Convergence of higher derivatives of random polynomials with independent roots

TL;DR

This paper proves that the zeros of high-order derivatives of random polynomials with independent roots tend to follow the original root distribution, even for derivative orders nearly linear in the polynomial degree, extending previous limits.

Contribution

It establishes the convergence of zeros of derivatives of random polynomials to the original measure for derivative orders up to nearly linear in degree, surpassing the prior logarithmic barrier.

Findings

Zeros of derivatives converge to original measure for large classes of distributions.

Convergence holds for derivative orders up to o(n/log n).

Robustness under perturbations of roots is demonstrated.

Abstract

Let $\mu$ be a probability measure on $\mathbb C$, and let $P_n$ be the random polynomial whose zeros are sampled independently from $\mu$. We study the asymptotic distribution of zeros of high-order derivatives of $P_n$. We show that, for large classes of measures $\mu$, the empirical distribution of zeros of the $k$-th derivative converges back to $\mu$ for all derivative orders $k=o(n/\log n)$. This includes all discrete measures and a broad family of measures satisfying a mild dimension-nondegeneracy condition. We further establish a robustness result showing that, for arbitrary $\mu$, even after adding a vanishing proportion of roots drawn from a dimension-nondegenerate perturbation, the derivative zero measures still converge back to $\mu$. These results break the previously known logarithmic barrier on the order of differentiation and demonstrate that the limiting root distribution is preserved under differentiation of order growing nearly linearly with the degree.