Random Matrices: Theory and Applications

Tensor products of M random unitary matrices of size N from the circular unitary ensemble are investigated. We show that the spectral statistics of the tensor product of random matrices becomes Poissonian if M = 2, N become large or M become large and N = 2.

We derive simple linear, inhomogeneous recurrences for the variance of the index by utilizing the fact that the generating function for the distribution of the number of positive eigenvalues of a Gaussian unitary ensemble is a τ-function of the fourth Painlevé equation. From this we deduce a simple summation formula, several integral representations and finally an exact hypergeometric function evaluation for the variance.

We prove that for the eigenvectors matrix of a Wigner matrix, under some moments conditions, the bivariate random process

In the present paper we obtain the large N asymptotics of the partition function Z_N of the six-vertex model with domain wall boundary conditions on the critical line between the disordered and antiferroelectric phases. Using the weights a = 1 - x, b = 1 + x, c = 2, |x| < 1, we prove that, as N → ∞, Z_N = CF^N2N^1/12(1 + O(N^-1)), where F is given by an explicit expression in x and the x-dependency in C is determined. This result reproduces and improves the one given in the physics literature by Bogoliubov, Kitaev and Zvonarev [Boundary polarization in the six-vertex model, Phys. Rev. E65 (2002) 026126]. Furthermore, we prove that the free energy exhibits an infinite-order phase transition between the disordered and antiferroelectric phases. Our proofs are based on the large N asymptotics for the underlying orthogonal polynomials which involve a non-analytical weight function, the Deift–Zhou non-linear steepest descent method to the corresponding Riemann–Hilbert problem, and the Toda equation for the tau-function.

We study the global fluctuations for linear statistics of the form as n → ∞, for C¹ functions f, and λ₁, …, λ_n being the eigenvalues of a (general) β-Jacobi ensemble. The fluctuation from the mean turns out to be given asymptotically by a Gaussian process. We compute the covariance matrix for the process and show that it is diagonalized by a shifted Chebyshev polynomial basis; in addition, we analyze the deviation from the predicted mean for polynomial test functions, and we obtain a law of large numbers.