Zeros of polynomial powers under the heat flow

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 μ 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 λ ⱼ of P, and μ ₜ approximates the superposition of semicircle laws around λ ⱼ. Then for arbitrary time, the support of μ ₜ forms intricate curves, which merge as t grows, until for large time, the limit distribution μ ₜ approaches a widespread semicircle law through the initial center of mass of the λ ⱼ. We further show that the Stieltjes transform of μ 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 μ ₜ is available.