An entropy for Boolean independence

TL;DR

The paper defines Boolean entropy via large deviations for two random-matrix models that realize asymptotic Boolean independence, showing that the rate functions converge to a logarithmic energy form and identifying $\Gamma(\mu) = \int \log x^2 \, d\mu(x)$ as the Boolean entropy. It proves LDPs for the Gaussian Symmetric Block model and the Conditioned GUE model with speeds $pn$ and $NM$, respectively, and demonstrates that the entropy is maximized by the Rademacher distribution and monotone along the Boolean CLT. A refined scaling reveals a joint large deviation principle for split mass near $\pm\sqrt{2}$, yielding precise convergence results to the two-point limit and clarifying the role of the logarithmic energy in Boolean independence. Collectively, these results establish a Boolean-entropy framework paralleling classical and free entropy, with concrete rate-function analyses and implications for spectral models under partial traces.

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 Cébron & 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.

Figures (1)

• Figure 1: $y=\ell(x)$

Theorems & Definitions (39)

• Remark 1.1
• Theorem 1.2: Symmetric Block model
• Theorem 1.3: Conditioned GUE model
• Remark 1.4
• Theorem 1.5
• Proposition 2.1
• proof
• Corollary 2.2
• Corollary 3.1
• proof