Dimer-dimer correlations at the rough-smooth boundary

TL;DR

This paper analyzes the transition region between rough and smooth phases in large periodic dimer models, providing uniform asymptotics for dimer correlations in the two-periodic Aztec diamond.

Contribution

It derives uniform asymptotics for dimer-dimer correlations in the rough-smooth transition region of the two-periodic Aztec diamond, advancing understanding of phase boundaries.

Findings

Asymptotic formulas for dimer correlations in the transition region

Application of inverse Kasteleyn matrix formula to asymptotics

Results extend to related infinite dimer models

Abstract

Three phases of macroscopic domains have been seen for large but finite periodic dimer models; these are known as the frozen, rough and smooth phases. The transition region between the frozen and rough region has received a lot of attention for the last twenty years and recently work has been underway to understand the rough-smooth transition region in the case of the two-periodic Aztec diamond. We compute uniform asymptotics for dimer-dimer correlations of the two-periodic Aztec diamond when the dimers lie in the rough-smooth transition region. These asymptotics rely on a formula found in [5] for the inverse Kasteleyn matrix, they also apply to a related infinite dimer model.