These lectures begin with an introduction to random matrix theory and their connection with integrable systems of the Painlevé type. Following [1], we give newly simplified proofs of the representation of some of the basic distributions in terms of Fredholm determinants. These ideas will be illustrated in the context of the distribution of the largest eigenvalue in the three ensembles GOE, GUE and GSE. In the edge scaling limit, these distributions are all expressible in terms of a certain Painlevé II transcendent [2-5]. These same distribution functions arise in combinatorics [6,7,8] and random growth models [9]. Some of these last connections will be discussed along with the needed fundational combinatorics, e.g. Ch. 7 in [10].
Lectures will be taken from the references below. Obviously, we will not cover the content of each paper, but rather provide the student with an overview of the issues involved. Detailed proofs will be given in the case of the Airy kernel following [5].
Lectures 1 & 2:
We prove that the distribution function of the
largest eigenvalue in the Gaussian Unitary Ensemble (GUE) in the
edge scaling limit is expressible in terms of Painleve II. Our
goal in these two lectures is to concentrate on this important
example of the connection between random matrix theory and integrable
systems, and in so doing to introduce the newcomer to the subject
as a whole. These two lectures will follow the preprint solv-int/9901004
on the Los Alamos Archives.
Lectures 3 & 4:
Applications of random matrix theory to
combinatorial problems of the Robinson-Schensted-Knuth (RSK) type.
We explain the RSK correspondence between random permutations (or
random words) and Young tableaux. Using this correspondence
we show that certain Toeplitz determinants are the generating
functions for the distribution function of the length of the
longest increasing subsequence in random permutations/words.
The connection of these distributions with those of Lectures 1 & 2
will be explained. These lectures will be of an overview nature.
Basic references are work of Baik, Deift, Johansson, Rains and
others that can be found on the Mathematics section of the Los
Alamos Archives.