Nikolai Kuchumov: Limit Shapes and Harmonic Tricks: elliptic parametrization of limit shape of multiply-connected Aztec diamond

Abstract:

In this talk, we will discuss random domino tilings and methods for analyzing their typical behavior in three parts. First, we will recall the random tiling model, height function, the Arctic circle theorem, and the variational approach for it. Next, we will introduce the tangent plane method, a technique for parametrizing limit shapes using harmonic functions, developed by Istvan Prause and Richard Kenyon in 2020. Finally, we will analyze the multiply-connected version of the Aztec diamond. The main tool in this part will be elliptic functions, which give us the limit shape of random domino tilings of the Aztec diamond with a hole.