Random Functions via Dyson Brownian Motion: Progress and Problems

TL;DR

This paper extends the Dyson Brownian Motion algorithm to efficiently generate locally smooth random functions, identifies stability issues over large distances, and proposes a modification to mitigate unstable growth for modeling landscapes in string theory.

Contribution

It introduces an efficient extension of DBM for generating local random functions and addresses stability challenges for global applications.

Findings

Generated random functions exhibit unstable growth over distance.

The proposed modification partially suppresses growth, improving stability.

Application to cosmological axionic potentials demonstrates practical relevance.

Abstract

We develope a computationally efficient extension of the Dyson Brownian Motion (DBM) algorithm to generate random function in C2 locally. We further explain that random functions generated via DBM show an unstable growth as the traversed distance increases. This feature restricts the use of such functions considerably if they are to be used to model globally defined ones. The latter is the case if one uses random functions to model landscapes in string theory. We provide a concrete example, based on a simple axionic potential often used in cosmology, to highlight this problem and also offer an ad hoc modification of DBM that suppresses this growth to some degree.