Integrable Systems Seminar
Thursday, November 7, 2002, 4:00pm, 218 Physics
Pavel Bleher (Indiana University, Indianapolis)
Double scaling limit in random matrix models: The Riemann-Hilbert approach

Abstract:

The double scaling limit in a random matrix model describes asymptotics of eigenvalue correlation functions at a critical point, in the limit when simultaneously the size N of the matrix approaches infinity and the vector v of parameters of the model approaches the critical value v_c, with an appropriate relation between v-v_c and N. We will discuss the double scaling limit at a critical point which corresponds to a bifurcation of an interval of the support of the equilibrium measure into two intervals. Our main result is that the double scaling limit is given in this case by solutions to a linear system of differential equations associated with Painlev\'e II. Our approach is based on the Riemann-Hilbert problem. Time permitting, we will discuss a construction of a nonlinear hierarchy of integrable differential equations of the Painlev\'e II type, which describes the double scaling limits of a higher order. The talk is based on joint works with A. Its and B. Eynard.

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