Limit operators for circular ensembles

TL;DR

This paper studies the convergence of powers of random unitary matrices decomposed into reflections, showing they approach a flow of operators with eigenvalues following a sine-kernel distribution and eigenvectors as Gaussian fields.

Contribution

It introduces a novel flow of operators derived from products of random reflections, linking finite-dimensional random matrices to infinite-dimensional spectral processes.

Findings

Eigenvalues converge to a sine-kernel point process

Eigenvectors become Gaussian random fields

Flow provides a new example of a random operator with sine-kernel spectrum

Abstract

It is known that a unitary matrix can be decomposed into a product of reflections, one for each dimension, and the Haar measure on the unitary group pushes forward to independent uniform measures on the reflections. We consider the sequence of unitary matrices given by successive products of random reflections. In this coupling, we show that powers of the sequence of matrices converge in a suitable sense to a flow of operators which acts on a random vector space. The vector space has an explicit description as a subspace of the space of sequences of complex numbers. The eigenvalues of the matrices converge almost surely to the eigenvalues of the flow, which are distributed in law according to a sine-kernel point process. The eigenvectors of the matrices converge almost surely to vectors which are distributed in law as Gaussian random fields on a countable set. This flow gives the first example of a random operator with a spectrum distributed according to a sine-kernel point process which is naturally constructed from finite dimensional random matrix theory.