  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.
