Zeros of polynomial powers under the heat flow

Abstract

We study the zero evolution of high powers of polynomials 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 holomorphic heat flow operator: For small time, zeros of the heat-evolved polynomial start to spread out from the initial zeros $\lambda_j$ of $P$, and $\mu_t$ approximates the superposition of semicircle laws around $\lambda_j$. Then for arbitrary time, the support of $\mu_t$ forms intricate curves, which merge as $t$ grows, until for large time, the limit distribution $\mu_t$ approaches a widespread semicircle law through the initial center of mass of the $\lambda_j$. We further show that the Stieltjes transform of $\mu_t$ 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.