Height fluctuations in non-integrable classical dimers

TL;DR

This paper proves that the height fluctuations in non-integrable classical dimer models on the square lattice behave like a massless Gaussian free field, using innovative methods combining discrete holomorphicity and renormalization group techniques.

Contribution

It introduces a novel approach to analyze non-integrable models at criticality, establishing the microscopic origin of sine-Gordon field theory in this context.

Findings

Height function fluctuations match Gaussian free field predictions

New technique combining discrete holomorphicity with renormalization group

Applicable to other non-integrable critical models

Abstract

We rigorously establish the asymptotic equivalence between the height function of interacting dimers on the square lattice and the massless Gaussian free field. Our theorem explains the microscopic origin of the sine-Gordon field theory description away from the free fermion point, which has previously been elusive. We use a novel technique, based on the combination of discrete holomorphicity with exact, constructive, renormalization group methods, which has the potential of being applicable to a variety of other non-integrable models at or close to criticality.