Advances in Extreme Value Statistics Based on the Free Probability Theory

Abstract

Free probability theory was established by Dan Voiculescu in the 1980s to resolve an open problem in operator algebra theory. The random-matrix theory has undergone active development in recent years, attracting considerable attention for its connections to diverse fields, such as quantum information theory and deep learning. Concurrently, various extensions and deformations have been proposed as part of foundational research for Voiculescu’s free probability theory. In 2006, Ben Arous and Voiculescu proposed a theoretical framework for analyzing the spectral maximum of freely independent, noncommutative random variables. This framework is considered the free probability analog of classical extreme value statistics, which has evolved into the free extreme value theory. In this study, we review advances in this theory, incorporating the author’s most recent findings.