Complexity of Gaussian random fields with isotropic increments: critical points with given indices
Abstract
We study the landscape complexity of the Hamiltonian $X_N(x) +\frac\mu2 \x\^2,$ 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2022
 arXiv:
 arXiv:2206.13834
 Bibcode:
 2022arXiv220613834A
 Keywords:

 Mathematics  Probability;
 Mathematical Physics
 EPrint:
 Submission arXiv:2007.07668 was updated and split into two articles. This submission corresponds to the second part of arXiv:2007.07668v2. 50 pages