Gumbel Central Limit Theorem for Max-Min and Min-Max

TL;DR

This paper establishes universal Gumbel distribution limit-laws for the Max-Min and Min-Max of large random matrices, linking random-matrix theory and extreme-value theory, with broad applications in science and engineering.

Contribution

It introduces novel, universal limit-laws for Max-Min and Min-Max of large matrices, revealing Gumbel statistics regardless of entry distributions.

Findings

Gumbel distribution emerges as the limit law for Max-Min and Min-Max.

Limit-laws are universal, independent of matrix entry distributions.

Results have broad applicability in physical sciences and engineering.

Abstract

The Max-Min and Min-Max of matrices arise prevalently in science and engineering. However, in many real-world situations the computation of the Max-Min and Min-Max is challenging as matrices are large and full information about their entries is lacking. Here we take a statistical-physics approach and establish limit-laws -- akin to the Central Limit Theorem -- for the Max-Min and Min-Max of large random matrices. The limit-laws intertwine random-matrix theory and extreme-value theory, couple the matrix-dimensions geometrically, and assert that Gumbel statistics emerge irrespective of the matrix-entries' distribution. Due to their generality and universality, as well as their practicality, these novel results are expected to have a host of applications in the physical sciences and beyond.