  We study Fredholm determinants of the Painlev\'e II and Painlev\'e XXXIV
kernels. In certain critical unitary random matrix ensembles, these
determinants describe special gap probabilities of eigenvalues. We obtain
Tracy-Widom formulas for the Fredholm determinants, which are explicitly given
in terms of integrals involving a family of distinguished solutions to the
coupled Painlev\'e II system in dimension four. Moreover, the large gap
asymptotics for these Fredholm determinants are derived, where the constant
terms are given explicitly in terms of the Riemann zeta-function.
