Extreme matrices or how an exponential map links classical and free extreme laws

TL;DR

This paper introduces an exponential map linking classical and free extreme laws using a thinning approach, providing explicit formulas, a formalism for extreme matrices, and demonstrating their equivalence to classical Peak-Over-Threshold methods.

Contribution

It develops a novel exponential mapping connecting classical and free extreme laws and introduces an extreme matrix formalism with explicit calculations for various ensembles.

Findings

Explicit exponential formula linking classical and free extreme laws.

Development of an extreme matrix formalism for refined analysis.

Demonstration of equivalence between free extreme laws and classical Peak-Over-Threshold method.

Abstract

Using the proposed by us thinning approach to describe extreme matrices, we find an explicit exponentiation formula linking classical extreme laws of Fr\'echet, Gumbel and Weibull given by Fisher-Tippet-Gnedenko classification and free extreme laws of free Fr\'echet, free Gumbel and free Weibull by Ben Arous and Voiculescu [1]. We also develop an extreme random matrix formalism, in which refined questions about extreme matrices can be answered. In particular, we demonstrate explicit calculations for several more or less known random matrix ensembles, providing examples of all three free extreme laws. Finally, we present an exact mapping, showing the equivalence of free extreme laws to the Peak-Over-Threshold method in classical probability.