Double scaling limit in the random matrix model: The Riemann-Hilbert approach

We prove the existence of the double scaling limit in the unitary matrix model with quartic interaction, and we show that the correlation functions in the double scaling limit are expressed in terms of the integrable kernel determined by the psi function for the Hastings-McLeod solution to the Painleve II equation. The proof is based on the Riemann-Hilbert approach, and the central point of the proof is an analysis of analytic semiclassical asymptotics for the psi function at the critical point in the presence of four coalescing turning points. (C) 2003 Wiley Periodicals, Inc.