Anti-concentration applied to roots of randomized derivatives of polynomials

Abstract

Let $(Z^{(n)}_k)_{1 \leq k \leq n}$ be a random set of points and let $\mu_n$ be its \emph{empirical measure}: $$\mu_n = \frac{1}{n} \sum_{k=1}^n \delta_{Z^{(n)}_k}. $$ Let $$P_n(z) := (z - Z^{(n)}_1)\cdots (z - Z^{(n)}_n)\quad \text{and}\quad Q_n (z) := \sum_{k=1}^n \gamma^{(n)}_k \prod_{1 \leq j \leq n, j \neq k} (z- Z^{(n)}_j), $$ where $(\gamma^{(n)}_k)_{1 \leq k \leq n}$ are independent, i.i.d. random variables with Gamma distribution of parameter $\beta/2$, for some fixed $\beta > 0$. We prove that in the case where $\mu_n$ almost surely tends to $\mu$ when $n \rightarrow \infty$, the empirical measure of the complex zeros of the \emph{randomized derivative} $Q_n$ also converges almost surely to $\mu$ when $n$ tends to infinity. Furthermore, for $k = o(n / \log n)$, we obtain that the zeros of the $k-$th \emph{randomized derivative} of $P_n$ converge to the limiting measure $\mu$ in the same sense. We also derive the same conclusion for a variant of the randomized derivative related to the unit circle.