Crystallization of random trigonometric polynomials

TL;DR

This paper investigates how repeated differentiation of random trigonometric polynomials leads their roots to become evenly spaced, revealing non-Gaussian asymptotic distributions and modeling a one-dimensional crystallization process.

Contribution

It provides a precise asymptotic analysis of root distributions during the differentiation process, modeling crystallization in a simplified mathematical setting.

Findings

Roots become evenly spaced with repeated differentiation

Distribution of roots around crystalline configuration is non-Gaussian

Asymptotic behavior of root distribution is characterized

Abstract

We give a precise measure of the rate at which repeated differentiation of a random trigonometric polynomial causes the roots of the function to approach equal spacing. This can be viewed as a toy model of crystallization in one dimension. In particular we determine the asymptotics of the distribution of the roots around the crystalline configuration and find that the distribution is not Gaussian.