  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.
