Extreme value statistics of smooth Gaussian random fields
Article first published online: 15 APR 2011
© 2011 The Authors Monthly Notices of the Royal Astronomical Society © 2011 RAS
Monthly Notices of the Royal Astronomical Society
Volume 414, Issue 3, pages 2436–2445, July 2011
How to Cite
Colombi, S., Davis, O., Devriendt, J., Prunet, S. and Silk, J. (2011), Extreme value statistics of smooth Gaussian random fields. Monthly Notices of the Royal Astronomical Society, 414: 2436–2445. doi: 10.1111/j.1365-2966.2011.18563.x
- Issue published online: 21 JUN 2011
- Article first published online: 15 APR 2011
- Accepted 2011 February 18. Received 2011 February 17; in original form 2010 December 22
- methods: analytical;
- methods: statistical;
- large-scale structure of Universe
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.