Asymptotics of the partition function of a random matrix model

Pavel M. Bleher[1]; Alexander Its

  • [1] Indiana University-Purdue University Indianapolis, department of mathematical sciences, 402 N. Blackford Street, Indianapolis IN 46202 (USA)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 6, page 1943-2000
  • ISSN: 0373-0956

Abstract

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We prove a number of results concerning the large N asymptotics of the free energy of a random matrix model with a polynomial potential. Our approach is based on a deformation of potential and on the use of the underlying integrable structures of the matrix model. The main results include the existence of a full asymptotic expansion in even powers of N of the recurrence coefficients of the related orthogonal polynomials for a one-cut regular potential and the double scaling asymptotics of the free energy for a singular quartic potential. We also prove the analyticity of the coefficients of the asymptotic expansions of the recurrence coefficients and the free energy, with respect to the coefficients of the potential, and the one-sided analyticity of the recurrent coefficients and the free energy for a one-cut singular potential.

How to cite

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M. Bleher, Pavel, and Its, Alexander. "Asymptotics of the partition function of a random matrix model." Annales de l’institut Fourier 55.6 (2005): 1943-2000. <http://eudml.org/doc/116239>.

@article{M2005,
abstract = {We prove a number of results concerning the large $N$ asymptotics of the free energy of a random matrix model with a polynomial potential. Our approach is based on a deformation of potential and on the use of the underlying integrable structures of the matrix model. The main results include the existence of a full asymptotic expansion in even powers of $N$ of the recurrence coefficients of the related orthogonal polynomials for a one-cut regular potential and the double scaling asymptotics of the free energy for a singular quartic potential. We also prove the analyticity of the coefficients of the asymptotic expansions of the recurrence coefficients and the free energy, with respect to the coefficients of the potential, and the one-sided analyticity of the recurrent coefficients and the free energy for a one-cut singular potential.},
affiliation = {Indiana University-Purdue University Indianapolis, department of mathematical sciences, 402 N. Blackford Street, Indianapolis IN 46202 (USA)},
author = {M. Bleher, Pavel, Its, Alexander},
journal = {Annales de l’institut Fourier},
keywords = {Matrix Models; orthogonal polynomials; partition function},
language = {eng},
number = {6},
pages = {1943-2000},
publisher = {Association des Annales de l'Institut Fourier},
title = {Asymptotics of the partition function of a random matrix model},
url = {http://eudml.org/doc/116239},
volume = {55},
year = {2005},
}

TY - JOUR
AU - M. Bleher, Pavel
AU - Its, Alexander
TI - Asymptotics of the partition function of a random matrix model
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 6
SP - 1943
EP - 2000
AB - We prove a number of results concerning the large $N$ asymptotics of the free energy of a random matrix model with a polynomial potential. Our approach is based on a deformation of potential and on the use of the underlying integrable structures of the matrix model. The main results include the existence of a full asymptotic expansion in even powers of $N$ of the recurrence coefficients of the related orthogonal polynomials for a one-cut regular potential and the double scaling asymptotics of the free energy for a singular quartic potential. We also prove the analyticity of the coefficients of the asymptotic expansions of the recurrence coefficients and the free energy, with respect to the coefficients of the potential, and the one-sided analyticity of the recurrent coefficients and the free energy for a one-cut singular potential.
LA - eng
KW - Matrix Models; orthogonal polynomials; partition function
UR - http://eudml.org/doc/116239
ER -

References

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