## Asymptotics of a Fredholm determinant corresponding to the first bulk critical universality class in random matrix models.(English)Zbl 1290.15004

Summary: We study the determinant $$\det(I-K_{\text{PII}})$$ of an integrable Fredholm operator $$K_{\text{PII}}$$ acting on the interval $$(-s, s)$$ whose kernel is constructed out of the $${\Psi}$$-function associated with the Hastings-McLeod solution of the second Painlevé equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the unitary ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann-Hilbert method, we evaluate the large $$s$$-asymptotics of $$\det(I-K_{\text{PII}})$$.

### MSC:

 15A15 Determinants, permanents, traces, other special matrix functions 15A18 Eigenvalues, singular values, and eigenvectors 15B52 Random matrices (algebraic aspects) 15B57 Hermitian, skew-Hermitian, and related matrices 47A53 (Semi-) Fredholm operators; index theories 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
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