Convergence of higher derivatives of random polynomials with independent roots

TL;DR

This work proves that the zeros of high-order derivatives of random polynomials with i.i.d. roots from a general measure \mu converge back to the root distribution, for derivative orders k that can grow nearly linearly with the degree n (specifically k = o(n/\log n)) in broad settings. The authors split the analysis into discrete and dimension-nondegenerate measures, developing a two-pronged approach: a multiplicity-based argument for discrete measures and a potential-theoretic/anti-concentration framework for dimension-nondegenerate measures via the auxiliary polynomial S_n and Jensen-type bounds. They extend the range of applicability beyond fixed k by establishing convergence in probability under mild regularity (generic measures) and a Doeblin condition, and they show robustness to vanishing perturbations of the root set, where a small fraction α_n of roots from a dimension-nondegenerate perturbation does not alter the limiting distribution. Collectively, these results break the logarithmic barrier and demonstrate that the limiting zero distribution is preserved under differentiation of order nearly linear in the degree, with implications for the study of random polynomials and complex potential theory.

Abstract

Let $μ$ be a probability measure on $\mathbb C$, and let $P_n$ be the random polynomial whose zeros are sampled independently from $μ$. We study the asymptotic distribution of zeros of high-order derivatives of $P_n$. We show that, for large classes of measures $μ$, the empirical distribution of zeros of the $k$-th derivative converges back to $μ$ 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 $μ$, even after adding a vanishing proportion of roots drawn from a dimension-nondegenerate perturbation, the derivative zero measures still converge back to $μ$. 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.

Figures (1)

• Figure 1: Zeros of $P_n$ (black), $P_n'$ (blue), and $P_n^{(5)}$ (red) for ten independent samples. Left: $100$ roots are drawn from $\mu$ (uniform on the unit circle) and $10$ from $\nu$ (uniform on a small disk of radius $0.1$ centered at $3$). Right: all $110$ roots are drawn from $\mu$.

Theorems & Definitions (14)

• Conjecture 1.1
• Definition 1.2: Dimension-nondegenerate measures
• Theorem 1.3: Main theorem
• Theorem 1.4
• Theorem 1.5
• Proposition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4: Anti-concentration bound
• proof : Proof of Lemma \ref{['lm:upper']}