Random tessellations associated with max-stable random fields

With any max-stable random process eta on X = Z(d) or R-d, we associate a random tessellation of the parameter space X. The construction relies on the Poisson point process representation of the max-stable process eta which is seen as the pointwise maximum of a random collection of functions Phi = {phi(i), i >= 1}. The tessellation is constructed as follows: two points x, y is an element of X are in the same cell if and only if there exists a function phi is an element of Phi that realizes the maximum eta at both points x and y, that is, phi (x) = eta(x) and phi(y) = eta(y). We characterize the distribution of cells in terms of coverage and inclusion probabilities. Most interesting is the stationary case where the asymptotic properties of the cells are strongly related to the ergodic and mixing properties of the max-stable process eta and to its conservative/dissipative and positive/null decompositions.