Zeros of random polynomials and their higher derivatives

We consider zeros of higher derivatives of various random polynomials and show that their limiting empirical measures agree with those of roots of corresponding random polynomials. Examples of random polynomials include those whose roots are given by i.i.d. random variables and those whose zeros are nearly all deterministic except for a fixed number of random roots. As an application, we show that such a phenomenon holds for the random polynomials whose roots follow the distribution of the 2D Coulomb gas ensemble.