Abstract
We generate random functions locally via a novel generalization of Dyson Brownian motion, such that the functions are in a desired differentiability class Ck, 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 (0D∼ 10). 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.
Highlights
We focus on the first point, the generation of random functions V ∈ Ck, which provides the foundation for concrete applications, such as the computation of cosmological perturbations, or further refinements, such as the incorporation of bounds mentioned in point 2
As in Dyson Brownian motion (DBM), the Hessian of these functions evaluated at well separated points is a random matrix in the Gaussian orthogonal ensemble (GOE)
Potentials V ∈ C2 are of interest to us, since they enable the search for minima as well as the study of cosmological perturbations and the computation of the power-spectrum
Summary
Inflationary observables depend only on properties of the potential in the vicinity of the trajectory, which motivated Marsh et al [45] to develop a computationally economical approach to generate random potentials locally by defining random functions around a path Γ in field space: for any Γ, given the value of the potential V , gradient V and Hessian V ≡ H at a point p0, the values of the potential and the gradient vector at a nearby point p1 can be obtained to leading order by means of a Taylor expansion. The distribution of the Hessian matrix at well-separated points (i.e. separated by several units of a characteristic correlation length Λh) can be restricted to any desired distribution; if Wigner’s Gaussian Orthogonal Ensemble (GOE) is chosen, as in [45], the elements of the Hessian undergo Dyson Brownian motion (DBM) [46]. We focus on the first point, the generation of random functions V ∈ Ck, which provides the foundation for concrete applications, such as the computation of cosmological perturbations, or further refinements, such as the incorporation of bounds mentioned in point 2. The latter topics are the subject of companion publications (in preparation)
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