Zeros of polynomial powers under the heat flow

TL;DR

This paper investigates how the zeros of high powers of polynomials evolve under heat flow, revealing a transition from semicircular distributions to complex curves and a final semicircle law, characterized by a self-consistent equation.

Contribution

It provides a detailed analysis of the zero distribution evolution for polynomial powers under heat flow, including the derivation of a self-consistent equation and Burgers' equation for the limit distribution.

Findings

Zeros spread into semicircular distributions at small times

Complex curves form and merge during evolution

Zero distribution approaches a semicircle law at large times

Abstract

We study the evolution of zeros of high polynomial powers under the heat flow. For any fixed polynomial $P(z)$, we prove that the empirical zero distribution of its heat-evolved $n$-th power converges to a distribution on the complex plane as $n$ tends to infinity. We describe this limit distribution $\mu_t$ as a function of the time parameter $t$ of the heat evolution: For small time, zeros start to spread out in approximately semicircular distributions, then intricate curves start to form and merge, until for large time, the zero distribution approaches a widespread semicircle law through the initial center of mass. The Stieltjes transform of the limit distribution $\mu_t$ satisfies a self-consistent equation and a Burgers' equation. The present paper deals with general complex-rooted polynomials for which, in contrast to the real-rooted case, no free-probabilistic representation for $\mu_t$ is available.