523 research outputs found
Determinantal formulae and loop equations
We prove that the correlations functions, generated by the determinantal
process of the Christoffel-Darboux kernel of an arbitrary order 2 ODE, do
satisfy loop equations.Comment: Latex, 35 pages few misprints correcte
Correlation Functions of Complex Matrix Models
For a restricted class of potentials (harmonic+Gaussian potentials), we
express the resolvent integral for the correlation functions of simple traces
of powers of complex matrices of size , in term of a determinant; this
determinant is function of four kernels constructed from the orthogonal
polynomials corresponding to the potential and from their Cauchy transform. The
correlation functions are a sum of expressions attached to a set of fully
packed oriented loops configurations; for rotational invariant systems,
explicit expressions can be written for each configuration and more
specifically for the Gaussian potential, we obtain the large expansion ('t
Hooft expansion) and the so-called BMN limit.Comment: latex BMN.tex, 7 files, 6 figures, 30 pages (v2 for spelling mistake
and added reference) [http://www-spht.cea.fr/articles/T05/174
Operator Product Expansion on a Fractal: The Short Chain Expansion for Polymer Networks
We prove to all orders of renormalized perturbative polymer field theory the
existence of a short chain expansion applying to polymer solutions of long and
short chains. For a general polymer network with long and short chains we show
factorization of its partition sum by a short chain factor and a long chain
factor in the short chain limit. This corresponds to an expansion for short
distance along the fractal perimeter of the polymer chains connecting the
vertices and is related to a large mass expansion of field theory.
The scaling of the second virial coefficient for bimodal solutions is
explained. Our method also applies to the correlations of the multifractal
measure of harmonic diffusion onto an absorbing polymer. We give a result for
expanding these correlations for short distance along the fractal carrier of
the measure.Comment: 28 pages, revtex, 4 Postscript figures, 3 latex emlines pictures.
Replacement eliminates conflict with a blob resul
The sine-law gap probability, Painlev\'e 5, and asymptotic expansion by the topological recursion
The goal of this article is to rederive the connection between the Painlev\'e
integrable system and the universal eigenvalues correlation functions of
double-scaled hermitian matrix models, through the topological recursion
method. More specifically we prove, \textbf{to all orders}, that the WKB
asymptotic expansions of the -function as well as of determinantal
formulas arising from the Painlev\'e Lax pair are identical to the large
double scaling asymptotic expansions of the partition function and
correlation functions of any hermitian matrix model around a regular point in
the bulk. In other words, we rederive the "sine-law" universal bulk asymptotic
of large random matrices and provide an alternative perturbative proof of
universality in the bulk with only algebraic methods. Eventually we exhibit the
first orders of the series expansion up to .Comment: 37 pages, 1 figure, published in Random Matrices: Theory and
Application
Ratios of characteristic polynomials in complex matrix models
We compute correlation functions of inverse powers and ratios of characteristic polynomials for random matrix models with complex eigenvalues. Compact expressions are given in terms of orthogonal polynomials in the complex plane as well as their Cauchy transforms, generalizing previous expressions for real eigenvalues. We restrict ourselves to ratios of characteristic polynomials over their complex conjugate
Non-Commutative Complete Mellin Representation for Feynman Amplitudes
We extend the complete Mellin (CM) representation of Feynman amplitudes to
the non-commutative quantum field theories. This representation is a versatile
tool. It provides a quick proof of meromorphy of Feynman amplitudes in
parameters such as the dimension of space-time. In particular it paves the road
for the dimensional renormalization of these theories. This complete Mellin
representation also allows the study of asymptotic behavior under rescaling of
arbitrary subsets of external invariants of any Feynman amplitude.Comment: 14 pages, no figur
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