Anti-concentration applied to roots of randomized derivatives of polynomials

Let (Z^ (n) ₖ) ₁ ₊ ₍ be a random set of points and let ₙ be its empirical measure: ₙ = 1n ₊=₁ⁿ ₙ^ (₍) ₖ. Let Pₙ (z): = (z - Z^ (n) ₁) (z - Z^ (n) ₙ) and Qₙ (z): = ₊=₁ⁿ ^ (n) ₖ ₁ ₉ ₍, ₉ ₊ (z- Z^ (n) ⱼ), where (^ (n) ₖ) ₁ ₊ ₍ are independent, i. i. d. random variables with Gamma distribution of parameter /2, for some fixed > 0. We prove that in the case where ₙ almost surely tends to when n, the empirical measure of the complex zeros of the randomized derivative Qₙ also converges almost surely to when n tends to infinity. Furthermore, for k = o (n / n), we obtain that the zeros of the k-th randomized derivative of Pₙ converge to the limiting measure in the same sense. We also derive the same conclusion for a variant of the randomized derivative related to the unit circle.