Extreme value statistics of smooth Gaussian random fields

作者：Colombi Stephane^*; Davis Olaf; Devriendt Julien; Prunet Simon; Silk Joe

来源：Monthly Notices of the Royal Astronomical Society, 2011, 414(3): 2436-2445.

DOI：10.1111/j.1365-2966.2011.18563.x

• methods: analytical
• methods: statistical
• large-scale structure of Universe

We consider the Gumbel or extreme value statistics describing the distribution function p(G)(nu(max)) of the maximum values of a random field nu 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 nu >= nu(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 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.

• 出版日期2011-7