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Double scaling limit in the random matrix model: The Riemann-Hilbert approach. (English) Zbl 1032.82014
Summary: 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 Painlevé 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.

##### MSC:
 82B31 Stochastic methods applied to problems in equilibrium statistical mechanics 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics 30E25 Boundary value problems in the complex plane 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies 60E05 Probability distributions: general theory
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