Infinite random matrices and ergodic measures

TL;DR

This paper introduces a family of invariant measures on infinite Hermitian matrices, linking their ergodic decomposition to determinantal point processes and explicitly computing the correlation kernel, including the sine kernel case.

Contribution

It provides a new framework connecting infinite matrix measures with determinantal processes and explicitly derives the correlation kernels, including classical cases.

Findings

Decomposition of measures into ergodic components described by determinantal point processes.

Explicit computation of the correlation kernel for the point process.

Identification of the sine kernel as a special case within this family.

Abstract

We introduce and study a 2-parameter family of unitarily invariant probability measures on the space of infinite Hermitian matrices. We show that the decomposition of a measure from this family on ergodic components is described by a determinantal point process on the real line. The correlation kernel for this process is explicitly computed. At certain values of parameters the kernel turns into the well-known sine kernel which describes the local correlation in Circular and Gaussian Unitary Ensembles. Thus, the random point configuration of the sine process is interpreted as the random set of ``eigenvalues'' of infinite Hermitian matrices distributed according to the corresponding measure.