2013/11/13 by Olivier Marchal, Marchal, Olivier, Bertrand Eynard +4
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #hep-th #math-ph #math.MP #math.PR #nlin.SI
paper · pdf · doi:10.48550/arxiv.1311.3217
37 pages, 1 figure, published in Random Matrices: Theory and Applications
arxiv created 2014/08/19 · arxiv updated 2014/08/20
The goal of this article is to rederive the connection between the Painlevé 5 integrable system and the universal eigenvalues correlation functions of double-scaled hermitian matrix models, through the topological recursion method. More specifically we prove, to all orders, that the WKB asymptotic expansions of the τ-function as well as of determinantal formulas arising from the Painlevé 5 Lax pair are identical to the large N double scaling asymptotic expansions of the partition function and correlation functions of any hermitian matrix model around a regular point in the bulk. In other words, we rederive the "sine-law" universal bulk asymptotic of large random matrices and provide an alternative perturbative proof of universality in the bulk with only algebraic methods. Eventually we exhibit the first orders of the series expansion up to O(N-5).