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 Painlev\'e II equation. The proof is based on the Riemann-Hilbert approach.