Central limit theorem for lozenge tilings with curved limit shape

TL;DR

This paper proves a central limit theorem for the fluctuations of lozenge tilings with curved limit shapes, showing they converge to a Gaussian Free Field in the absence of frozen regions.

Contribution

It establishes the first general result demonstrating Gaussian Free Field fluctuations for arbitrary non-frozen limit shapes in lozenge tilings.

Findings

Fluctuations converge to a Gaussian Free Field variant

Constructed domains with prescribed limit shapes and fluctuations

Extended understanding of fluctuation behavior beyond special cases

Abstract

It has been well known for a long time that the height function of random lozenge tilings of large domains follow a law of large number and possible limits called dimer limit shapes are well understood. For the next order, it is expected that fluctuations behave like version of a Gaussian Free field, at least away from some special "frozen" regions. However despite being one of the main questions in the domain for 20 years, only special cases have been obtained. In this paper we show that for any specified limit shape with no frozen region, one can construct a sequence of domains whose height functions converge to that limit shape and where the height fluctuation converge to a variant of the Gaussian Free Field.