7,970 research outputs found
Condensates and instanton - torus knot duality. Hidden Physics at UV scale
We establish the duality between the torus knot superpolynomials or the
Poincar\'e polynomials of the Khovanov homology and particular condensates in
-deformed 5D supersymmetric QED compactified on a circle with 5d
Chern-Simons(CS) term. It is explicitly shown that -instanton contribution
to the condensate of the massless flavor in the background of four-observable,
exactly coincides with the superpolynomial of the torus knot where
- is the level of CS term. In contrast to the previously known results, the
particular torus knot corresponds not to the partition function of the gauge
theory but to the particular instanton contribution and summation over the
knots has to be performed in order to obtain the complete answer. The
instantons are sitting almost at the top of each other and the physics of the
"fat point" where the UV degrees of freedom are slaved with point-like
instantons turns out to be quite rich. Also also see knot polynomials in the
quantum mechanics on the instanton moduli space. We consider the different
limits of this correspondence focusing at their physical interpretation and
compare the algebraic structures at the both sides of the correspondence. Using
the AGT correspondence, we establish a connection between superpolynomials for
unknots and q-deformed DOZZ factors.Comment: v2: text substantially improve
Asymptotic zero distribution of Jacobi-Pi\~neiro and multiple Laguerre polynomials
We give the asymptotic distribution of the zeros of Jacobi-Pi\~neiro
polynomials and multiple Laguerre polynomials of the first kind. We use the
nearest neighbor recurrence relations for these polynomials and a recent result
on the ratio asymptotics of multiple orthogonal polynomials. We show how these
asymptotic zero distributions are related to the Fuss-Catalan distribution.Comment: 19 pages, 2 figures. Some minor corrections and four new references
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Sub-Wavelength Plasmonic Crystals: Dispersion Relations and Effective Properties
We obtain a convergent power series expansion for the first branch of the
dispersion relation for subwavelength plasmonic crystals consisting of
plasmonic rods with frequency-dependent dielectric permittivity embedded in a
host medium with unit permittivity. The expansion parameter is , where is the norm of a fixed wavevector, is the period of
the crystal and is the wavelength, and the plasma frequency scales
inversely to , making the dielectric permittivity in the rods large and
negative. The expressions for the series coefficients (a.k.a., dynamic
correctors) and the radius of convergence in are explicitly related to
the solutions of higher-order cell problems and the geometry of the rods.
Within the radius of convergence, we are able to compute the dispersion
relation and the fields and define dynamic effective properties in a
mathematically rigorous manner. Explicit error estimates show that a good
approximation to the true dispersion relation is obtained using only a few
terms of the expansion. The convergence proof requires the use of properties of
the Catalan numbers to show that the series coefficients are exponentially
bounded in the Sobolev norm
Spectral density of generalized Wishart matrices and free multiplicative convolution
We investigate the level density for several ensembles of positive random
matrices of a Wishart--like structure, , where stands for a
nonhermitian random matrix. In particular, making use of the Cauchy transform,
we study free multiplicative powers of the Marchenko-Pastur (MP) distribution,
, which for an integer yield Fuss-Catalan
distributions corresponding to a product of independent square random
matrices, . New formulae for the level densities are derived
for and . Moreover, the level density corresponding to the
generalized Bures distribution, given by the free convolution of arcsine and MP
distributions is obtained. We also explain the reason of such a curious
convolution. The technique proposed here allows for the derivation of the level
densities for several other cases.Comment: 10 latex pages including 4 figures, Ver 4, minor improvements and
references updat
Challenges of beta-deformation
A brief review of problems, arising in the study of the beta-deformation,
also known as "refinement", which appears as a central difficult element in a
number of related modern subjects: beta \neq 1 is responsible for deviation
from free fermions in 2d conformal theories, from symmetric omega-backgrounds
with epsilon_2 = - epsilon_1 in instanton sums in 4d SYM theories, from
eigenvalue matrix models to beta-ensembles, from HOMFLY to super-polynomials in
Chern-Simons theory, from quantum groups to elliptic and hyperbolic algebras
etc. The main attention is paid to the context of AGT relation and its possible
generalizations.Comment: 20 page
Singularity analysis, Hadamard products, and tree recurrences
We present a toolbox for extracting asymptotic information on the
coefficients of combinatorial generating functions. This toolbox notably
includes a treatment of the effect of Hadamard products on singularities in the
context of the complex Tauberian technique known as singularity analysis. As a
consequence, it becomes possible to unify the analysis of a number of
divide-and-conquer algorithms, or equivalently random tree models, including
several classical methods for sorting, searching, and dynamically managing
equivalence relationsComment: 47 pages. Submitted for publicatio
Random graph states, maximal flow and Fuss-Catalan distributions
For any graph consisting of vertices and edges we construct an
ensemble of random pure quantum states which describe a system composed of
subsystems. Each edge of the graph represents a bi-partite, maximally entangled
state. Each vertex represents a random unitary matrix generated according to
the Haar measure, which describes the coupling between subsystems. Dividing all
subsystems into two parts, one may study entanglement with respect to this
partition. A general technique to derive an expression for the average
entanglement entropy of random pure states associated to a given graph is
presented. Our technique relies on Weingarten calculus and flow problems. We
analyze statistical properties of spectra of such random density matrices and
show for which cases they are described by the free Poissonian
(Marchenko-Pastur) distribution. We derive a discrete family of generalized,
Fuss-Catalan distributions and explicitly construct graphs which lead to
ensembles of random states characterized by these novel distributions of
eigenvalues.Comment: 37 pages, 24 figure
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