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 μₙ be its empirical measure: μₙ = 1/n ∑ₖ₌₁ⁿ δ_Z⁽ⁿ⁾ₖ. Let Pₙ(z) := (z - Z⁽ⁿ⁾₁)⋯ (z - Z⁽ⁿ⁾ₙ) and Qₙ (z) := ∑ₖ₌₁ⁿ 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 μₙ almost surely tends to μ when n → ∞, the empirical measure of the complex zeros of the randomized derivative Qₙ 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 Pₙ 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.