Anti-concentration applied to roots of randomized derivatives of polynomials

Let $(Z_k^{(n)}{)}_{1\le k\le n}$ be a random set of points and let ${\mu }_n$ be its \emph{empirical measure}: $${\mu }_n=\frac{1}{n}\sum _{k=1}^n{\delta }_{Z_k^{(n)}}\mathrm{.}$$ Let $$P_n(z):=(z-Z_1^{(n)})\cdots (z-Z_n^{(n)})\text{and}{Q}_n(z):=\sum _{k=1}^n{\gamma }_k^{(n)}\prod _{1\le j\le n,j\mathrm{\ne }k}(z-Z_j^{(n)}),$$ where $({\gamma }_k^{(n)}{)}_{1\le k\le n}$ are independent, i.i.d. random variables with Gamma distribution of parameter $\beta \mathrm{/}2$, for some fixed $\beta >0$. We prove that in the case where ${\mu }_n$ almost surely tends to $\mu$ when $n\to \mathrm{\infty }$, 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 $k=o(n\mathrm{/}\log n)$, 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.