203 research outputs found
A random matrix decimation procedure relating to
Classical random matrix ensembles with orthogonal symmetry have the property
that the joint distribution of every second eigenvalue is equal to that of a
classical random matrix ensemble with symplectic symmetry. These results are
shown to be the case of a family of inter-relations between eigenvalue
probability density functions for generalizations of the classical random
matrix ensembles referred to as -ensembles. The inter-relations give
that the joint distribution of every -st eigenvalue in certain
-ensembles with is equal to that of another
-ensemble with . The proof requires generalizing a
conditional probability density function due to Dixon and Anderson.Comment: 19 pages, 1 figur
Correlations for the Dyson Brownian motion model with Poisson initial conditions
The circular Dyson Brownian motion model refers to the stochastic dynamics of
the log-gas on a circle. It also specifies the eigenvalues of certain
parameter-dependent ensembles of unitary random matrices. This model is
considered with the initial condition that the particles are non-interacting
(Poisson statistics). Jack polynomial theory is used to derive a simple exact
expression for the density-density correlation with the position of one
particle specified in the initial state, and the position of one particle
specified at time , valid for all .
The same correlation with two particles specified in the initial state is
also derived exactly, and some special cases of the theoretical correlations
are illustrated by comparison with the empirical correlations calculated from
the eigenvalues of certain parameter-dependent Gaussian random matrices.
Application to fluctuation formulas for time displaced linear statistics in
made.Comment: 17 pgs., 2 postscript fig
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
Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges
For the orthogonal-unitary and symplectic-unitary transitions in random
matrix theory, the general parameter dependent distribution between two sets of
eigenvalues with two different parameter values can be expressed as a
quaternion determinant. For the parameter dependent Gaussian and Laguerre
ensembles the matrix elements of the determinant are expressed in terms of
corresponding skew-orthogonal polynomials, and their limiting value for
infinite matrix dimension are computed in the vicinity of the soft and hard
edges respectively. A connection formula relating the distributions at the hard
and soft edge is obtained, and a universal asymptotic behaviour of the two
point correlation is identified.Comment: 37 pgs., 1fi
Exact and asymtotic formulas for overdamped Brownian dynamics
Exact and asymptotic formulas relating to dynamical correlations for
overdamped Brownian motion are obtained. These formulas include a
generalization of the -sum rule from the theory of quantum fluids, a formula
relating the static current-current correlation to the static density-density
correlation, and an asymptotic formula for the small- behaviour of the
dynamical structure factor. Known exact evaluations of the dynamical
density-density correlation in some special models are used to illustrate and
test the formulas.Comment: 18 pages,LaTe
Interlaced particle systems and tilings of the Aztec diamond
Motivated by the problem of domino tilings of the Aztec diamond, a weighted
particle system is defined on lines, with line containing
particles. The particles are restricted to lattice points from 0 to , and
particles on successive lines are subject to an interlacing constraint. It is
shown that marginal distributions for this particle system can be computed
exactly. This in turn is used to give unified derivations of a number of
fundamental properties of the tiling problem, for example the evaluation of the
number of distinct configurations and the relation to the GUE minor process. An
interlaced particle system associated with the domino tiling of a certain half
Aztec diamond is similarly defined and analyzed.Comment: 17 pages, 4 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
Expanded Vandermonde powers and sum rules for the two-dimensional one-component plasma
The two-dimensional one-component plasma (2dOCP) is a system of mobile
particles of the same charge on a surface with a neutralising background.
The Boltzmann factor of the 2dOCP at temperature can be expressed as a
Vandermonde determinant to the power . Recent advances in
the theory of symmetric and anti-symmetric Jack polymonials provide an
efficient way to expand this power of the Vandermonde in their monomial basis,
allowing the computation of several thermodynamic and structural properties of
the 2dOCP for values up to 14 and equal to 4, 6 and 8. In this
work, we explore two applications of this formalism to study the moments of the
pair correlation function of the 2dOCP on a sphere, and the distribution of
radial linear statistics of the 2dOCP in the plane
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
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
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