By Vladimir Kazakov (auth.), Vadim Malyshev, Anatoly Vershik (eds.)

ISBN-10: 1402007930

ISBN-13: 9781402007934

ISBN-10: 9401005753

ISBN-13: 9789401005753

New and extraordinary effects bought lately from a thorough learn of asymptotic combinatorics have ended in a brand new, larger point of knowing of comparable difficulties: the idea of integrable platforms, the Riemann-Hilbert challenge, asymptotic illustration conception, spectra of random matrices, combinatorics of younger diagrams and diversifications, or even a few facets of quantum box theory.

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INTRODUCTION TO MATRIX MODELS E. fr) Laboratoire de Physique Theorique Ecole Normale Superieure 24 rue Lhomond 15231 Paris Cedex 05 Prance* 1. ples of physical problellls involving randolll Illatrices Nuclear levels. In 1951 E. Wigner [1] proposed, as a first approach to the understanding of the structure of the eigenstates of complex nuclei, to sustitute to the Schrodinger equation, in which the forces between the nucleons are not very well-known, and whose solution requires drastic simplifying assumptions, a simple random Hamiltonian drawn from a Gaussian ensemble.

P('\') . \). It is immediate to verify that the solution is fez) = ~V'(z) - P(z)Vz 2 - a2 (36) in which P is a polynomial. (We limit here ourselves to even potentials: the cut is the symmetric interval (-a, +a). For some potentials the cut will break up into several segments. We do not discuss these cases, but the method is staightforward; they can lead to different string equations but the matter content of the corresponding "gravity" is unclear). 33 The polynomial P and the parameter a are uniquely determined by demanding that J(z) falls-off for large z as liz.

And Zamolodchikov, A. B. (1988) Mod. Phys. Lett. A, 3, 819. David, F. (1988) Mod. Phys. Lett. A, 3, 1651. Distler, J. and Kawai, H. (1989) Nucl. Phys. B, 231, 509. Polyakov, A. (1981) Phys. Lett. B, 163, 207. , Parisi, G. -B. (1978) Commun. Math. , 69, 35. Brezin, E. and Kazakov, V. (1990) Phys. Lett. B, 236, 144. Douglas, M. and Shenker, S. (1990) Nucl. Phys. B, 335, 635. Gross, D. J. and Migdal, A. A. (1990) Phys. Rev. , 64,127. Douglas, M. (1990) Phys. Lett. B, 238, 213. Gel'fand, L. M. and Dikii, L.

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