Research project

Integrable systems and exactly solvable models are important parts of mathematical physics. These models have a rich algebraic or combinatorial structure that allows for special methods to solve them. Often there are special solutions that have relations to orthogonal polynomials and other special functions.

Lattice models provide an important class of examples, with the first steps taken in the 1930-1940s with the solution of the 2 dimensional Ising model on a square lattice. Since the 80’s the field really took off when powerful tools were applied to critical phenomenon, especially in two dimensions. The last twenty years have witnessed a remarkable convergence of results and interests among pure mathematicians and mathematical/statistical physicists.

A second class of examples comes from random matrix theory. Eigenvalues of matrices with random entries typically repel each other and their behavior is in many respects very different from that of independent points. The simplest random matrix model is the Gaussian Unitary Ensemble (GUE) where the matrices are Hermitian with independent complex Gaussian entries. This model, as well as many others, is exactly solvable as a determinantal point process in which the correlation functions can be expressed as determinants. Gap probabilities are Fredholm determinants.

The mathematical tools for many lattice models and random matrix ensembles are similar, consisting of

• Symmetry considerations by means of group theory and representation theory
• Exact formulas with Toeplitz, Hankel and Fredholm determinants
• Special functions and orthogonal polynomials
• Asymptotic analysis by means of steepest descent

The field of integrable systems is immensely vast, and in this project we can only focus on a number of specialized topics within this domain. The choice of topics is guided by the expertise and interest of the partners and the prospect of joint collaborations between the different groups. The main topics are random matrices, lattice models, and orthogonal polynomials. The project is thus naturally divided into three work packages.

• Work package 1: Random Matrices
• Work package 2: Lattice Models
• Work package 3: Orthogonal Polynomials and Special Functions