In order to have a better understanding of finite random matrices with non-Gaussian entries, we study the 1/N expansion of local eigenvalue statistics in both the bulk and at the hard edge of the spectrum of random matrices. This gives valuable information about the smallest singular value not seen in universality laws. In particular, we show the dependence on the fourth moment (or the kurtosis...

We discuss the problem of adding random matrices, which enable us to study Hamiltonians consisting of a deterministic term plus a random term. Using a diagrammatic approach and introducing the concept of “gluon connectedness,” we calculate the density of energy levels for a wide class of probability distributions governing the random term, thus generalizing a result obtained recently by Brézin,...

In this paper the relationship between the problem of constructing the ground state energy for the quantum quartic oscillator and the problem of computing mean eigenvalue of large positively de nite random hermitean matrices is established. This relationship enables one to present several more or less closed expressions for the oscillator energy. One of such expressions is given in the form of ...

In this paper we present an analytic method for calculating the transition probability between two random Gaussian matrices with given eigenvalue spectra in the context of Dyson Brownian motion. We show that in the Coulomb gas language, in large N limit, memory of the initial state is preserved in the form of a universal linear potential acting on the eigenvalues. We compute the likelihood of a...

1. Wigner Matrices 2 2. Wigner’s Semicircle Law 4 3. Markov’s Inequality and Convergence of Expectation 6 4. First Proof of Wigner’s Semicircle Law 7 4.1. Convergence of matrix moments in Expectation 7 4.2. Convergence of Matrix Moments in Probability 12 4.3. Weak Convergence 15 5. Removing Moment Assumptions 17 6. Largest Eigenvalue 21 7. The Stiletjes Transform 31 8. The Stieltjes Transform a...

We study Hamiltonians consisting of a deterministic term plus a random term. Using a diagrammatic approach and introducing the concept of " gluon connected-ness, " we calculate the density of energy levels for a wide class of probability distributions governing the random term, thus generalizing a result obtained recently by Brézin, Hikami, and Zee. The method used here may be applied to a broa...

We investigate random, discrete Schrödinger operators which arise naturally in the theory of random matrices, and depend parametrically on Dyson’s Coulomb gas inverse temperature β. They are similar to the class of “critical” random Schrödiner operators with random potentials which diminish as |x|− 12 . We show that as a function of β they undergo a transition from a regime of (power-law) local...