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

## Alexander Its

(Joint series with Pavel Bleher)

**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.