Its, Alexander R.Bothner, Thomas JoachimBleher, Pavel, 1947-Tarasov, VitalyEremenko, AlexandreMukhin, Evgeny2013-11-062013-11-062013-11-06https://hdl.handle.net/1805/3655http://dx.doi.org/10.7912/C2/2394Indiana University-Purdue University Indianapolis (IUPUI)We study the one-parameter family of determinants $det(I-\gamma K_{PII}),\gamma\in\mathbb{R}$ of an integrable Fredholm operator $K_{PII}$ acting on the interval $(-s,s)$ whose kernel is constructed out of the $\Psi$-function associated with the Hastings-McLeod solution of the second Painlev\'e equation. In case $\gamma=1$, 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 $\det(I-\gamma K_{PII})$ for all values of the real parameter $\gamma$.en-USIntegrable operators, Riemann-Hilbert approach, Deift-Zhou method, asymptotical analysis of Fredholm determinantsFredholm equations -- Numerical solutionsFredholm operators -- ResearchLinear operatorsRiemann-Hilbert problemsRandom matricesIntegral equations -- Numerical solutionsStructural dynamics -- Mathematical modelsEigenvalues -- ResearchOperator theory