We show the density of eigenvalues for three classes of random matrix ensembles is determinantal. First we derive the density of eigenvalues of product of k independent n×n matrices with i.i.d. complex Gaussian entries with a few of matrices being inverted. In second example we calculate the same for (compatible) product of rectangular matrices with i.i.d. Gaussian entries and in last example we calculate for product of independent truncated unitary random matrices. We derive exact expressions for limiting expected empirical spectral distributions of above mentioned ensembles.

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In this article, we study critical points (zeros of derivative) of random polynomials. Take two deterministic sequences {an} n≥1 and {bn} n≥1 of complex numbers whose limiting empirical measures are same. By choosing ξn = an or bn with equal probability, define the sequence of polynomials by Pn(z) = (z − ξ1) . . . (z − ξn). We show that the limiting measure of zeros and critical points agree for this sequence of random polynomials under some assumption. We also prove a similar result for triangular array of numbers. A similar result for zeros of generalized derivative (can be thought as random rational function) is also proved. Pemantle and Rivin initiated the study of critical points of random polynomials. Kabluchko proved the result considering the zeros to be i.i.d. random variables.

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In this article we study the limiting empirical measure of zeros of higher derivatives for sequences of random polynomials. We show that these measures agree with the limiting empirical measure of zeros of corresponding random polynomials. Various models of random polynomials are considered by introducing randomness through multiplying a factor with a random zero or removing a zero at random for a given sequence of deterministic polynomials. We also obtain similar results for random polynomials whose zeros are given by i.i.d. random variables. As an application, we show that these phenomenon appear for random polynomials whose zeros are given by the 2D Coulomb gas density.