Large N asymptotics in random matrices. The Riemann-Hilbert approach

Pavel Bleher

(Joint series with Alexander Its)

Abstract:

Lecture 1. (Thursday June 23, A. Its)
The setting and elements of the general theory of Riemann-Hilbert factorization problems. Riemann-Hilbert representation of orthogonal polynomials and matrix models.

Lecture 2. (Friday June 24, A. Its)
Semiclassical asymptotics of orthogonal polynomials via the nonlinear steepest descent and isomonodromy methods I. Quartic exponential weight; one-cut case. Painlevé equations

Lecture 3. (Monday June 27, A. Its)
Semiclassical asymptotics of orthogonal polynomials via the nonlinear steepest descent and isomonodromy methods II. Quatric exponential weight; two-cut case. Sketch of the analysis in the general multi-cut case. Airy-kernel and Sine-kernel universality classes. The Tracy-Widom distribution function.

Lecture 4. (Monday, June 27, P. Bleher)
Large N asymptotics of orthogonal polynomials with exponential weights: the Riemann-Hilbert approach. Construction of a parametrix for the RH problem.

Lecture 5. (Tuesday June 28, A. Its)
The Painlevé equations and matrix models. The Hastings-McLeod second Painlevé transcendent; the Riemann-Hilbert representation, asymptotics, and the connection formulae.

Lecture 6. (Tuesday, June 28, P. Bleher)
Double scaling limits of orthogonal polynomials and their applications to random matrix models.

Lecture 7. (Wednesday, June 29, P. Bleher)
Large N asymptotics of the partition function of random matrix models. Critical exponents and double scaling asymptotics of the free energy.

Lecture 8. ((Thursday, June 30, P. Bleher)
Large N asymptotics of a random matrix model with external source. Multiple orthogonal polynomials.

Lecture 9. (Friday, July 1, P. Bleher)
Double scaling limit of the random matrix model with external source.