Asymptotics for a special solution of the thirty fourth Painleve equation

In a previous paper we studied the double scaling limit of unitary random matrix ensembles of the form Z_{n,N}^{-1} |\det M|^{2\alpha} e^{-N \Tr V(M)} dM with \alpha > -1/2. The factor |\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 computed the limiting eigenvalue correlation kernel in the double scaling limit as n, N \to \infty such that n^{2/3}(n/N-1) = O(1) by using the Deift-Zhou steepest descent method for the Riemann-Hilbert problem for polynomials on the line orthogonal with respect to the weight |x|^{2\alpha} e^{-NV(x)}. Our main attention was on the construction of a local parametrix near the origin by means of the \psi-functions associated with a distinguished solution u_{\alpha} 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. In this paper we compute the asymptotic behavior of u_{\alpha}(s) as s \to \pm \infty. We conjecture that this asymptotics characterizes u_{\alpha} and we present supporting arguments based on the asymptotic analysis of a one-parameter family of solutions of the Painleve XXXIV equation which includes u_{\alpha}. We identify this family as the family of tronquee solutions of the thirty fourth Painleve equation.