Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques and applications to graphical enumeration

By Nicholas M. Ercolani, K. Mclaughlin. 2002

209 citations recorded by the source corpus.

We study the partition function from random matrix theory using a well known connection to orthogonal polynomials, and a recently developed Riemann-Hilbert approach to the computation of detailed asymptotics for these orthogonal polynomials. We obtain the first proof of a complete large N expansion for the partition function, for a general class of probability measures on matrices, originally conjectured by Bessis, Itzykson, and Zuber. We prove that the coefficients in the asymptotic expansion are analytic functions of parameters in the original probability measure, and that they are generating functions for the enumeration of labelled maps according to genus and valence. Central to the analysis is a large N expansion for the mean density of eigenvalues, uniformly valid on the entire real axis.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/0696fa35b2ba38f5f8ee75f7c059f36f2b0fc0d1

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In Discrete Mathematics and Combinatorics

1. Arno B. J. Kuijlaars
2. Alice Guionnet
3. Édouard Maurel-Segala
4. Benoît Collins
5. Walter Van Assche
6. K. Mclaughlin
7. Gaëtan Borot
8. J. Gustavsson

In Mathematical Physics

1. Édouard Maurel-Segala
2. Alice Guionnet
3. Benoît Collins
4. Gaëtan Borot
5. J. Gustavsson
6. Arno B. J. Kuijlaars
7. Walter Van Assche
8. Maarten Vanlessen

In Statistics and Probability

1. Édouard Maurel-Segala
2. Alice Guionnet
3. Arno B. J. Kuijlaars
4. Benoît Collins
5. Gaëtan Borot
6. J. Gustavsson
7. Walter Van Assche
8. Maarten Vanlessen

Citations in the corpus

• Second order asymptotics for matrix models (2006, methodology): “In [10], the idea was to develop a Riemann–Hilbert approach based on precise asymptotics of orthogonal polynomials.”
• Gaussian fluctuations of eigenvalues in the GUE (2004, background): “2) in [6] applied to the hermitian case says that”
• Asymptotic Expansion of β Matrix Models in the One-cut Regime (2011, methodology): “FIK92], for which the large N asymptotics have been intensively studied [BI99,DKM +97,DKM 99b,DKM+99a] with the steepest descent method introduced in [DZ95]. As a consequence, Ercolani and McLaughlin [EM03] were able to prove the existence of a 1=N2 expansion of lnZV N;=2 . However, the topological ”
• The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [-1,1] (2001, methodology): “er with an application of the steepest descent method of Deift and Zhou, introduced in [15] and further developed in [4, 14, 16] and the papers cited before. See [10, 13, 25] for an introduction, and [5, 9, 17, 18, 19, 23, 26, 27, 28, 29, 30, 37, 41, 45, 46] for the latest developments. A different m”
• The rate of convergence for spectra of GUE and LUE matrix ensembles (2005, unknown): “[9] N.M. Ercolani and K.D.T.-R. McLaughlin: “Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques, and applications to grafical enumeration”, Int.”
• First Order Asymptotics of Matrix Integrals; A Rigorous Approach Towards the Understanding of Matrix Models (2002, background): “where CP (g) enumerates some maps with genus g. Such an expansion was proved to hold in the one matrix case by K. McLaughlin and N. Ercolani [ 12 ].”
• Asymptotics of unitary and orthogonal matrix integrals (2006, background): “Such ‘topological’ expansions were also shown to hold in the large N limit, and then to match with the formal expansion on a mathematical level of rigor by two authors [16,17,23] and previously in the one matrix case in [1,2] and [12].”