Flow of the zeros of polynomials under iterated differentiation

For a monic polynomial $Q_n$ of degree $n$, let $Q_{n, k}$ be its $k$-th derivative normalized to be monic. Under the only assumption that the sequence $\{Q_n\}$ has a weak* limiting zero distribution (an empirical distribution of zeros) represented by a probability measure $\mu_0$ with compact support in the complex plane, we show that as $n, k \rightarrow \infty$ such that $k / n \rightarrow t \in(0,1)$, the Cauchy transform of the normalized zero-counting measure of the polynomials $Q_{n, k}$ converges in a neighborhood of infinity to an analytic function, uniquely determined by $\mu_0$ and $t$, that can be written as the Cauchy transform of a measure $\mu_t$, not necessarily uniquely determined unless $\mu_0$ is supported on the real line. The family of these Cauchy transforms and, when well defined, the corresponding measures $\mu_t$, $t \in(0,1)$, whose dependence on the parameter $t$ can be interpreted as a flow of the zeros under iterated differentiation, has several interesting connections with the inviscid Burgers equation, the fractional free convolution of $\mu_0$, or a nonlocal diffusion equation governing the density of $\mu_t$ on $\mathbb R$. We provide an elementary and unified approach that not only recovers, but also explains various phenomena observed in prior works - from Burgers-type PDEs to free probability limits.