Infinitesimal Operators and the Distribution of Anticommutators and Commutators

TL;DR

This paper develops a framework for analyzing infinitesimal operators in free probability, revealing new relationships between infinitesimal distributions, Boolean cumulants, and matrix models exhibiting various forms of independence.

Contribution

It introduces a method to find non-commutative distributions involving infinitesimal operators and links Boolean cumulants to infinitesimal moments, connecting to matrix models and the Markov-Krein transform.

Findings

Joint infinitesimal distribution yields Boolean cumulants.

Boolean cumulants can be expressed as infinitesimal moments.

Connections to matrix models with asymptotic Boolean and monotone independence.

Abstract

In an infinitesimal probability space we consider operators which are infinitesimally free and one of which is infinitesimal, in that all its moments vanish. Many previously analysed random matrix models are captured by this framework. We show that there is a simple way of finding non-commutative distributions involving infinitesimal operators and apply this to the commutator and anticommutator. We show the joint infinitesimal distribution of an operator and an infinitesimal idempotent gives us the Boolean cumulants of the given operator. We also show that Boolean cumulants can be expressed as infinitesimal moments thus giving matrix models which exhibit asymptotic Boolean independence and monotone independence. Finally we demonstrate a connection to the Markov-Krein transform.