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