Course Integrable Probability 18-19

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V5F1 ADVANCED TOPICS IN PROBABILITY THEORY

Time: Wednesday 12-14 in N0.008 and Friday 14-16 in N0.003; Endenicher Allee

60, Neubau.

Lecturer: Alexey Bufetov (alexey.bufetov AT gmail.com)

Prerequisites: Basic Probability (e.g., what is a random variable ?), basic Linear

Algebra (e.g. what is a determinant ?). The rest will be defined and explained.

Preliminary plan of the course:

In the first part we will cover

• Introduction

• Wigner ensemble: symmetric matrices with independent entries. Law of Large

Numbers for eigenvalues: Semicircle law. Moment method. Wishart ensemble

and Marcenko-Pastur distribution.

• Symmetric polynomials. Schur polynomials and their properties.

• Domino tilings of the Aztec diamond. Number of tilings.

• Global limit behavior of a measure determined by a Schur generating function.

• Global limit behavior of domino tilings of the Aztec diamond

• Limit behavior of more complicated lozenge and domino tiling models.

• Schur-Weyl and Plancherel probability measures on Young diagrams. Global limit behavior of random Young diagrams under these measures.

In the second part of the course we will cover some (but not all) of the following

topics. The choice will depend on preferences of students.

• Limit shape for arbitrary tiling models: Variational principle.

• Free probability. Voiculescu’s theorem for the spectrum of the sum of indepen-

dent matrices.

• Determinantal processes in classical matrix ensembles and in tiling problems.

Local limit behavior.

• Global fluctuations of the limit shape in matrix ensembles and in tiling prob-

lems. Gaussian Free Field.

• Extreme characters of the infinite symmetric group and of the infinite-

dimensional unitary group. Thoma’s and Edrei-Voiculescu’s classifications.

Vershik-Kerov’s approach.

Suggested literature

• G. Anderson, A. Guionnet and O. Zeitouni: An Introduction to Random Matri-

ces; available from the O. Zeitouni’s webpage http://www.wisdom.weizmann.

• T. Tao: Topics in random matrix theory; available at https://terrytao.

• D. Romik: The Surprising Mathematics of Longest Increasing Subsequences.

• A. Borodin, V. Gorin: Lectures on Integrable Probability; https://arxiv.

• A. Bufetov, V. Gorin: Representations of classical Lie groups and quantized

(more references will be provided during the course).