Statistics of Stationary Points of Random Finite Polynomial Potentials

TL;DR

This paper develops a reliable numerical method to find and certify all stationary points of multivariate random polynomial potentials, providing new insights into their statistical properties and implications for physics and mathematics.

Contribution

It introduces a combined numerical polynomial homotopy continuation and certification approach to accurately identify all stationary points of Gaussian random polynomial potentials.

Findings

All stationary points can be reliably computed and certified.

Distribution of stationary points follows specific statistical patterns.

Implications for string theory landscapes and random system mathematics.

Abstract

The stationary points (SPs) of the potential energy landscapes (PELs) of multivariate random potentials (RPs) have found many applications in many areas of Physics, Chemistry and Mathematical Biology. However, there are few reliable methods available which can find all the SPs accurately. Hence, one has to rely on indirect methods such as Random Matrix theory. With a combination of the numerical polynomial homotopy continuation method and a certification method, we obtain all the certified SPs of the most general polynomial RP for each sample chosen from the Gaussian distribution with mean 0 and variance 1. While obtaining many novel results for the finite size case of the RP, we also discuss the implications of our results on mathematics of random systems and string theory landscapes.