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

Jean-Marc Azaïs (1) , Céline Delmas (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 MIAT INRAE - Unité de Mathématiques et Informatique Appliquées de Toulouse

Let 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.

Mots clés: Critical points; GOE matrices; Kac-Rice formula; Gaussian fields

Domaines: Probabilités [math.PR]

Jean-Marc Azaïs, Céline Delmas. Mean number and correlation function of critical points of isotropic Gaussian fields and some results on GOE random matrices. Stochastic Processes and their Applications, 2022, 150, pp.411-445. ⟨10.1016/j.spa.2022.04.013⟩. ⟨hal-03663330⟩