Mean number and correlation function of critical points of isotropic Gaussian fields

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

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

Let X = {X(t) : t ∈ R N } be an isotropic Gaussian random field with real values. In a first part we study the mean number of critical points of X with index k using random matrices tools. We obtain an exact expression for the probability density of the eigenvalue of rank k of a N-GOE matrix. We deduce some exact expressions for the mean number of critical points with a given index. In a second part we study 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 observed. The correlation function between maxima (or minima) depends on the dimension of the ambient space.

Mots clés: Random matrices; Point processes; Critical points; GOE matrices; Isotropic Gaussian fields; Kac-Rice for- mula

Domaines: Probabilités [math.PR]

Jean-Marc Azaïs, Céline Delmas. Mean number and correlation function of critical points of isotropic Gaussian fields. 2019. ⟨hal-02317445v1⟩