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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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