Level spacing distribution of critical random matrix ensembles

We consider unitary invariant random matrix ensembles that obey spectral statistics different from the Wigner-Dyson statistics, including unitary ensembles with slowly $\left(\sim {\log }^{2}x\right)$ growing potentials and the finite-temperature Fermi gas model. If the deformation parameters in these matrix ensembles are small, the asymptotically translational-invariant region in the spectral bulk is universally governed by a one-parameter generalization of the sine kernel. We provide an analytic expression for the distribution of the eigenvalue spacings of this universal asymptotic kernel, which is a hybrid of the Wigner-Dyson and the Poisson distributions, by determining the Fredholm determinant of the universal kernel in terms of a Painlevé VI transcendental function.