Perfect t-Embeddings of Uniformly Weighted Aztec Diamonds and Tower Graphs

Abstract

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 [9]. This confirms the prediction of [10]. 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.