  We consider the smallest eigenvalue distributions of some Freud unitary
ensembles, that is, the probabilities that all the eigenvalues of the Hermitian
matrices from the ensembles lie in the interval $(t,\infty)$. This problem is
related to the Hankel determinants generated by the Freud weights with a jump
discontinuity. By using Chen and Ismail's ladder operator approach, we obtain
the discrete systems for the recurrence coefficients of the corresponding
orthogonal polynomials. This enables us to derive the large $n$ asymptotics of
the recurrence coefficients via Dyson's Coulomb fluid approach. We finally
obtain the large $n$ asymptotics of the Hankel determinants and that of the
probabilities from their relations to the recurrence coefficients and with the
aid of some recent results in the literature.
