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.

Abstract. We study the hole probabilities of the infinite Ginibre ensemble X∞, a determinantal point process on the complex plane with the kernelwith respect to the Lebesgue measure on the complex plane. Let U be an open subset of open unit disk D and X∞(rU ) denote the number of points of X∞ that fall in rU . Then, under some conditions on U , we show thatwhere ∅ is the empty set andis the space of all compactly supported probability measures with support in U c . Using potential theory, we give an explicit formula for R U , the minimum possible energy of a probability measure compactly supported on U c under logarithmic potential with a quadratic external field. Moreover, we calculate R U explicitly for some special sets like annulus, cardioid, ellipse, equilateral triangle and half disk.

Abstract. We show that Lyapunov exponents and stability exponents are equal in the case of product of i.i.d isotropic(also known as bi-unitarily invariant) random matrices. We also derive aysmptotic distribution of singular values and eigenvalues of these product random matrices. Moreover, Lyapunov exponents are distinct, unless the random matrices are random scalar multiples of Haar unitary matrices or orthogonal matrices. As a corollary of above result, we show probability that product of n i.i.d real isotropic random matrices has all eigenvalues real goes to one as n → ∞. Also, in the proof of a lemma, we observe that a real (complex) Ginibre matrix can be written as product of a random lower triangular matrix and an independent truncated Haar orthogonal (unitary) matrix. Definitions and introductionLet M 1 , M 2 , . . . be sequence of i.i.d random matrices of order d. Define σ n to be diagonal matrix with singular values of product matrix P n = M 1 M 2 ..M n in the diagonal in decreasing order and similarly λ n to be diagonal matrix with eigenvalues of P n in the diagonal in decreasing order of absolute values, for n = 1, 2, . . .. Let |λ n | 1 n and |σ n | 1 n denote diagonal matrices with non-negative n-th roots of absolute values of diagonal entries of λ n and σ n in the diagonal, respectively.Define σ := lim n→∞ |σ n | 1 n and λ := lim n→∞ |λ n | 1 n , if the limits exist. Then diagonal elements of ln σ and ln λ are called Lyapunov exponents and stability exponents for products of i.i.d random matrices, respectively. In other words, they are rates of exponential growth(or decay) of singular values and eigenvalues of product matrices P n , respectively as n → ∞.We consider both real and complex random matrices in this paper.For the sake of simplicity, we restrict ourselves mostly to complex random matrices . But all the definitions and statements, along with proofs, carry over immediately to real case (by replacing everywhere unitary matrices by orthogonal matrices). Definition 1. A random matrix M is said to be isotropic if probability distribution of U M V is same as that of M , for all unitary matrices U, V .They also go by the names of bi-unitarily invariant and rotation invariant random matrices. It follows from definition that distribution of U M V is same as that of M , if U, V are Haar distributed random unitary matrices independent of M and each other. M = P DQ be the singular value decomposition of M , then U M V =

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