Taro Nagao

Asymptotic correlations at the spectrum edge of random matrices

Abstract Edge correlation functions among the eigenvalues are evaluated for Laguerre and Jacobi ensembles of real-symmetric and self-dual quaternion random matrices of infinite dimension. Universal and non-universal behavior of the eigenvalue correlations is found.

Nonuniversal correlations for random matrix ensembles

The eigenvalue correlations of the generalized Gaussian and Laguerre random matrix ensembles are calculated exactly. The fluctuations are shown to be nonuniversal in certain intervals of the spectrum. A physical example from quantum transport where such nonuniversal effects occur is discussed.

Laguerre ensembles of random matrices: Nonuniversal correlation functions

The eigenvalue correlation functions of the Laguerre ensembles (exponential ensembles) of real symmetric and self‐dual quaternion random matrices are calculated. It is shown that in certain intervals of the spectrum the renormalized correlations are not universal. A physical example in the transmission spectrum of a disordered conductor where such nonuniversality occurs is given.

Eigenvalue distribution of random matrices at the spectrum edge

Local eigenvalue correlations are exactly evaluated at the edge region for random matrix ensembles related to orthogonal polynomials on a finite real interval. We define a determinant of a quaternion matrix and utilize it for the evaluation. It is shown that there is an exact equivalence between the local correlations for Jacobi and Laguerre ensembles of random matrices.

Exactly solvable models and finite size corrections

Recent developments in the theory of exactly solvable models are reviewed. Particular attention is paid to the finite size corrections to the Bethe ansatz equations. Baxter’s formula which relates a 2-dimensional statistical model with a 1-dimensional spin model is extended into the finite temperature case. A combination of this extension and the theory of finite size corrections gives a systematic method to evaluate low temperature expansions of physical quantities. Applications of the method to 1-dimensional quantum spin models are discussed. Throughout this paper, the usefulness of the soliton theory should be observed.

An Integration Method on Generalized Circular Ensembles

Dysons circular ensembles of random matrices have been successfully applied to account for the statistical properties of complex spectra. We present an integration method on generalized circular ensembles. In order to express the correlation functions of unitary ensembles, Szegos orthogonal polynomials on the unit circle are used. Further, we introduce skew orthogonal polynomials on the unit circle to derive the correlation functions of orthogonal and symplectic ensembles.

Quantum integrable systems

Abstract We report some recent results related to quantum integrable systems. The Baxter formula is generalized into the case of finite temperature. We reformulate the quantum inverse scattering method for 1D quantum systems with long-range interactions. Extensions of the Gaudin model and integrability of the Calogero model are discussed.

Correlation Functions for Jastrow-Product Wave Functions

Correlation functions are evaluated for the Jastrow-product variational wave functions in one dimension. Discrete version of the integration method for random matrices is introduced for the evaluation. The results are expressed in terms of orthogonal polynomials on a discrete measure.

Thermodynamics of Particle Systems Related to Random Matrices

In the theory of random matrices, the eigenvalue statistics of Hamiltonian ensembles is reduced to one-dimensional classical statistical mechanics of logarithmically interacting particles. Correspo...In the theory of random matrices, the eigenvalue statistics of Hamiltonian ensembles is reduced to one-dimensional classical statistical mechanics of logarithmically interacting particles. Corresponding to one-body external potentials, there may be infinite number of ensembles related to orthogonal polynomials. Gaussian ensembles, which are related to the Hermite polynomials, are usually adopted. We choose general classical orthogonal polynomials and get a wider class of statistical ensembles. The partition functions for these ensembles are given by Selbergs integral formula. We discuss the thermodynamic limit of this formula and evaluate the free energies, the internal energies, the entropies and the specific heats.

Level Statistics of Discrete Schrödinger Equations and Orthogonal Polynomials

One-dimensional discrete Schrodinger equations (tight binding models) are analyzed from the viewpoint of level statistics. The energy levels of them are identical to the zeroes of the corresponding orthogonal polynomials. The level densities and local level correlations are calculated in the cases related to classical orthogonal polynomials. The level densities are the same as those of the random matrix ensembles which correspond to the same classical orthogonal polynomials. The local correlations of levels show perfect rigidity.