Resumen
In a previous paper we studied the double scaling limit of unitary random matrix ensembles of the form with a > -1/2. The factor | det M|2a induces critical eigenvalue behaviour 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 ? 8 such that n2/3(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|2ae-NV(x). Our main attention was on the construction of a local parametrix near the origin by means of the ?-functions associated with a distinguished solution ua of the Painlevé XXXIV equation. This solution is related to a particular solution of the Painlevé II equation, which, however, is different from the usual Hastings–McLeod solution. In this paper we compute the asymptotic behaviour of ua(s) as s ? ±8. We conjecture that this asymptotics characterizes ua and we present supporting arguments based on the asymptotic analysis of a one-parameter family of solutions of the Painlevé XXXIV equation which includes ua. We identify this family as the family of tronquée solutions of the thirty fourth Painlevé equation.