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.