2013

Asymptotic expansion of beta matrix models in the multi-cut regime

Borot, Gaëtan, Guionnet, Alice

Understand

We establish the asymptotic expansion in $\beta$ matrix models with a confining, off-critical potential, in the regime where the support of the equilibrium measure is a union of segments.

We first address the case where the filling fractions of these segments are fixed, and show the existence of a $1/N$ expansion.
We then study the asymptotics of the sum over the filling fractions, to obtain the full asymptotic expansion for the initial problem in the multi-cut regime.
In particular, we identify the fluctuations of the linear statistics and show that they are approximated in law by the sum of a Gaussian random variable and an independent Gaussian discrete random variable with oscillating center.

Built on

F. Nevanlinna and R. Nevanlinna, Über die Eigenschaften einer analytischen Funktion in der Umgebung einer singulären Stelle oder Linie , Acta Soc. Sci. Fennica 5
1922
Earlier work this paper cites.
S. Bergman and M. Schiffer, Kernel functions and elliptic differential equations in mathematical physics , Academic Press Inc., New York, NY (1953)
1953
Earlier work this paper cites.
J. Fay, Theta functions on Riemann surfaces , Lecture Notes in Mathematics, vol. 352, Springer, Berlin, 1970
1970
Earlier work this paper cites.
R. Nevanlinna, Analytic functions , Grundlehren der mathematischen Wissenschaften 162
1970
Earlier work this paper cites.
D. Mumford, Tata lectures on Theta , Modern Birkhäuser Classics, Birkhäuser, Boston, 1984, Volume I (no. 28), II (no. 43), III (no. 97)
1984
Earlier work this paper cites.
J.L. Cardy and I. Peschel, Finite-size dependence of the free energy in two-dimensional critical systems , Nucl. Phys. B 300
1988
Earlier work this paper cites.
J. Jurkiewicz, Chaotic behaviour in one-matrix model , Phys. Lett. B 261
1991
Earlier work this paper cites.
I. Krichever, The Tau function of the universal Whitham hierarchy, matrix models and topological field theories , Commun. Pure Appl. Math. 47
1994
Earlier work this paper cites.
A. Boutet de Monvel, L. Pastur, and M. Shcherbina, On the statistical mechanics approach in the random matrix theory. Integrated density of states , J. Stat. Phys. 79
1995
Earlier work this paper cites.
P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the mKdV equation , Ann. Math. 137
1995
Earlier work this paper cites.
G. Akemann, Higher genus correlators for the hermitian matrix model with multiple cuts , Nucl. Phys. B 482
1996
Earlier work this paper cites.
G. Ben Arous and A. Guionnet, Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy , Probab. Th. Rel. Fields 108
1997
Earlier work this paper cites.
P. Deift, T. Kriecherbauer, K.T.-R. McLaughlin, S. Venakides, and X. Zhou, Asymptotics for polynomials orthogonal with respect to varying exponential weights , Int. Math. Res. Not. 16
1997
Earlier work this paper cites.
K. Johansson, On fluctuations of eigenvalues of random Hermitian matrices , Duke Math. J. 91
1998
Earlier work this paper cites.
M. Adler and E. Horozov and P. van Moerbeke, The Pfaff lattice and skew-orthogonal polynomials , Int. Math. Res. Not. 11
1999
Earlier work this paper cites.
P. Deift, Orthogonal polynomials and random matrices: a Riemann–Hilbert approach , Courant Lecture Notes in Mathematics, vol. 3, New York University Courant Institute of Mathematical Sciences, New York, 1999
1999
Earlier work this paper cites.
B. Dubrovin, Differential geometry of moduli spaces and its applications to soliton equations and to topological conformal field theory , Surveys in Differential Geometry, Vol. IV, 213–238 (1999); Preprint n.117, Scuola Normale Superiore, Pisa
1999
Earlier work this paper cites.
G. Bonnet, F. David, and B. Eynard, Breakdown of universality in multi-cut matrix models , J. Phys. A 33
2000
Earlier work this paper cites.
S. Albeverio, L. Pastur, and M. Shcherbina, On the 1 / N 1/N expansion for some unitary invariant ensembles of random matrices , Commun. Math. Phys. 224
2001
Earlier work this paper cites.

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2010
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G. Borot, Quelques problèmes de géométrie énumérative, de matrices aléatoires, d’intégrabilité, étudiés via la géometrie des surfaces de Riemann , 2011, Thèse de Doctorat, Université d’Orsay, math-ph/1110.1493
Original
2011
Later among the works it cites.
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2011
Later among the works it cites.
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Original
2011
Later among the works it cites.
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2012
Later among the works it cites.
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Original
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Later among the works it cites.
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Original
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