Local random potentials of high differentiability to model the Landscape

TL;DR

This paper introduces a method to generate high-differentiability random potentials locally, useful for modeling complex landscapes in string theory and cosmology, with feasible numerical implementation in high-dimensional spaces.

Contribution

A novel local generation technique for smooth random potentials with Hessians in the GOE, enabling modeling of string landscapes and cosmological phenomena in high dimensions.

Findings

Potentials are generated with desired differentiability and Gaussian orthogonal ensemble Hessians.

Numerical examples demonstrate properties of the generated potentials.

Method facilitates studies of high-dimensional landscapes in cosmology.

Abstract

We generate random functions locally via a novel generalization of Dyson Brownian motion, such that the functions are in a desired differentiability class, 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 are required/desirable to model string theoretical landscapes, for instance to compute cosmological perturbations (e.g., smooth first and second derivatives 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.