Heat flow and repeated differentiation of polynomials with i.i.d. roots

Abstract

How do the zeros of a polynomial evolve under the holomorphic heat flow or repeated differentiation? We study these two evolutions for random polynomials with i.i.d.~roots $z_1,\dots,z_n\simμ_0$, where $μ_0$ has a bounded compactly supported density in the complex plane. For the heat flow of small enough time $t>0$ and assuming Lipschitz continuous Stieltjes transform of $μ_0$, we prove the conjectured weak convergence of the empirical root distributions, in fact almost surely. We also identify the conjectured weak limit after $\lfloor tn \rfloor$ differentiations, for rotationally invariant initial distributions. Both limit distributions are given by push-forwards under explicit transport maps: The heat flow transforms of the circular law into the elliptic law, while differentiation moves surviving mass towards the origin. The common proof strategy crucially relies on recursion identities from leaving a factor out, together with Stieltjes transforms and concentration inequalities. It allows for generalizations beyond symmetric distributions, and quantifications of the statement that the limiting distribution $μ_0$ is preserved under $o(n)$ derivatives or heat flow of time $t_n\to 0$.