2,258 research outputs found
{\bf -Function Evaluation of Gap Probabilities in Orthogonal and Symplectic Matrix Ensembles}
It has recently been emphasized that all known exact evaluations of gap
probabilities for classical unitary matrix ensembles are in fact
-functions for certain Painlev\'e systems. We show that all exact
evaluations of gap probabilities for classical orthogonal matrix ensembles,
either known or derivable from the existing literature, are likewise
-functions for certain Painlev\'e systems. In the case of symplectic
matrix ensembles all exact evaluations, either known or derivable from the
existing literature, are identified as the mean of two -functions, both
of which correspond to Hamiltonians satisfying the same differential equation,
differing only in the boundary condition. Furthermore the product of these two
-functions gives the gap probability in the corresponding unitary
symmetry case, while one of those -functions is the gap probability in
the corresponding orthogonal symmetry case.Comment: AMS-Late
Boundary conditions associated with the Painlev\'e III' and V evaluations of some random matrix averages
In a previous work a random matrix average for the Laguerre unitary ensemble,
generalising the generating function for the probability that an interval at the hard edge contains eigenvalues, was evaluated in terms of
a Painlev\'e V transcendent in -form. However the boundary conditions
for the corresponding differential equation were not specified for the full
parameter space. Here this task is accomplished in general, and the obtained
functional form is compared against the most general small behaviour of
the Painlev\'e V equation in -form known from the work of Jimbo. An
analogous study is carried out for the the hard edge scaling limit of the
random matrix average, which we have previously evaluated in terms of a
Painlev\'e \IIId transcendent in -form. An application of the latter
result is given to the rapid evaluation of a Hankel determinant appearing in a
recent work of Conrey, Rubinstein and Snaith relating to the derivative of the
Riemann zeta function
Applications and generalizations of Fisher-Hartwig asymptotics
Fisher-Hartwig asymptotics refers to the large form of a class of
Toeplitz determinants with singular generating functions. This class of
Toeplitz determinants occurs in the study of the spin-spin correlations for the
two-dimensional Ising model, and the ground state density matrix of the
impenetrable Bose gas, amongst other problems in mathematical physics. We give
a new application of the original Fisher-Hartwig formula to the asymptotic
decay of the Ising correlations above , while the study of the Bose gas
density matrix leads us to generalize the Fisher-Hartwig formula to the
asymptotic form of random matrix averages over the classical groups and the
Gaussian and Laguerre unitary matrix ensembles. Another viewpoint of our
generalizations is that they extend to Hankel determinants the Fisher-Hartwig
asymptotic form known for Toeplitz determinants.Comment: 25 page
Asymptotic corrections to the eigenvalue density of the GUE and LUE
We obtain correction terms to the large N asymptotic expansions of the
eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of
random N by N matrices, both in the bulk of the spectrum and near the spectral
edge. This is achieved by using the well known orthogonal polynomial expression
for the kernel to construct a double contour integral representation for the
density, to which we apply the saddle point method. The main correction to the
bulk density is oscillatory in N and depends on the distribution function of
the limiting density, while the corrections to the Airy kernel at the soft edge
are again expressed in terms of the Airy function and its first derivative. We
demonstrate numerically that these expansions are very accurate. A matching is
exhibited between the asymptotic expansion of the bulk density, expanded about
the edge, and the asymptotic expansion of the edge density, expanded into the
bulk.Comment: 14 pages, 4 figure
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.
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
Probabilistic Weyl laws for quantized tori
For the Toeplitz quantization of complex-valued functions on a
-dimensional torus we prove that the expected number of eigenvalues of
small random perturbations of a quantized observable satisfies a natural Weyl
law. In numerical experiments the same Weyl law also holds for ``false''
eigenvalues created by pseudospectral effects.Comment: 33 pages, 3 figures, v2 corrected listed titl
Joint distribution of the first and second eigenvalues at the soft edge of unitary ensembles
The density function for the joint distribution of the first and second
eigenvalues at the soft edge of unitary ensembles is found in terms of a
Painlev\'e II transcendent and its associated isomonodromic system. As a
corollary, the density function for the spacing between these two eigenvalues
is similarly characterized.The particular solution of Painlev\'e II that arises
is a double shifted B\"acklund transformation of the Hasting-McLeod solution,
which applies in the case of the distribution of the largest eigenvalue at the
soft edge. Our deductions are made by employing the hard-to-soft edge
transitions to existing results for the joint distribution of the first and
second eigenvalue at the hard edge \cite{FW_2007}. In addition recursions under
of quantities specifying the latter are obtained. A Fredholm
determinant type characterisation is used to provide accurate numerics for the
distribution of the spacing between the two largest eigenvalues.Comment: 26 pages, 1 Figure, 2 Table
Free expansion of impenetrable bosons on one-dimensional optical lattices
We review recent exact results for the free expansion of impenetrable bosons
on one-dimensional lattices, after switching off a confining potential. When
the system is initially in a superfluid state, far from the regime in which the
Mott-insulator appears in the middle of the trap, the momentum distribution of
the expanding bosons rapidly approaches the momentum distribution of
noninteracting fermions. Remarkably, no loss in coherence is observed in the
system as reflected by a large occupation of the lowest eigenstate of the
one-particle density matrix. In the opposite limit, when the initial system is
a pure Mott insulator with one particle per lattice site, the expansion leads
to the emergence of quasicondensates at finite momentum. In this case,
one-particle correlations like the ones shown to be universal in the
equilibrium case develop in the system. We show that the out-of-equilibrium
behavior of the Shannon information entropy in momentum space, and its contrast
with the one of noninteracting fermions, allows to differentiate the two
different regimes of interest. It also helps in understanding the crossover
between them.Comment: 21 pages, 14 figures, invited brief revie
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