Anti-concentration applied to roots of randomized derivatives of polynomials

Let (Z^(n)_k)_1 ≤ k ≤ n be a random set of points and let μ_n be its empirical measure: μ_n = 1/n∑_k=1^n δ_Z^(n)_k. Let P_n(z) := (z - Z^(n)_1)⋯ (z - Z^(n)_n) and Q_n (z) := ∑_k=1^n γ^(n)_k ∏_1 ≤ j ≤ n, j ≠ k (z- Z^(n)_j), where (γ^(n)_k)_1 ≤ k ≤ n are independent, i.i.d. random variables with Gamma distribution of parameter β/2, for some fixed β > 0. We prove that in the case where μ_n almost surely tends to μ when n →∞, the empirical measure of the complex zeros of the randomized derivative Q_n also converges almost surely to μ when n tends to infinity. Furthermore, for k = o(n / log n), we obtain that the zeros of the k-th randomized derivative of P_n converge to the limiting measure μ in the same sense. We also derive the same conclusion for a variant of the randomized derivative related to the unit circle.