110 research outputs found
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
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
Mixed correlation function and spectral curve for the 2-matrix model
We compute the mixed correlation function in a way which involves only the
orthogonal polynomials with degrees close to , (in some sense like the
Christoffel Darboux theorem for non-mixed correlation functions). We also
derive new representations for the differential systems satisfied by the
biorthogonal polynomials, and we find new formulae for the spectral curve. In
particular we prove the conjecture of M. Bertola, claiming that the spectral
curve is the same curve which appears in the loop equations.Comment: latex, 1 figure, 55 page
Double scaling limits of random matrices and minimal (2m,1) models: the merging of two cuts in a degenerate case
In this article, we show that the double scaling limit correlation functions
of a random matrix model when two cuts merge with degeneracy (i.e. when
for arbitrary values of the integer ) are the same as the
determinantal formulae defined by conformal models. Our approach
follows the one developed by Berg\`{e}re and Eynard in \cite{BergereEynard} and
uses a Lax pair representation of the conformal models (giving
Painlev\'e II integrable hierarchy) as suggested by Bleher and Eynard in
\cite{BleherEynard}. In particular we define Baker-Akhiezer functions
associated to the Lax pair to construct a kernel which is then used to compute
determinantal formulae giving the correlation functions of the double scaling
limit of a matrix model near the merging of two cuts.Comment: 37 pages, 4 figures. Presentation improved, typos corrected.
Published in Journal Of Statistical Mechanic
Fractional statistic
We improve Haldane's formula which gives the number of configurations for
particles on states in a fractional statistic defined by the coupling
. Although nothing is changed in the thermodynamic limit, the new
formula makes sense for finite with integer and A
geometrical interpretation of fractional statistic is given in terms of
''composite particles''.Comment: flatex hald.tex, 3 files Submitted to: Phys. Rev.
Four Loop Massless Propagators: a Numerical Evaluation of All Master Integrals
We present numerical results which are needed to evaluate all non-trivial
master integrals for four-loop massless propagators, confirming the recent
analytic results of[1]and evaluating an extra order in \ep expansion for each
master integral
More on ambiguities in the pole mass
The relation between the pole quark mass and the -renormalized
mass is governed by an infrared renormalon singularity, which leads to an
ambiguity of order in the definition of the pole mass. We use
the renormalization group and heavy quark effective theory to determine the
exact nature of this singularity up to an overall normalization. In the abelian
gauge theory, the normalization is computed partially to next-to-leading order
in the flavour expansion.Comment: LaTeX, 10 pages, Figures appended as uu-encoded file, UM-TH-94-3
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