I will discuss a recent work which concerns the critical behavior of eigenvalues in ensembles 1/Z(n,N) det M(2 alpha) exp(-N Tr V(M)) dM with alpha greater than -1/2, where the factor det M(2alpha) induces critical eigenvalue behavior near the origin. Supposing that the limiting mean eigenvalue density associated with V is regular, and that the origin is a right endpoint of its support, one can compute (using the Deift-Zhou steepest-descent method) the limiting eigenvalue correlation kernel in the double scaling limit as n, N to infinity such that n(2/3) (n/N-1) = O(1). It turns out that the limiting kernel can be described through a distinguished solution of the thirty fourth Painleve equation. This solution is related to a particular solution of the Painleve II equation, which however is different from the usual Hastings-McLeod solution. The talk is based on joint work with Alexander Its and Arno Kuijlaars.