Extreme value statistics of smooth random Gaussian fields

TL;DR

This paper derives analytical estimates for the extreme value distribution of smooth Gaussian random fields in two and three dimensions, linking it to topological measures like the Euler Characteristic, and compares predictions with numerical simulations.

Contribution

It introduces a novel analytical approach to estimate Gumbel extreme value statistics for smooth Gaussian fields, connecting it to topological characteristics and extending to non-Gaussian fields.

Findings

Analytical estimates match numerical simulations well.

Gumbel statistics relate to the Euler Characteristic of the field.

Convergence to Gumbel form improves with weaker correlations.

Abstract

We consider the Gumbel or extreme value statistics describing the distribution function p_G(x_max) of the maximum values of a random field x within patches of fixed size. We present, for smooth Gaussian random fields in two and three dimensions, an analytical estimate of p_G which is expected to hold in a regime where local maxima of the field are moderately high and weakly clustered. When the patch size becomes sufficiently large, the negative of the logarithm of the cumulative extreme value distribution is simply equal to the average of the Euler Characteristic of the field in the excursion x > x_max inside the patches. The Gumbel statistics therefore represents an interesting alternative probe of the genus as a test of non Gaussianity, e.g. in cosmic microwave background temperature maps or in three-dimensional galaxy catalogs. It can be approximated, except in the remote positive tail, by a negative Weibull type form, converging slowly to the expected Gumbel type form for infinitely large patch size. Convergence is facilitated when large scale correlations are weaker. We compare the analytic predictions to numerical experiments for the case of a scale-free Gaussian field in two dimensions, achieving impressive agreement between approximate theory and measurements. We also discuss the generalization of our formalism to non-Gaussian fields.