Non-crossing Annular Pairings and The Infinitesimal Distribution of the GOE

TL;DR

This paper develops a combinatorial framework using non-crossing partitions to analyze the infinitesimal distribution of the GOE, revealing new insights into infinitesimal freeness and cumulants.

Contribution

It introduces a combinatorial approach to describe the infinitesimal distribution of GOE and explores the asymptotic infinitesimal freeness of Wishart matrices.

Findings

Infinitesimal moments are described by non-crossing partitions.

Independent Wishart matrices exhibit asymptotic infinitesimal freeness.

Independent GOE ensembles lack asymptotic infinitesimal freeness.

Abstract

We present a combinatorial approach to the infinitesimal distribution of the Gaussian orthogonal ensemble (GOE). In particular we show how the infinitesimal moments are described by non-crossing partitions, but not of type B. We demonstrate the asymptotic infinitesimal freeness of independent complex Wishart matrices. With the combinatorial picture we can easily compute the infinitesimal cumulants of the GOE and demonstrate the lack of asymptotic infinitesimal freeness of independent GOE ensembles. I have added a new figure and some additional explanatory remarks. This update corrects a number of typographical errors; I am grateful to the readers who brought these to my attention.