133 research outputs found
Pfaffian Expressions for Random Matrix Correlation Functions
It is well known that Pfaffian formulas for eigenvalue correlations are
useful in the analysis of real and quaternion random matrices. Moreover the
parametric correlations in the crossover to complex random matrices are
evaluated in the forms of Pfaffians. In this article, we review the
formulations and applications of Pfaffian formulas. For that purpose, we first
present the general Pfaffian expressions in terms of the corresponding skew
orthogonal polynomials. Then we clarify the relation to Eynard and Mehta's
determinant formula for hermitian matrix models and explain how the evaluation
is simplified in the cases related to the classical orthogonal polynomials.
Applications of Pfaffian formulas to random matrix theory and other fields are
also mentioned.Comment: 28 page
Symmetrized models of last passage percolation and non-intersecting lattice paths
It has been shown that the last passage time in certain symmetrized models of
directed percolation can be written in terms of averages over random matrices
from the classical groups , and . We present a theory of
such results based on non-intersecting lattice paths, and integration
techniques familiar from the theory of random matrices. Detailed derivations of
probabilities relating to two further symmetrizations are also given.Comment: 21 pages, 5 figure
A generalized plasma and interpolation between classical random matrix ensembles
The eigenvalue probability density functions of the classical random matrix
ensembles have a well known analogy with the one component log-gas at the
special couplings \beta = 1,2 and 4. It has been known for some time that there
is an exactly solvable two-component log-potential plasma which interpolates
between the \beta =1 and 4 circular ensemble, and an exactly solvable
two-component generalized plasma which interpolates between \beta = 2 and 4
circular ensemble. We extend known exact results relating to the latter --- for
the free energy and one and two-point correlations --- by giving the general
(k_1+k_2)-point correlation function in a Pfaffian form. Crucial to our working
is an identity which expresses the Vandermonde determinant in terms of a
Pfaffian. The exact evaluation of the general correlation is used to exhibit a
perfect screening sum rule.Comment: 21 page
Two-dimensional one-component plasma on a Flamm's paraboloid
We study the classical non-relativistic two-dimensional one-component plasma
at Coulomb coupling Gamma=2 on the Riemannian surface known as Flamm's
paraboloid which is obtained from the spatial part of the Schwarzschild metric.
At this special value of the coupling constant, the statistical mechanics of
the system are exactly solvable analytically. The Helmholtz free energy
asymptotic expansion for the large system has been found. The density of the
plasma, in the thermodynamic limit, has been carefully studied in various
situations
A method to calculate correlation functions for random matrices of odd size
The calculation of correlation functions for random matrix
ensembles, which can be carried out using Pfaffians, has the peculiar feature
of requiring a separate calculation depending on the parity of the matrix size
N. This same complication is present in the calculation of the correlations for
the Ginibre Orthogonal Ensemble of real Gaussian matrices. In fact the methods
used to compute the , N odd, correlations break down in the case of N
odd real Ginibre matrices, necessitating a new approach to both problems. The
new approach taken in this work is to deduce the , N odd correlations
as limiting cases of their N even counterparts, when one of the particles is
removed towards infinity. This method is shown to yield the correlations for N
odd real Gaussian matrices.Comment: 20 pages, corrected typo
Moments of vicious walkers and M\"obius graph expansions
A system of Brownian motions in one-dimension all started from the origin and
conditioned never to collide with each other in a given finite time-interval
is studied. The spatial distribution of such vicious walkers can be
described by using the repulsive eigenvalue-statistics of random Hermitian
matrices and it was shown that the present vicious walker model exhibits a
transition from the Gaussian unitary ensemble (GUE) statistics to the Gaussian
orthogonal ensemble (GOE) statistics as the time is going on from 0 to .
In the present paper, we characterize this GUE-to-GOE transition by presenting
the graphical expansion formula for the moments of positions of vicious
walkers. In the GUE limit , only the ribbon graphs contribute and the
problem is reduced to the classification of orientable surfaces by genus.
Following the time evolution of the vicious walkers, however, the graphs with
twisted ribbons, called M\"obius graphs, increase their contribution to our
expansion formula, and we have to deal with the topology of non-orientable
surfaces. Application of the recent exact result of dynamical correlation
functions yields closed expressions for the coefficients in the M\"obius
expansion using the Stirling numbers of the first kind.Comment: REVTeX4, 11 pages, 1 figure. v.2: calculations of the Green function
and references added. v.3: minor additions and corrections made for
publication in Phys.Rev.
A real quaternion spherical ensemble of random matrices
One can identify a tripartite classification of random matrix ensembles into
geometrical universality classes corresponding to the plane, the sphere and the
anti-sphere. The plane is identified with Ginibre-type (iid) matrices and the
anti-sphere with truncations of unitary matrices. This paper focusses on an
ensemble corresponding to the sphere: matrices of the form \bY= \bA^{-1} \bB,
where \bA and \bB are independent matrices with iid standard
Gaussian real quaternion entries. By applying techniques similar to those used
for the analogous complex and real spherical ensembles, the eigenvalue jpdf and
correlation functions are calculated. This completes the exploration of
spherical matrices using the traditional Dyson indices .
We find that the eigenvalue density (after stereographic projection onto the
sphere) has a depletion of eigenvalues along a ring corresponding to the real
axis, with reflective symmetry about this ring. However, in the limit of large
matrix dimension, this eigenvalue density approaches that of the corresponding
complex ensemble, a density which is uniform on the sphere. This result is in
keeping with the spherical law (analogous to the circular law for iid
matrices), which states that for matrices having the spherical structure \bY=
\bA^{-1} \bB, where \bA and \bB are independent, iid matrices the
(stereographically projected) eigenvalue density tends to uniformity on the
sphere.Comment: 25 pages, 3 figures. Added another citation in version
Level-Spacing Distributions and the Bessel Kernel
The level spacing distributions which arise when one rescales the Laguerre or
Jacobi ensembles of hermitian matrices is studied. These distributions are
expressible in terms of a Fredholm determinant of an integral operator whose
kernel is expressible in terms of Bessel functions of order . We derive
a system of partial differential equations associated with the logarithmic
derivative of this Fredholm determinant when the underlying domain is a union
of intervals. In the case of a single interval this Fredholm determinant is a
Painleve tau function.Comment: 18 pages, resubmitted to make postscript compatible, no changes to
manuscript conten
Correlation Functions for \beta=1 Ensembles of Matrices of Odd Size
Using the method of Tracy and Widom we rederive the correlation functions for
\beta=1 Hermitian and real asymmetric ensembles of N x N matrices with N odd.Comment: 15 page
On Eigenvalues of the sum of two random projections
We study the behavior of eigenvalues of matrix P_N + Q_N where P_N and Q_N
are two N -by-N random orthogonal projections. We relate the joint eigenvalue
distribution of this matrix to the Jacobi matrix ensemble and establish the
universal behavior of eigenvalues for large N. The limiting local behavior of
eigenvalues is governed by the sine kernel in the bulk and by either the Bessel
or the Airy kernel at the edge depending on parameters. We also study an
exceptional case when the local behavior of eigenvalues of P_N + Q_N is not
universal in the usual sense.Comment: 14 page
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