Domino Tilings of the Aztec Diamond in Random Environment and Schur Generating Functions

TL;DR

This paper investigates the asymptotic fluctuations of domino tilings of the Aztec diamond in random environments, revealing different fluctuation behaviors depending on the variance scale of the weights, and employs Schur generating functions for analysis.

Contribution

It introduces a novel analysis of domino tilings in random environments using Schur generating functions, extending existing laws of large numbers and CLTs for particle systems.

Findings

Fluctuations governed by Gaussian Free Field plus Brownian motion at critical variance scale.

Fluctuations are purely Brownian motion at fixed weight distribution.

Different fluctuation scales depend on the variance decay rate of the weights.

Abstract

We study the asymptotic behavior of random domino tilings of the Aztec diamond of size $M$ in a random environment, where the environment is a one-periodic sequence of i.i.d. random weights attached to domino positions (i.e., to the edges of the underlying portion of the square grid). We consider two cases: either the variance of the weights decreases at a critical scale $1/M$, or the distribution of the weights is fixed. In the former case, the unrescaled fluctuations of the domino height function are governed by the sum of a Gaussian Free Field and an independent Brownian motion. In the latter case, we establish fluctuations on the much larger scale $\sqrt M$, given by the Brownian motion alone. To access asymptotic fluctuations in random environment, we employ the method of Schur generating functions. Moreover, we substantially extend the known Law of Large Numbers and Central Limit Theorems for particle systems via Schur generating functions in order to apply them to our setting. These results might be of independent interest.