Roots of polynomials under repeated differentiation and repeated applications of fractional differential operators

We start with a random polynomial $P^N(z)$ of degree $N$ with independent coefficients. We then consider a new polynomial $P_t^N$ obtained by $\lceil Nt\rceil$ applications of a fractional differential operator of the form $z^a(d\mathrm{/}dz{)}^b,$ where $a$ and $b$ are real numbers. When $b>0,$ we compute the limiting root distribution ${\mu }_t$ of $P_t^N$ as $N\to \mathrm{\infty }\mathrm{.}$ We show that ${\mu }_t$ is the push-forward of the limiting root distribution of $P^N$ under a transport map $T_t$. The map $T_t$ is defined by flowing along the characteristic curves of a PDE satisfied by the log potential of ${\mu }_t\mathrm{.}$ In the special case of repeated differentiation, our results may be interpreted as saying that the roots evolve radially $\mathit{\text{with}}\mathit{\text{constant}}\mathit{\text{speed}}$ until they hit the origin, at which point, they cease to exist. For general $a$ and $b,$ the transport map $T_t$ has a free probability interpretation as multiplication of an $R$-diagonal operator by an $R$-diagonal \textquotedblleft transport operator.\textquotedblright As an application, we obtain a push-forward characterization of the free self-convolution semigroup $\oplus$ of radial measures on $\mathbb{C}$. We also consider the case $b<0,$ which includes the case of repeated integration. More complicated behavior of the roots can occur in this case.