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Uspekhi Mat. Nauk, 2006, Volume 61, Issue 3(369), Pages 93–156 (Mi umn1754)  

This article is cited in 3 scientific papers (total in 3 papers)

Solving matrix models in $1/N$-expansion

L. O. Chekhovab

a Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We describe properties of most general multisupport solutions to one-matrix models. We begin with the one-matrix model in the presence of hard walls, i.e., in the case where the eigenvalue support is confined to several fixed intervals of the real axis. We then consider the eigenvalue model, which generalizes the one-matrix model to the Dyson gas case. We show that in all these cases, the structure of the solution at the leading order is described by semiclassical, or generalized Whitham–Krichever hierarchies. Derivatives of tau-functions for these solutions are associated with families of Riemann surfaces (spectral curves with possible double points) and satisfy the Witten–Dijkgraaf–Verlinde–Verlinde equations. We develop the diagrammatic technique for finding correlation functions and free energy of these models in all orders of the 't Hooft expansion in the reciprocal matrix size. In all cases, these quantities can be formulated in terms of strucutures associated with the spectral curves.

DOI: https://doi.org/10.4213/rm1754

Full text: PDF file (1086 kB)
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English version:
Russian Mathematical Surveys, 2006, 61:3, 483–543

Bibliographic databases:

Document Type: Article
UDC: 514.753.2
MSC: Primary 81T10; Secondary 81T18, 81T40, 32G15, 37K10, 14H15, 41A60
Received: 28.05.2006

Citation: L. O. Chekhov, “Solving matrix models in $1/N$-expansion”, Uspekhi Mat. Nauk, 61:3(369) (2006), 93–156; Russian Math. Surveys, 61:3 (2006), 483–543

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. O. Shishanin, “Phases of the Goldstone multitrace matrix model in the large-$N$ limit”, Theoret. and Math. Phys., 152:3 (2007), 1258–1265  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. Matveev V.B., “30 years of finite-gap integration theory”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366:1867 (2008), 837–875  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. O. Marchal, “One-cut solution of the $\beta$ ensembles in the Zhukovsky variable”, J. Stat. Mech, 2012 (2012), P01011  crossref  isi  elib  scopus
  • Успехи математических наук Russian Mathematical Surveys
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