Renormalization group theory for finite-size scaling in extreme statistics

TL;DR

This paper develops a renormalization group framework to analyze finite-size effects and convergence in extreme value statistics, revealing universal behaviors and stability properties akin to statistical physics.

Contribution

It introduces a detailed RG theory for extreme statistics, showing how finite-size corrections and convergence rates can be derived from fixed points and perturbations.

Findings

Finite-size shape corrections can be obtained from linearized RG near fixed points.

Stable and unstable perturbations determine convergence and divergence behaviors.

Marginally stable perturbations are common in the Gumbel class.

Abstract

We present a renormalization group (RG) approach to explain universal features of extreme statistics, applied here to independent, identically distributed variables. The outlines of the theory have been described in a previous Letter, the main result being that finite-size shape corrections to the limit distribution can be obtained from a linearization of the RG transformation near a fixed point, leading to the computation of stable perturbations as eigenfunctions. Here we show details of the RG theory which exhibit remarkable similarities to the RG known in statistical physics. Besides the fixed points explaining universality, and the least stable eigendirections accounting for convergence rates and shape corrections, the similarities include marginally stable perturbations which turn out to be generic for the Fisher-Tippett-Gumbel class. Distribution functions containing unstable perturbations are also considered. We find that, after a transitory divergence, they return to the universal fixed line at the same or at a different point depending on the type of perturbation.