Extreme value statistics and some applications in statistical physics

TL;DR

This paper explores extreme value statistics (EVS) in complex correlated systems relevant to statistical physics, highlighting deviations from classical EVS in systems like random walks, random matrices, and disordered media.

Contribution

It extends EVS analysis to strongly correlated systems in statistical physics, illustrating applications to models like the Random Energy Model and KPZ universality class.

Findings

Classical EVS does not apply to correlated systems.

EVS provides insights into disordered systems and interfaces.

Applications include random matrices and polymers.

Abstract

These notes are based on lectures delivered by G. Schehr at the XVIth School on Fundamental Problems in Statistical Physics (FPSP), held in Oropa (Italy) from 30 June to 11 July 2025. After a brief introduction to extreme value statistics (EVS) for independent and identically distributed (IID) random variables, we discuss several paradigmatic examples of strongly correlated systems where classical extreme value theory no longer applies. In particular, we focus on time series generated by random walks and Brownian motion, as well as on eigenvalue statistics in random matrix theory. Emphasis is placed on applications of EVS to fundamental problems in statistical physics and disordered systems, including the Random Energy Model, stochastic search problems, as well as fluctuating interfaces, and directed polymers in random media within the Kardar-Parisi-Zhang universality class.