44 research outputs found
Painlev\'e V and time dependent Jacobi polynomials
In this paper we study the simplest deformation on a sequence of orthogonal
polynomials, namely, replacing the original (or reference) weight
defined on an interval by It is a well-known fact that under
such a deformation the recurrence coefficients denoted as and
evolve in according to the Toda equations, giving rise to the
time dependent orthogonal polynomials, using Sogo's terminology. The resulting
"time-dependent" Jacobi polynomials satisfy a linear second order ode. We will
show that the coefficients of this ode are intimately related to a particular
Painlev\'e V. In addition, we show that the coefficient of of the
monic orthogonal polynomials associated with the "time-dependent" Jacobi
weight, satisfies, up to a translation in the Jimbo-Miwa -form of
the same while a recurrence coefficient is up to a
translation in and a linear fractional transformation
These results are found
from combining a pair of non-linear difference equations and a pair of Toda
equations. This will in turn allow us to show that a certain Fredholm
determinant related to a class of Toeplitz plus Hankel operators has a
connection to a Painlev\'e equation
QCD in One Dimension at Nonzero Chemical Potential
Using an integration formula recently derived by Conrey, Farmer and
Zirnbauer, we calculate the expectation value of the phase factor of the
fermion determinant for the staggered lattice QCD action in one dimension. We
show that the chemical potential can be absorbed into the quark masses; the
theory is in the same chiral symmetry class as QCD in three dimensions at zero
chemical potential. In the limit of a large number of colors and fixed number
of lattice points, chiral symmetry is broken spontaneously, and our results are
in agreement with expressions based on a chiral Lagrangian. In this limit, the
eigenvalues of the Dirac operator are correlated according to random matrix
theory for QCD in three dimensions. The discontinuity of the chiral condensate
is due to an alternative to the Banks-Casher formula recently discovered for
QCD in four dimensions at nonzero chemical potential. The effect of temperature
on the average phase factor is discussed in a schematic random matrix model.Comment: Latex, 23 pages and 5 figures; Added two references and corrected
several typo
Introduction to Random Matrices
These notes provide an introduction to the theory of random matrices. The
central quantity studied is where is the integral
operator with kernel 1/\pi} {\sin\pi(x-y)\over x-y} \chi_I(y). Here
and is the characteristic function
of the set . In the Gaussian Unitary Ensemble (GUE) the probability that no
eigenvalues lie in is equal to . Also is a tau-function
and we present a new simplified derivation of the system of nonlinear
completely integrable equations (the 's are the independent variables)
that were first derived by Jimbo, Miwa, M{\^o}ri, and Sato in 1980. In the case
of a single interval these equations are reducible to a Painlev{\'e} V
equation. For large we give an asymptotic formula for , which is
the probability in the GUE that exactly eigenvalues lie in an interval of
length .Comment: 44 page
Entanglement in Quantum Spin Chains, Symmetry Classes of Random Matrices, and Conformal Field Theory
We compute the entropy of entanglement between the first spins and the
rest of the system in the ground states of a general class of quantum
spin-chains. We show that under certain conditions the entropy can be expressed
in terms of averages over ensembles of random matrices. These averages can be
evaluated, allowing us to prove that at critical points the entropy grows like
as , where and are determined explicitly. In an important class of systems,
is equal to one-third of the central charge of an associated Virasoro algebra.
Our expression for therefore provides an explicit formula for the
central charge.Comment: 4 page
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
The Probability of an Eigenvalue Number Fluctuation in an Interval of a Random Matrix Spectrum
We calculate the probability to find exactly eigenvalues in a spectral
interval of a large random matrix when this interval contains eigenvalues on average. The calculations exploit an analogy to the
problem of finding a two-dimensional charge distribution on the interface of a
semiconductor heterostructure under the influence of a split gate.Comment: 4 pages, postscrip
Eigenvalue correlations on Hyperelliptic Riemann surfaces
In this note we compute the functional derivative of the induced charge
density, on a thin conductor, consisting of the union of g+1 disjoint
intervals, with respect to an external
potential. In the context of random matrix theory this object gives the
eigenvalue fluctuations of Hermitian random matrix ensembles where the
eigenvalue density is supported on J.Comment: latex 2e, seven pages, one figure. To appear in Journal of Physics
Gaussian Fluctuation in Random Matrices
Let be the number of eigenvalues, in an interval of length , of a
matrix chosen at random from the Gaussian Orthogonal, Unitary or Symplectic
ensembles of by matrices, in the limit . We prove that has a Gaussian distribution when . This theorem, which
requires control of all the higher moments of the distribution, elucidates
numerical and exact results on chaotic quantum systems and on the statistics of
zeros of the Riemann zeta function. \noindent PACS nos. 05.45.+b, 03.65.-wComment: 13 page
The two-dimensional two-component plasma plus background on a sphere : Exact results
An exact solution is given for a two-dimensional model of a Coulomb gas, more
general than the previously solved ones. The system is made of a uniformly
charged background, positive particles, and negative particles, on the surface
of a sphere. At the special value of the reduced inverse
temperature, the classical equilibrium statistical mechanics is worked out~:
the correlations and the grand potential are calculated. The thermodynamic
limit is taken, and as it is approached the grand potential exhibits a
finite-size correction of the expected universal form.Comment: 23 pages, Plain Te