7,308 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
Analogies between random matrix ensembles and the one-component plasma in two-dimensions
The eigenvalue PDF for some well known classes of non-Hermitian random
matrices --- the complex Ginibre ensemble for example --- can be interpreted as
the Boltzmann factor for one-component plasma systems in two-dimensional
domains. We address this theme in a systematic fashion, identifying the plasma
system for the Ginibre ensemble of non-Hermitian Gaussian random matrices ,
the spherical ensemble of the product of an inverse Ginibre matrix and a
Ginibre matrix , and the ensemble formed by truncating unitary
matrices, as well as for products of such matrices. We do this when each has
either real, complex or real quaternion elements. One consequence of this
analogy is that the leading form of the eigenvalue density follows as a
corollary. Another is that the eigenvalue correlations must obey sum rules
known to characterise the plasma system, and this leads us to a exhibit an
integral identity satisfied by the two-particle correlation for real quaternion
matrices in the neighbourhood of the real axis. Further random matrix ensembles
investigated from this viewpoint are self dual non-Hermitian matrices, in which
a previous study has related to the one-component plasma system in a disk at
inverse temperature , and the ensemble formed by the single row and
column of quaternion elements from a member of the circular symplectic
ensemble.Comment: 25 page
Asymptotics of finite system Lyapunov exponents for some random matrix ensembles
For products of random matrices of size , there is a
natural notion of finite Lyapunov exponents . In the
case of standard Gaussian random matrices with real, complex or real quaternion
elements, and extended to the general variance case for , methods known
for the computation of are used to
compute the large form of the variances of the exponents. Analogous
calculations are performed in the case that the matrices making up are
products of sub-blocks of random unitary matrices with Haar measure.
Furthermore, we make some remarks relating to the coincidence of the Lyapunov
exponents and the stability exponents relating to the eigenvalues of .Comment: 15 page
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
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