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From Brezin-Hikami to Harer-Zagier formulas for Gaussian correlators · Around

2010

From Brezin-Hikami to Harer-Zagier formulas for Gaussian correlators

Morozov, A., Shakirov, Sh.

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Brezin-Hikami contour-integral representation of exponential multidensities in finite N Hermitian matrix model is a remarkable implication of the old Hermitian-Kontsevich duality.

It is also a simplified version of Okounkov's formulas for the same multidensities in the cubic Kontsevich model and of Nekrasov calculus for LMNS integrals, a central piece of the modern studies of AGT relations.
In this paper we use Brezin-Hikami representation to derive explicit expressions for the Harer-Zagier multidensities (from arXiv:0906.0036): the only known exhaustive generating functions of all-genera Gaussian correlators which are fully calculable and expressed in terms of elementary functions.
Using the Brezin-Hikami contour integrals, we rederive the 1-point function of Harer and Zagier and the 2-point arctangent function of arXiv:0906.0036.

Built on

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M. Petkovsek, H. Wilf and D. Zeilberger, ”A = B” , 1996
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A.Morozov, Phys.Usp.(UFN) 35
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N.Nekrasov, Adv.Theor.Math.Phys. 7
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A.Alexandrov, A.Mironov and A.Morozov, Int.J.Mod.Phys. A19
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E.Brezin and S.Hikami, JHEP 0710(2007)096, arXiv:0709.3378; Commun.Math.Phys.283(2008)507-521, arXiv:0708.2210; arXiv:cond-mat/9804024
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2008
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Sh.Shakirov and A.Morozov, JHEP 0912(2009)003, arXiv:0906.0036
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2009
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A.Mironov, A.Morozov and And.Morozov, arXiv:1003.5752
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Cited in the paper.

Then

L.Chekhov and Yu.Makeenko, Phys.Lett. B278 (1992) 271-278, arXiv:hep-th/9202006 S. Kharchev and A. Marshakov, hep-th/9210072; S. Kharchev and A. Marshakov, Int. J. Mod. Phys. A 10
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2009
Later among the works it cites.
D.Gaiotto, arXiv:0908.0307 L.Alday, D.Gaiotto and Y.Tachikawa, Lett.Math.Phys. 91
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2009
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Vl.Dotsenko and V.Fateev, Nucl.Phys. B240
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2009
Later among the works it cites.
A.Okounkov, arXiv:math/0101201 A.Alexandrov, A.Mironov, A.Morozov and P.Putrov, Int.J.Mod.Phys. A24
Original
2009
Later among the works it cites.

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