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