×
Marchal, O.; Eynard, B.; Bergère, M.

The sine-law gap probability, Painlevé 5, and asymptotic expansion by the topological recursion. (English) Zbl 1314.15027

Random Matrices Theory Appl. 3, No. 3, Article ID 1450013, 41 p. (2014).
The authors investigate the connection between the Painlevé 5 integrable system and the universal eigenvalues correlation functions of double-scaled Hermitian matrix models. The results are obtained by applying the topological recursion method. It is shown that the WKB asymptotic expansions of the tau-function as well as of determinantal formulas arising from the Painlevé 5 Lax pair are identical to the large double scaling asymptotic expansions of the partition function and correlation functions of any Hermitian matrix model around a regular point in the bulk. As a result, the “sine-law” universal bulk asymptotic of large random matrices is given and an alternative perturbative proof of universality in the bulk with algebraic methods is provided.
Reviewer: Taras Bodnar (Berlin)

Cited in 7 Documents

MSC:

15B52 Random matrices (algebraic aspects)
60B20 Random matrices (probabilistic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
14H70 Relationships between algebraic curves and integrable systems

Keywords:

random matrix theory; Painlevé 5 integrable system; double-scaled Hermitian matrix model; eigenvalue; topological recursion method
Cite Review PDF
Full Text: DOI arXiv

References:

[1] DOI: 10.1007/s002200100531 · Zbl 1038.82039 · doi:10.1007/s002200100531
[2] DOI: 10.1016/0550-3213(93)90476-6 · Zbl 1043.81636 · doi:10.1016/0550-3213(93)90476-6
[3] DOI: 10.1088/0305-4470/36/12/314 · Zbl 1053.15017 · doi:10.1088/0305-4470/36/12/314
[4] DOI: 10.1002/cpa.10065 · Zbl 1032.82014 · doi:10.1002/cpa.10065
[5] Borot G., J. Stat. Mech. 2011 pp 56– (2011)
[6] DOI: 10.1007/s00220-012-1619-4 · Zbl 1344.60012 · doi:10.1007/s00220-012-1619-4
[7] DOI: 10.1007/978-1-4612-1532-5_6 · doi:10.1007/978-1-4612-1532-5_6
[8] DOI: 10.1088/1742-5468/2005/10/P10006 · doi:10.1088/1742-5468/2005/10/P10006
[9] Eynard B., J. High Energy Phys. 2009 pp 03003– (2009)
[10] DOI: 10.4310/CNTP.2007.v1.n2.a4 · Zbl 1161.14026 · doi:10.4310/CNTP.2007.v1.n2.a4
[11] DOI: 10.1088/1126-6708/2007/06/058 · doi:10.1088/1126-6708/2007/06/058
[12] DOI: 10.1214/009117907000000141 · Zbl 1129.15020 · doi:10.1214/009117907000000141
[13] DOI: 10.1016/0167-2789(81)90013-0 · Zbl 1194.34167 · doi:10.1016/0167-2789(81)90013-0
[14] DOI: 10.1016/0167-2789(81)90021-X · Zbl 1194.34166 · doi:10.1016/0167-2789(81)90021-X
[15] DOI: 10.1063/1.2794560 · Zbl 1152.81499 · doi:10.1063/1.2794560
[16] DOI: 10.1088/1742-5468/2011/04/P04013 · doi:10.1088/1742-5468/2011/04/P04013
[17] DOI: 10.1088/1742-5468/2012/01/P01011 · doi:10.1088/1742-5468/2012/01/P01011
[18] Mehta M. L., Pure and Applied Mathematics 142, in: Random Matrices (2004)
[19] DOI: 10.1007/BFb0021444 · doi:10.1007/BFb0021444
[20] C. Tracy and H. Widom, Statistical Physics on the Eve of the 21st Century: In Honour of J. B. McGuire on the Occasion of his 65th Birthday (World Scientific, 1999) pp. 230–239.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.