2,603 research outputs found
Matrix integrals as Borel sums of Schur function expansions
The partition function for unitary two matrix models is known to be a double
KP tau-function, as well as providing solutions to the two dimensional Toda
hierarchy. It is shown how it may also be viewed as a Borel sum regularization
of divergent sums over products of Schur functions in the two sequences of
associated KP flow variables.Comment: PlainTex file. 8 pgs. Based on talk by J. Harnad at the workshop:
Symmetry and Perturbation Theory 2002, Cala Gonoone (Sardinia), May 1-26,
2002. To appear in proceedings. (World Scientific, Singapore, eds. S. Abenda,
G. Gaeta). Typographical correction made to formula (2.7) to include
previously omitted powers of r and
Scalar products of symmetric functions and matrix integrals
We present relations between Hirota-type bilinear operators, scalar products
on spaces of symmetric functions and integrals defining matrix model partition
functions. Using the fermionic Fock space representation, a proof of the
expansion of an associated class of KP and 2-Toda tau functions in
a series of Schur functions generalizing the hypergeometric series is given and
related to the scalar product formulae. It is shown how special cases of such
-functions may be identified as formal series expansions of partition
functions. A closed form exapnsion of in terms of Schur
functions is derived.Comment: LaTex file. 15 pgs. Based on talks by J. Harnad and A. Yu. Orlov at
the workshop: Nonlinear evolution equations and dynamical systems 2002, Cadiz
(Spain) June 9-16, 2002. To appear in proceedings. (Minor typographical
corrections added, abstract expanded
Fermionic construction of partition functions for two-matrix models and perturbative Schur function expansions
A new representation of the 2N fold integrals appearing in various two-matrix
models that admit reductions to integrals over their eigenvalues is given in
terms of vacuum state expectation values of operator products formed from
two-component free fermions. This is used to derive the perturbation series for
these integrals under deformations induced by exponential weight factors in the
measure, expressed as double and quadruple Schur function expansions,
generalizing results obtained earlier for certain two-matrix models. Links with
the coupled two-component KP hierarchy and the two-component Toda lattice
hierarchy are also derived.Comment: Submitted to: "Random Matrices, Random Processes and Integrable
Systems", Special Issue of J. Phys. A, based on the Centre de recherches
mathematiques short program, Montreal, June 20-July 8, 200
Fermionic construction of partition function for multi-matrix models and multi-component TL hierarchy
We use -component fermions to present -fold
integrals as a fermionic expectation value. This yields fermionic
representation for various -matrix models. Links with the -component
KP hierarchy and also with the -component TL hierarchy are discussed. We
show that the set of all (but two) flows of -component TL changes standard
matrix models to new ones.Comment: 16 pages, submitted to a special issue of Theoretical and
Mathematical Physic
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