Local random potentials of high differentiability to model the Landscape
We generate random functions locally via a novel generalization of Dyson Brownian motion, such that the functions are in a desired differentiability class C{sup k}, while ensuring that the Hessian is a member of the Gaussian orthogonal ensemble (other ensembles might be chosen if desired). Potentials in such higher differentiability classes (k≥2) are required/desirable to model string theoretical landscapes, for instance to compute cosmological perturbations (e.g., k=2 for the power-spectrum) or to search for minima (e.g., suitable de Sitter vacua for our universe). Since potentials are created locally, numerical studies become feasible even if the dimension of field space is large (D∼100). In addition to the theoretical prescription, we provide some numerical examples to highlight properties of such potentials; concrete cosmological applications will be discussed in companion publications.