Lipschitz Lectures

Integrable probability and Macdonald processes

Ivan Corwin (Clay Mathematics Institute and Massachusetts Institute of Technology)


Date: January 9 - 24, 2013
Time: Wednesdays 10 - 12 a.m., Thursdays 9 - 11 a.m.
Location: Lipschitz Lecture Hall, Bonn University, Endenicher Allee 60, Bonn

Abstract

The Kardar-Parisi-Zhang (KPZ) equation and universality class was introduced in 1986. It is believed to describe the long-time/large-scale behavior of a wide class of mathematical and physical systems including driven lattice gases, randomly growing interfaces, directed polymers, and certain noisy stochastic PDEs.
Macdonald symmetric functions were introduced a year later in 1987. They generalized the Schur, Jack and Hall-Littlewood functions and have found many applications in representation theory, algebraic geometry, combinatorics, knot theory and quantum algebra.


In a 2011 joint work with Alexei Borodin, we brought together these two developments into the theory of Macdonald processes. The main subject of these lectures is this theory and its implications and applications. In particular we will see how, using the structural properties of Macdonald symmetric functions, it is possible to (1) construct many interesting Markov dynamics which are described by Macdonald processes, and (2) write simple formulas for a rich class of expectations of observables of these processes.


Via asymptotics of the dynamics of (1) and associated formulas of (2) we are able to access to certain universal statistics associated with the KPZ equation and universality class. Some of these arose in the work of Baik-Deift-Johansson and Johansson in 1999, and the more recent work of Tracy-Widom in 2008 and of Amir-Corwin-Quastel and Sasamoto-Spohn in 2010.


In the lectures we will also develop connections between Macdonald processes and certain generalizations of the delta Bose gas, a rigorous version of the physics replica trick for directed polymers, the asymmetric simple exclusion process, the geometric lifting of the Robinson-Schensted-Knuth correspondence, and random matrix theory.