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

TL;DR

This paper analyzes the mean number and correlation functions of critical points in isotropic Gaussian fields, linking them to GOE random matrices, and reveals dimension-dependent attraction or repulsion behaviors among critical points.

Contribution

It provides exact formulas for the distribution of eigenvalues of GOE matrices and applies these to derive critical point statistics and their interactions in Gaussian fields.

Findings

Exact eigenvalue density for GOE matrices derived.

Critical points exhibit attraction in dimensions greater than two.

Strong repulsion observed between maxima and minima.

Abstract

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.