Extreme value statistics of smooth Gaussian random fields

Journal article

We consider the Gumbel or extreme value statistics describing the distribution function pG(νmax) of the maximum values of a random field ν within patches of fixed size. We present, for smooth Gaussian random fields in two and three dimensions, an analytical estimate of pG 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 ν≥νmax 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 3D galaxy catalogues. 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. © 2011 The Authors. Monthly Notices of the Royal Astronomical Society © 2011 RAS.

Authors: Colombi, S; Davis, O; Devriendt, J; Prunet, S; Silk, J

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Oxford University Press
• Journal: Monthly Notices of the Royal Astronomical Society
• Volume: 414
• Issue: 3
• Pages: 2436-2445
• Publication date: 2011-01-01
• DOI: 10.1111/j.1365-2966.2011.18563.x
• EISSN: 1365-2966
• ISSN: 0035-8711
• Language: English
• Keywords: Large-scale structure of Universe; Methods: analytical; Methods: statistical