  We consider the Gaussian unitary ensemble perturbed by a Fisher-Hartwig
singularity simultaneously of both root type and jump type. In the critical
regime where the singularity approaches the soft edge, namely, the edge of the
support of the equilibrium measure for the Gaussian weight, the asymptotics of
the Hankel determinant and the recurrence coefficients, for the orthogonal
polynomials associated with the perturbed Gaussian weight, are obtained and
expressed in terms of a family of smooth solutions to the Painlev\'{e} XXXIV
equation and the $\sigma$-form of the Painlev\'{e} II equation. In addition, we
further obtain the double scaling limit of the distribution of the largest
eigenvalue in a thinning procedure of the conditioning Gaussian unitary
ensemble, and the double scaling limit of the correlation kernel for the
critical perturbed Gaussian unitary ensemble. The asymptotic properties of the
Painlev\'{e} XXXIV functions and the $\sigma$-form of the Painlev\'{e} II
equation are also studied.
