Dynamics of roots of randomized derivative polynomials

In this paper, we study the asymptotic macroscopic behavior of the root sets of iterated, randomized derivatives of polynomials. The randomization depend on a parameter of inverse temperature $\beta \in (0, \infty]$, the case $\beta = \infty$ corresponding to the situation where one considers the derivative of polynomials, without randomization. Our constructions can be connected to random matrix theory: in particular, as detailed in Section 2, for $\beta = 2$ and roots on the real line, we get the distribution of the eigenvalues of minors of unitarily invariant random matrices. We prove that the asymptotic macroscopic behavior of the roots, i.e. the hydrodynamic limit, does not depend on $\beta$, and coincides with what we obtain for the non-randomized iterated derivatives, i.e. for $\beta = \infty$. Since recent results obtained for iterated derivations show that the limiting dynamics is governed by a non-local and non-linear PDE, we can transfer this information to the macroscopic behavior of the randomized setting. Our proof is completely explicit and relies on the analysis of increments in a triangular bivariate Markov chain.