A Universality Property of Gaussian Analytic Functions

We consider random analytic functions defined on the unit disk of the complex plane as power series such that the coefficients are i.i.d., complex valued random variables, with mean zero and unit variance. For the case of complex Gaussian coefficients, Peres and Vir\'ag showed that the zero set forms a determinantal point process with the Bergman kernel. We show that for general choices of random coefficients, the zero set is asymptotically given by the same distribution near the boundary of the disk, which expresses a universality property. The proof is elementary and general.Comment: 7 pages. In the new version we shortened the proof. The original arXiv submission is longer and more self-containe

A multilinear algebra proof of the Cauchy-Binet formula and a multilinear version of Parseval's identity

We give a short proof of the Cauchy-Binet determinantal formula using multilinear algebra by first generalizing it to an identity {\em not} involving determinants. By extending the formula to abstract Hilbert spaces we obtain, as a corollary, a generalization of the classical Parseval identity.Comment: 9 pages, 2 diagram

A note on the tensor product of two random unitary matrices

In this note we consider the point process of eigenvalues of the tensor product of two independent random unitary matrices of size m x m and n x n. When n becomes large, the process behaves like the superposition of m independent sine processes. When m and n go to infinity, we obtain the Poisson point process in the limit

The largest root of random Kac polynomials is heavy tailed

We prove that the largest and smallest root in modulus of random Kac polynomials have a non-universal behavior. They do not converge towards the edge of the support of the limiting distribution of the zeros. This non-universality is surprising as the large deviation principle for the empirical measure is universal. This is in sharp contrast with random matrix theory where the large deviation principle is non-universal but the fluctuations of the largest eigenvalue are universal. We show that the modulus of the largest zero is heavy tailed, with a number of finite moments bounded from above by the behavior at the origin of the distribution of the coefficients. We also prove that the random process of the roots of modulus smaller than one converges towards a limit point process. Finally, in the case of complex Gaussian coefficients, we use the work of Peres and Vir{\'a}g [PV05] to obtain explicit formulas for the limiting objects

Derivation of an eigenvalue probability density function relating to the Poincare disk

A result of Zyczkowski and Sommers [J.Phys.A, 33, 2045--2057 (2000)] gives the eigenvalue probability density function for the top N x N sub-block of a Haar distributed matrix from U(N+n). In the case n \ge N, we rederive this result, starting from knowledge of the distribution of the sub-blocks, introducing the Schur decomposition, and integrating over all variables except the eigenvalues. The integration is done by identifying a recursive structure which reduces the dimension. This approach is inspired by an analogous approach which has been recently applied to determine the eigenvalue probability density function for random matrices A^{-1} B, where A and B are random matrices with entries standard complex normals. We relate the eigenvalue distribution of the sub-blocks to a many body quantum state, and to the one-component plasma, on the pseudosphere.Comment: 11 pages; To appear in J.Phys