Introduction

The purpose of this article is to survey recent interactions between statistical questions and integrable theory. Two types of questions will be tackled here:

• (i) Consider a random ensemble of matrices, with certain symmetry conditions to guarantee the reality of the spectrum and subjected to a given statistics. What is the probability that all its eigenvalues belong to a given subset E? What happens, when the size of the matrices gets very large? The probabilities here are functions of the boundary points Ci of E.

• (ii) What is the statistics of the length of the largest increasing sequence in a random permutation, assuming each permutation is equally probable? Here, one considers generating functions (over the size of the permutations) for the probability distributions, depending on the variable x.

The main emphasis of this article is to show that integrable theory serves as a useful tool for finding equations satisfied by these functions of x, and conversely the probabilities point the way to new integrable systems.

These questions are all related to integrals over spaces of matrices. Such spaces can be classical Lie groups or algebras, symmetric spaces or their tangent spaces. In infinite-dimensional situations, the “ ∞ -fold” integrals get replaced by Fredholm determinants.

During the last decade, astonishing discoveries have been made in a variety of directions. A first striking feature is that these probabilities are all related to Painleve equations or interesting generalizations. In this way, new and unusual distributions have entered the statistical world.