Random matrices, continuous circular systems and the triangular operator

TL;DR

This paper introduces a Hilbert space framework to analyze the joint *-distributions of complex Gaussian random matrices, connecting them to continuous circular and triangular operators, and provides a new proof for their *-moment formulas.

Contribution

It develops a Hilbert space approach using creation and annihilation operators to unify the understanding of circular and triangular operators in random matrix theory.

Findings

Unified framework for circular and triangular operators

Explicit connection between random matrices and operator models

Bijective proof of *-moment formulas for triangular operators

Abstract

We present a Hilbert space approach to the limit joint *-distributions of complex independent Gaussian random matrices. For that purpose, we use a suitably defined family of creation and annihilation operators living in some direct integral of Hilbert spaces. These operators are decomposed in terms of continuous circular systems of operators acting between the fibers of the considered Hilbert space direct integral. In the case of square matrices with i.i.d. entries, we obtain the circular operators of Voiculescu, whereas in the case of upper-triangular matrices with i.i.d. entries, we obtain the triangular operators of Dykema and Haagerup. We apply this approach to give a bijective proof of a formula for *-moments of the triangular operator, using the enumeration formula of Chauve, Dulucq and Rechnizter for alternating ordered rooted trees.