1,011 research outputs found
Towards a full solution of the large N double-scaled SYK model
We compute the exact, all energy scale, 4-point function of the large
double-scaled SYK model, by using only combinatorial tools and relating the
correlation functions to sums over chord diagrams. We apply the result to
obtain corrections to the maximal Lyapunov exponent at low temperatures. We
present the rules for the non-perturbative diagrammatic description of
correlation functions of the entire model. The latter indicate that the model
can be solved by a reduction of a quantum deformation of SL, that
generalizes the Schwarzian to the complete range of energies.Comment: 52+28 pages, 14 figures; v2: references revised, typos corrected,
changed normalization of SL(2)_q 6j symbo
Difference Sturm--Liouville problems in the imaginary direction
We consider difference operators in on of the form where is the imaginary unit. The
domain of definiteness are functions holomorphic in a strip with some
conditions of decreasing at infinity. Problems of such type with discrete
spectra are well known (Meixner--Pollaszek, continuous Hahn, continuous dual
Hahn, and Wilson hypergeometric orthogonal polynomials).
We write explicit spectral decompositions for several operators with
continuous spectra. We also discuss analogs of 'boundary conditions' for such
operators.Comment: 27p
Symmetry Groups of Hypergeometric Series
Structures of symmetries of transformations for Holman-Biedenharn-Louck
hypergeometric series: terminating balanced series and
elliptic series are discussed. Namely the description of the
invariance groups and the classification all of possible transformations for
each types of hypergeometric series are given. Among them, a "periodic"
affine Coxeter group which seems to be new in the literature arises as an
invariance group for a class of series
Jacobian elliptic Kummer surfaces and special function identities
We derive formulas for the construction of all inequivalent Jacobian elliptic
fibrations on the Kummer surface of two non-isogeneous elliptic curves from
extremal rational elliptic surfaces by rational base transformations and
quadratic twists. We then show that each such decomposition yields a
description of the Picard-Fuchs system satisfied by the periods of the
holomorphic two-form as either a tensor product of two Gauss' hypergeometric
differential equations, an Appell hypergeometric system, or a GKZ differential
system. As the answer must be independent of the fibration used, identities
relating differential systems are obtained. They include a new identity
relating Appell's hypergeometric system to a product of two Gauss'
hypergeometric differential equations by a cubic transformation.Comment: 20 page
Dispersion and limit theorems for random walks associated with hypergeometric functions of type BC
The spherical functions of the noncompact Grassmann manifolds
over the (skew-)fields with rank and dimension parameter can be described
as Heckman-Opdam hypergeometric functions of type BC, where the double coset
space is identified with the Weyl chamber of
type B. The corresponding product formulas and Harish-Chandra integral
representations were recently written down by M. R\"osler and the author in an
explicit way such that both formulas can be extended analytically to all real
parameters , and that associated commutative convolution
structures on exist. In this paper we introduce moment functions
and the dispersion of probability measures on depending on and
study these functions with the aid of this generalized integral representation.
Moreover, we derive strong laws of large numbers and central limit theorems for
associated time-homogeneous random walks on where the moment
functions and the dispersion appear in order to determine drift vectors and
covariance matrices of these limit laws explicitely. For integers , all
results have interpretations for -invariant random walks on the
Grassmannians .
Besides the BC-cases we also study the spaces , which are related to Weyl chambers of type A, and for which corresponding
results hold. For the rank-one-case , the results of this paper are
well-known in the context of Jacobi-type hypergroups on .Comment: Extended version of arXiv:1205.4866; some corrections to prior
version. Accepted for publication in J. Theor. Proba
On the distribution of exponential functionals for Levy processes with jumps of rational transform
We derive explicit formulas for the Mellin transform and the distribution of
the exponential functional for Levy processes with rational Laplace exponent.
This extends recent results by Cai and Kou on the processes with
hyper-exponential jumps [N. Cai and S. Kou "Prising Asian options under a
general jump diffusion model", (2010)].Comment: 11 page
Distinct parts partitions without sequences
Partitions without sequences of consecutive integers as parts have been
studied recently by many authors, including Andrews, Holroyd, Liggett, and
Romik, among others. Their results include a description of combinatorial
properties, hypergeometric representations for the generating functions, and
asymptotic formulas for the enumeration functions. We complete a similar
investigation of partitions into distinct parts without sequences, which are of
particular interest due to their relationship with the Rogers-Ramanujan
identities. Our main results include a double series representation for the
generating function, an asymptotic formula for the enumeration function, and
several combinatorial inequalities.Comment: 15 page
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