We consider the Gumbel or extreme value statistics describing thedistribution function p_G(x_max) of the maximum values of a random field xwithin patches of fixed size. We present, for smooth Gaussian random fields intwo and three dimensions, an analytical estimate of p_G which is expected tohold in a regime where local maxima of the field are moderately high and weaklyclustered. When the patch size becomes sufficiently large, the negative of thelogarithm of the cumulative extreme value distribution is simply equal to theaverage of the Euler Characteristic of the field in the excursion x > x_maxinside the patches. The Gumbel statistics therefore represents an interestingalternative probe of the genus as a test of non Gaussianity, e.g. in cosmicmicrowave background temperature maps or in three-dimensional galaxy catalogs.It can be approximated, except in the remote positive tail, by a negativeWeibull type form, converging slowly to the expected Gumbel type form forinfinitely large patch size. Convergence is facilitated when large scalecorrelations are weaker. We compare the analytic predictions to numericalexperiments for the case of a scale-free Gaussian field in two dimensions,achieving impressive agreement between approximate theory and measurements. Wealso discuss the generalization of our formalism to non-Gaussian fields.
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