5,034 research outputs found
Tridiagonal realization of the anti-symmetric Gaussian -ensemble
The Householder reduction of a member of the anti-symmetric Gaussian unitary
ensemble gives an anti-symmetric tridiagonal matrix with all independent
elements. The random variables permit the introduction of a positive parameter
, and the eigenvalue probability density function of the corresponding
random matrices can be computed explicitly, as can the distribution of
, the first components of the eigenvectors. Three proofs are given.
One involves an inductive construction based on bordering of a family of random
matrices which are shown to have the same distributions as the anti-symmetric
tridiagonal matrices. This proof uses the Dixon-Anderson integral from Selberg
integral theory. A second proof involves the explicit computation of the
Jacobian for the change of variables between real anti-symmetric tridiagonal
matrices, its eigenvalues and . The third proof maps matrices from the
anti-symmetric Gaussian -ensemble to those realizing particular examples
of the Laguerre -ensemble. In addition to these proofs, we note some
simple properties of the shooting eigenvector and associated Pr\"ufer phases of
the random matrices.Comment: 22 pages; replaced with a new version containing orthogonal
transformation proof for both cases (Method III
Exact calculation of the ground state single-particle Green's function for the quantum many body system at integer coupling
The ground state single particle Green's function describing hole propagation
is calculated exactly for the quantum many body system at integer
coupling. The result is in agreement with a recent conjecture of Haldane.Comment: Late
The averaged characteristic polynomial for the Gaussian and chiral Gaussian ensembles with a source
In classical random matrix theory the Gaussian and chiral Gaussian random
matrix models with a source are realized as shifted mean Gaussian, and chiral
Gaussian, random matrices with real , complex ( and
real quaternion ) elements. We use the Dyson Brownian motion model
to give a meaning for general . In the Gaussian case a further
construction valid for is given, as the eigenvalue PDF of a
recursively defined random matrix ensemble. In the case of real or complex
elements, a combinatorial argument is used to compute the averaged
characteristic polynomial. The resulting functional forms are shown to be a
special cases of duality formulas due to Desrosiers. New derivations of the
general case of Desrosiers' dualities are given. A soft edge scaling limit of
the averaged characteristic polynomial is identified, and an explicit
evaluation in terms of so-called incomplete Airy functions is obtained.Comment: 21 page
Normalization of the wavwfunction for the Calogero-Sutherland model with internal degrees of freedom
The exact normalization of a multicomponent generalization of the ground
state wavefunction of the Calogero-Sutherland model is conjectured. This result
is obtained from a conjectured generalization of Selberg's -dimensional
extension of the Euler beta integral, written as a trigonometric integral. A
new proof of the Selberg integral is given, and the method is used to provide a
proof of the mulicomponent generalization in a special two-component case.Comment: 16 pgs, latex, no figures, submitted Int. J. of Mod. Phys.
- …