13,179 research outputs found
The Onset of Phase Transitions in Condensed Matter and Relativistic QFT
Kibble and Zurek have provided a unifying causal picture for the appearance
of topological defects like cosmic strings or vortices at the onset of phase
transitions in relativistic QFT and condensed matter systems respectively.
There is no direct experimental evidence in QFT, but in condensed matter the
predictions are largely, but not wholly, supported in superfluid experiments on
liquid helium. We provide an alternative picture for the initial appearance of
strings/vortices that is commensurate with all the experimental evidence from
condensed matter and consider some of its implications for QFT.Comment: 37 pages, to be published in Condensed Matter Physics, 200
New Multicritical Random Matrix Ensembles
In this paper we construct a class of random matrix ensembles labelled by a
real parameter , whose eigenvalue density near zero behaves
like . The eigenvalue spacing near zero scales like
and thus these ensembles are representatives of a {\em
continous} series of new universality classes. We study these ensembles both in
the bulk and on the scale of eigenvalue spacing. In the former case we obtain
formulas for the eigenvalue density, while in the latter case we obtain
approximate expressions for the scaling functions in the microscopic limit
using a very simple approximate method based on the location of zeroes of
orthogonal polynomials.Comment: 15 pages, 3 figures; v2: version to appear in Nucl. Phys.
Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests
The main purpose of this article is to show how symmetry structures in
partial differential equations can be preserved in a discrete world and
reflected in difference schemes. Three different structure preserving
discretizations of the Liouville equation are presented and then used to solve
specific boundary value problems. The results are compared with exact solutions
satisfying the same boundary conditions. All three discretizations are on four
point lattices. One preserves linearizability of the equation, another the
infinite-dimensional symmetry group as higher symmetries, the third one
preserves the maximal finite-dimensional subgroup of the symmetry group as
point symmetries. A 9-point invariant scheme that gives a better approximation
of the equation, but significantly worse numerical results for solutions is
presented and discussed
Cauchy Biorthogonal Polynomials
The paper investigates the properties of certain biorthogonal polynomials
appearing in a specific simultaneous Hermite-Pade' approximation scheme.
Associated to any totally positive kernel and a pair of positive measures on
the positive axis we define biorthogonal polynomials and prove that their
zeroes are simple and positive. We then specialize the kernel to the Cauchy
kernel 1/{x+y} and show that the ensuing biorthogonal polynomials solve a
four-term recurrence relation, have relevant Christoffel-Darboux generalized
formulae, and their zeroes are interlaced. In addition, these polynomial solve
a combination of Hermite-Pade' approximation problems to a Nikishin system of
order 2. The motivation arises from two distant areas; on one side, in the
study of the inverse spectral problem for the peakon solution of the
Degasperis-Procesi equation; on the other side, from a random matrix model
involving two positive definite random Hermitian matrices. Finally, we show how
to characterize these polynomials in term of a Riemann-Hilbert problem.Comment: 38 pages, partially replaces arXiv:0711.408
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