Fourier transform on high-dimensional unitary groups with applications to random tilings

TL;DR

This paper develops a Fourier analysis framework on high-dimensional unitary groups to study asymptotic behaviors of random tilings and quantum walks, proving Gaussian fluctuations and covariance structures.

Contribution

It introduces an asymptotic Fourier transform approach on $U(N)$, connecting characters with probability measures, and applies this to prove conjectures on tiling fluctuations and quantum random walks.

Findings

Proves Gaussian fluctuations for height functions in random tilings.

Establishes the covariance structure predicted by Kenyon-Okounkov.

Derives a CLT for $U(N)$ quantum random walks with random initial data.

Abstract

A combination of direct and inverse Fourier transforms on the unitary group $U(N)$ identifies normalized characters with probability measures on $N$-tuples of integers. We develop the $N\to\infty$ version of this correspondence by matching the asymptotics of partial derivatives at the identity of logarithm of characters with Law of Large Numbers and Central Limit Theorem for global behavior of corresponding random $N$-tuples. As one application we study fluctuations of the height function of random domino and lozenge tilings of a rich class of domains. In this direction we prove the Kenyon-Okounkov's conjecture (which predicts asymptotic Gaussianity and exact form of the covariance) for a family of non-simply connected polygons. Another application is a central limit theorem for the $U(N)$ quantum random walk with random initial data.