Perfect t-embeddings of uniformly weighted Aztec diamonds and tower graphs

TL;DR

This paper develops perfect t-embeddings for uniformly weighted Aztec diamonds and tower graphs, proving their convergence to Gaussian free field fluctuations and confirming theoretical predictions with explicit integral formulas.

Contribution

It introduces new perfect t-embeddings for Aztec diamonds and tower graphs, and demonstrates their role in analyzing height fluctuation convergence to Gaussian free fields.

Findings

Proved convergence of height fluctuations to Gaussian free field

Derived explicit integral formulas for perfect t-embeddings

Established a transformation linking tower graph fluctuations to Aztec diamond fluctuations

Abstract

In this work we study a sequence of perfect t-embeddings of uniformly weighted Aztec diamonds. We show that these perfect t-embeddings can be used to prove convergence of gradients of height fluctuations to those of the Gaussian free field. In particular we provide a first proof of the existence of a model satisfying all conditions of the main theorem of arXiv:2109.06272. This confirms the prediction of arXiv:2002.07540. An important part of our proof is to exhibit exact integral formulas for perfect t-embeddings of uniformly weighted Aztec diamonds. In addition, we construct and analyze perfect t-embeddings of another sequence of uniformly weighted finite graphs called tower graphs. Although we do not check all technical assumptions of the mentioned theorem for these graphs, we use perfect t-embeddings to derive a simple transformation which identifies height fluctuations on the tower graph with those of the Aztec diamond.