Marked GUE-corners process in doubly periodic dimer models

TL;DR

This paper analyzes the asymptotic fluctuations of a family of periodic Aztec diamond dimer models near their turning points, showing they converge to a marked GUE-corners process scaled by rac{N} and constructed via Bernoulli marks.

Contribution

It establishes the convergence of fluctuations in periodic Aztec diamond dimer models to a marked GUE-corners process using advanced integral representations.

Findings

Fluctuations are described by a marked GUE-corners process.

Scaling by rac{N} captures the asymptotic behavior.

Bernoulli marks encode the periodicity in the limit.

Abstract

We study a family of periodically weighted Aztec diamond dimer models near their turning points. We establish that, asymptotically, as $N\rightarrow\infty$, their fluctuations there, scaled by $\sqrt{N}$, are described by a marked GUE-corners process. This limiting point process is constructed by assigning a Bernoulli mark independently to each particle in a realization of the GUE-corners process. The Bernoulli parameters associated with the random marks reflect the periodicity of the model in the limit. To prove this result we use a double-contour integral representation of the inverse Kasteleyn matrix on a higher-genus Riemann surface, which is well-suited for asymptotic analysis.