A CLT for Plancherel representations of the infinite-dimensional unitary group

TL;DR

This paper establishes a central limit theorem for traces of noncommutative monomials in Plancherel representations of the infinite-dimensional unitary group, revealing convergence to correlated Gaussian Free Fields.

Contribution

It introduces a new CLT for noncommutative traces in infinite-dimensional unitary group representations, linking to Gaussian Free Fields and spectral limits of Wigner matrices.

Findings

Traces converge to moments of Gaussian processes

Limiting process relates to Gaussian Free Fields

Connections to spectral limits of random matrices

Abstract

We study asymptotics of traces of (noncommutative) monomials formed by images of certain elements of the universal enveloping algebra of the infinite-dimensional unitary group in its Plancherel representations. We prove that they converge to (commutative) moments of a Gaussian process that can be viewed as a collection of simply yet nontrivially correlated two-dimensional Gaussian Free Fields. The limiting process has previously arisen via the global scaling limit of spectra for submatrices of Wigner Hermitian random matrices. This note is an announcement, proofs will appear elsewhere.