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 $U(N)$ 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.

66 pages, 10 figures

References in corpus (2)

• Gaussian fluctuations of characters of symmetric groups and of Young diagrams
• Gaussian fluctuations of Jack-deformed random Young diagrams

Cited by in corpus (7)

• A periodic hexagon tiling model and non-Hermitian orthogonal polynomials
• Matrix addition and the Dunkl transform at high temperature
• Product matrix processes as limits of random plane partitions
• Lozenge tilings and the Gaussian free field on a cylinder
• Airy Point Process via Supersymmetric Lifts
• Double Interlacing in Random Tiling Models
• Discrete symplectic fermions on double dimers and their Virasoro representation