Landscape k-complexity of isotropic centered Gaussian fields

In large dimension, we study the asymptotic behavior of the mean number of critical points with index k below a level u for an isotropic centered Gaussian random field defined on a family of subsets of $R^d$ depending on d. We prove the existence of three regimes depending on the speed of growth of the volume the parameter set. In the first regime the mean number of critical points decreases exponentially with the dimension. For the second regime, there exists a critical level $u_c$ such that the mean number of critical points with index k below a level u with $u > uc$ increases exponentially with the dimension d independently of the index k and decreases exponentially with d when $u < u_c$. In the third regime, there exists a layered structure depending on the level u considered and on the index $k$ of the critical points. This behavior is similar to the one encountered on the sphere by Auffinger et al. [5]. In the particular case of the Bargmann-Fock field, only two regimes coexist.