Mean number and correlation function of critical points of isotropic Gaussian fields and some results on GOE random matrices

International audienceLet X = {X(t) : t is an element of R-N} be an isotropic Gaussian random field with real values. The first part studies the mean number of critical points of X with index k using random matrices tools. An exact expression for the probability density of the kth eigenvalue of a N-GOE matrix is obtained. We deduce some exact expressions for the mean number of critical points with a given index. A second part studies the attraction or repulsion between these critical points. A measure is the correlation function. We prove attraction between critical points when N > 2, neutrality for N = 2 and repulsion for N = 1. The attraction between critical points that occurs when the dimension is greater than two is due to critical points with adjacent indexes. A strong repulsion between maxima and minima is proved. The correlation function between maxima (or minima) depends on the dimension of the ambient space