Critical edge behavior in unitary random matrix ensembles and the thirty fourth Painleve transcendent

We describe a new universality class for unitary invariant random matrix ensembles. It arises in the double scaling limit of ensembles of random $n \times n$Hermitianmatrices$Z_{n,N}^{-1} |\det M|^{2\alpha} e^{-N \Tr V(M)} dM$with$\alpha > -1/2$,wherethefactor$|\det M|^{2\alpha}$ induces critical eigenvalue behavior near the origin. Under the assumption that the limiting mean eigenvalue density associated with $V$ is regular, and that the origin is a right endpoint of its support, we compute the limiting eigenvalue correlation kernel in the double scaling limit as $n,N\to \mathrm{\infty }$ such that $n^{2\mathrm{/}3}(n\mathrm{/}N-1)=O(1)$. We use the Deift-Zhou steepest descent method for the Riemann-Hilbert problem for polynomials on the line orthogonal with respect to the weight $\mathrm{\mid }x{\mathrm{\mid }}^{2\alpha }{e}^{-NV(x)}$. Our main attention is on the construction of a local parametrix near the origin by means of the $\psi$-functions associated with a distinguished solution of the Painleve XXXIV equation. This solution is related to a particular solution of the Painleve II equation, which however is different from the usual Hastings-McLeod solution.