Complexity of Gaussian random fields with isotropic increments: critical points with given indices

We study the landscape complexity of the Hamiltonian $X_N(x)+\frac{\mu }{2}\mathrm{\parallel }x{\mathrm{\parallel }}^2,$ where $X_N$ is a smooth Gaussian process with isotropic increments on ${\mathbb{R}}^N$. This model describes a single particle on a random potential in statistical physics. We derive asymptotic formulas for the mean number of critical points of index $k$ with critical values in an open set as the dimension $N$ goes to infinity. In a companion paper, we provide the same analysis without the index constraint.