Split Two-Periodic Aztec Diamond

TL;DR

This paper introduces a modified Aztec diamond dimer model with broken periodicity, computes its correlation kernel, and analyzes its local asymptotics, revealing both similarities and differences with the standard two-periodic model.

Contribution

It extends existing methods to analyze a non-periodic Aztec diamond model, providing new insights into its correlation structure and asymptotic behavior.

Findings

Correlation kernel computed for the non-periodic model

Local asymptotics match the two-periodic model at leading order

Sub-leading order terms are affected by the broken periodicity

Abstract

Recent advancements have been made to understand the statistics of the Aztec diamond dimer model under general periodic weights. In this work we define a model that breaks periodicity in one direction by combining two different two-periodic weightings. We compute the correlation kernel for this Aztec diamond dimer model by extending the methods developed by Berggren and Duits (2019), which utilize the Eynard-Mehta theorem and a Wiener-Hopf factorization. From a form of the correlation kernel that is suitable for asymptotics, we compute the local asymptotics of the model in the different macroscopic regions present. We prove that the local asymptotics of the model agree with the typical two-periodic model in the highest order, however the sub-leading order terms are affected.