Jean-Marc Azaïs (Institut de mathématiques de Toulouse)

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

vendredi 20 décembre 2019, 9h30 - 10h30

Salle du conseil, espace Turing

Let $\cX= \{X(t) : t \in \R^N \} $ be an isotropic Gaussian random field with real values.
In a first part we study the mean number of critical points of $\cX$ with index $k$, above a level, 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 exact expressions for the mean number of critical points with a given index and their distribution as a function of their index.
In a second part we study attraction or repulsion between these critical points again as a function of their index. A measure is the correlation function.
We prove attraction between critical points when $N>2$, neutrality for $N=2$ and repulsion for $N=1$.
We prove that the attraction between critical points that occurs when the dimension is greater than 2 is due to attraction between critical points with adjacent indexes.
We prove a strong repulsion between maxima and minima. We study the correlation function between maxima (or minima). These results are part of an ARXIV paper by J.M. Azaïs and C. Delmas.\\