An entropy for Boolean independence

TL;DR

This paper introduces a Boolean entropy concept inspired by free entropy, analyzing two random matrix models where asymptotic Boolean independence occurs, and establishes properties like large deviation principles and maximization by Rademacher distribution.

Contribution

It defines a Boolean entropy parallel to free entropy and investigates its properties through two specific random matrix models, revealing key behaviors and maximizers.

Findings

Large deviation principles established for both models

Rate functions coincide up to scaling and are minimized by Rademacher distribution

Boolean entropy maximized by Rademacher distribution and monotone along Boolean CLT

Abstract

In this article, we aim to define a Boolean entropy notion parallel to the framework of free entropy proposed by Voiculescu. Motivated by the work of Lenczewski and the work of C\'ebron & Gillers, we mainly investigated two random matrix models (the Gaussian Symmetric Block model and the Conditioned GUE model), in which asymptotic Boolean independence appears. We showed a large deviation principle for both models. As a result, the two rate functions coincide up to scaling and are minimized by the Rademacher distribution. Therefore, we refer to the logarithmic integral in the rate function as Boolean entropy. Finally, we proved this logarithmic integral is maximized by the Rademacher distribution and monotone along the Boolean Central Limit Theorem.