Complexity of Gaussian Random Fields with Isotropic Increments

We study the energy landscape of a model of a single particle on a random potential, that is, we investigate the topology of level sets of smooth random fields on R-N of the form X-N (x)+ mu 2 parallel to x parallel to(2), where X-N is a Gaussian process with isotropic increments. We derive asymptotic formulas for the mean number of critical pointswith critical values in an open set as the dimension N goes to infinity. In a companion paper, we provide the same analysis for the number of critical points with a given index.

Automated summary

We investigate the topology of level sets of a smooth random field on R-N, which represents the energy landscape of a single particle on a random potential. Our study derives asymptotic formulas for the mean number of critical points with critical values in a given open set as the dimension N tends to infinity. A companion paper complements this analysis by providing the same study for the number of critical points with a given index.