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 INRA - Unité de Mathématiques et Informatique Appliquées de Toulouse

Let $\mathcal{X}= \{X(t) : t \in \mathbb{R}^N \} $ be an isotropic Gaussian random field with real values. In a first part we study the mean number of critical points of $\mathcal{X}$ with index $k$ using random matrices tools. We obtain an exact expression for the probability density of the $k$th eigenvalue 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 proved. The correlation function between maxima (or minima) depends on the dimension of the ambient space.

Mots clés: GOE matrices; Point processes; Random matrices; Critical points; Kac-Rice formula; Isotropic 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. 2020. ⟨hal-02317445v3⟩