Random Matrices: Theory and Applications

We consider an indexed class of real symmetric random matrices which generalize the symmetric Hankel and Reverse Circulant matrices. We show that the limiting spectral distribution of these matrices exists almost surely and the limit is continuous in the index. We also study other properties of the limit and, in particular, explicitly characterize it for a certain subclass of matrices as a mixture of the atomic distribution at zero and the symmetrized Rayleigh distribution.

The purpose of this paper is to study the eigenvalues of U^t where U is a large N×N random unitary matrix and t > 0. In particular we are interested in the typical times t for which all the eigenvalues are simultaneously close to 1 in different ways thus corresponding to recurrence times in the issue of quantum measurements. Our strategy consists in rewriting the problem as a random matrix integral and use loop equations techniques to compute the first-orders of the large N asymptotic. We also connect the problem to the computation of a large Toeplitz determinant whose symbol is the characteristic function of several arc segments of the unit circle. In particular in the case of a single arc segment we recover Widom's formula. Eventually we explain why the first return time is expected to converge toward an exponential distribution when N is large. Numerical simulations are provided along the paper to illustrate the results.

In this paper, we study two one-parameter families of random band Toeplitz matrices:

(1) a₀ = 0, {a₁, a₂, …} in A_n(t) are independent random variables and a_-i = a_i;

(2) a₀(t) = 0, {a₁(t), a₂(t), …} in B_n(t) are independent copies of the standard Brownian motion at time t and a_-i(t) = a_i(t).

As t varies, the empirical measures μ(A_n(t)) and μ(B_n(t)) are measure valued stochastic processes. The purpose of this paper is to study the fluctuations of μ(A_n(t)) and μ(B_n(t)) as n goes to ∞. Given a monomial f(x) = x^p with p ≥ 2, the corresponding rescaled fluctuations of μ(A_n(t)) and μ(B_n(t)) are

We consider the convergence of the eigenvalues to the support of the equilibrium measure in the β matrix models at criticality. We show a phase transition phenomenon, namely that, with probability one, all eigenvalues will fall in the support of the limiting spectral measure when β > 1, whereas this fails when β < 1.