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

Abstract

A combination of direct and inverse Fourier transforms on the unitary group $U\left(N\right)$ 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 the law of large numbers and the 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 conjecture (which predicts asymptotic Gaussianity and the exact form of the covariance) for a family of non-simply-connected polygons.

Another application is a central limit theorem for the $U\left(N\right)$ quantum random walk with random initial data.