3,294 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
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
Hypergeometric solutions to the q-Painlev\'e equation of type
We consider the q-Painlev\'e equation of type (a version of
q-Painlev\'e V equation) and construct a family of solutions expressible in
terms of certain basic hypergeometric series. We also present the determinant
formula for the solutions.Comment: 16 pages, IOP styl
Growth models, random matrices and Painleve transcendents
The Hammersley process relates to the statistical properties of the maximum
length of all up/right paths connecting random points of a given density in the
unit square from (0,0) to (1,1). This process can also be interpreted in terms
of the height of the polynuclear growth model, or the length of the longest
increasing subsequence in a random permutation. The cumulative distribution of
the longest path length can be written in terms of an average over the unitary
group. Versions of the Hammersley process in which the points are constrained
to have certain symmetries of the square allow similar formulas. The derivation
of these formulas is reviewed. Generalizing the original model to have point
sources along two boundaries of the square, and appropriately scaling the
parameters gives a model in the KPZ universality class. Following works of Baik
and Rains, and Pr\"ahofer and Spohn, we review the calculation of the scaled
cumulative distribution, in which a particular Painlev\'e II transcendent plays
a prominent role.Comment: 27 pages, 5 figure
The Ideal Conductor Limit
This paper compares two methods of statistical mechanics used to study a
classical Coulomb system S near an ideal conductor C. The first method consists
in neglecting the thermal fluctuations in the conductor C and constrains the
electric potential to be constant on it. In the second method the conductor C
is considered as a conducting Coulomb system the charge correlation length of
which goes to zero. It has been noticed in the past, in particular cases, that
the two methods yield the same results for the particle densities and
correlations in S. It is shown that this is true in general for the quantities
which depend only on the degrees of freedom of S, but that some other
quantities, especially the electric potential correlations and the stress
tensor, are different in the two approaches. In spite of this the two methods
give the same electric forces exerted on S.Comment: 19 pages, plain TeX. Submited to J. Phys. A: Math. Ge
Increasing subsequences and the hard-to-soft edge transition in matrix ensembles
Our interest is in the cumulative probabilities Pr(L(t) \le l) for the
maximum length of increasing subsequences in Poissonized ensembles of random
permutations, random fixed point free involutions and reversed random fixed
point free involutions. It is shown that these probabilities are equal to the
hard edge gap probability for matrix ensembles with unitary, orthogonal and
symplectic symmetry respectively. The gap probabilities can be written as a sum
over correlations for certain determinantal point processes. From these
expressions a proof can be given that the limiting form of Pr(L(t) \le l) in
the three cases is equal to the soft edge gap probability for matrix ensembles
with unitary, orthogonal and symplectic symmetry respectively, thereby
reclaiming theorems due to Baik-Deift-Johansson and Baik-Rains.Comment: LaTeX, 19 page
Correlations in two-component log-gas systems
A systematic study of the properties of particle and charge correlation
functions in the two-dimensional Coulomb gas confined to a one-dimensional
domain is undertaken. Two versions of this system are considered: one in which
the positive and negative charges are constrained to alternate in sign along
the line, and the other where there is no charge ordering constraint. Both
systems undergo a zero-density Kosterlitz-Thouless type transition as the
dimensionless coupling is varied through . In
the charge ordered system we use a perturbation technique to establish an
decay of the two-body correlations in the high temperature limit.
For , the low-fugacity expansion of the asymptotic
charge-charge correlation can be resummed to all orders in the fugacity. The
resummation leads to the Kosterlitz renormalization equations.Comment: 39 pages, 5 figures not included, Latex, to appear J. Stat. Phys.
Shortened version of abstract belo
Analytic solutions of the 1D finite coupling delta function Bose gas
An intensive study for both the weak coupling and strong coupling limits of
the ground state properties of this classic system is presented. Detailed
results for specific values of finite are given and from them results for
general are determined. We focus on the density matrix and concomitantly
its Fourier transform, the occupation numbers, along with the pair correlation
function and concomitantly its Fourier transform, the structure factor. These
are the signature quantities of the Bose gas. One specific result is that for
weak coupling a rational polynomial structure holds despite the transcendental
nature of the Bethe equations. All these new results are predicated on the
Bethe ansatz and are built upon the seminal works of the past.Comment: 23 pages, 0 figures, uses rotate.sty. A few lines added. Accepted by
Phys. Rev.
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