the moments of the height difference $h_{\boldsymbol{\xi}}-h_{\boldsymbol{\eta}}$ converge to those of the massless Gaussian Free Field (GFF), asymptotically as $|{\boldsymbol{\xi}}-{\boldsymbol{\eta}}|\to \infty$. We prove that the same holds for small non-zero interactions, as was conjectured in the theoretical physics literature. Remarkably, dimer-dimer correlation functions are instead not universal and decay with a critical exponent that depends on the interaction strength. Our proof is based on an exact representation of the model in terms of lattice interacting fermions, which are studied by constructive field theory methods. In the fermionic language, the height difference $h_{\boldsymbol{\xi}}-h_{\boldsymbol{\eta}}$ takes the form of a non-local operator, consisting of a sum of monomials along an {\it arbitrary} path connecting $\boldsymbol{\xi}$ and $\boldsymbol{\eta}$. As in the non-interacting case, this path-independence plays a crucial role in the proof.