Introductory description

In the past decade there has been an explosion of interest on the interface and relations between symmetric functions and probability. This has led, on the one hand, to remarkable resolutions of open problems in statistical mechanics and, on the other, to progress in the algebraic counterparts. The whole interaction has created a new and very active field named Integrable Probability.

This module will inform 4th year students on these novel developments and to equip them with the foundations on symmetric functions and probability that are required to enter this new field.

On the symmetric function side we will look into Schur functions and their various Macdonald generalisations. We will also look at the interplay with algebraic combinatorics and discuss Young tableaux and the Robinson-Schensted-Knuth correspondence. Connections with Crystal Theory of Kashiwara and Lustig will also be made as well as other fundamental notions from representation theory.

On the probability side we will see how the algebraic tools are combined with probabilistic notions to attack problems of random growth, in particular related to models in the Kardar-Parisi-Zhang (KPZ) universality. Other statistical mechanics models such as Ice and Vertex models will be discussed. We will explore how the interplay of probability and algebra leads to asymptotic analysis of the statistical of these models and their relation to statistics from Random Matrix Theory. We will also show how the probabilistic intuition leads to important algebraic identities such as Cauchy and Littlewood identities.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

1. basic symmetric functions
2. Schur functions
3. Young tableaux and Robinson-Schensted-Knuth correspondence
4. Macdonald functions
5. KPZ universality and integrable probabilistic models
6. Asymptotics analysis and Random Matrix (ie Tracy-WIdom) asymptotic laws

Learning outcomes

By the end of the module, students should be able to:

• have the foundations on algebraic combinatorics, symmetric functions and probability
• be able to use the above tools to derive probabilistic limit theorems
• perform asymptotic analysis
• analyse the minimiser of large deviation rate functions of basic examples and to provide interpretation of the possible occurrence of multiple minimiser.
• explain the role of the free energy in interacting systems and its link to stochastic modelling and provide different representations of the free energy for some basic examples
• estimate probabilities for interacting systems using Laplace integral techniques and basic understanding of Gibbs distributions.

Research element

The setting of this module is inspired and driven by current research developments. Students will be encourage to apply the knowledge into research problems

Subject specific skills

Students will learn fundamentals in symmetric functions, algebraic combinatorics , be acquainted with KPZ and other statistical mechanics models and be able to understand the interplay between all the above and use one field to solve problems in another.

Transferable skills

Develop ability to transfer knowledge from one scientific field to another and use intuition from one field to deepen understanding in another.