Plancherel representations of $U(\infty)$ and correlated Gaussian Free Fields

TL;DR

This paper investigates the asymptotic behavior of traces of noncommutative monomials in the Plancherel representations of $U( olinebreak)( olinebreak) olinebreak$ and shows they converge to correlated Gaussian Free Fields, linking representation theory with random matrix spectra.

Contribution

It establishes the convergence of traces of certain algebraic elements to Gaussian Free Fields, revealing new connections between infinite-dimensional group representations and random matrix theory.

Findings

Convergence of traces to Gaussian processes

Identification of the limiting process as correlated Gaussian Free Fields

Connection to spectra of Wigner 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. The results of the present work were announced in arXiv:1203.3010.