(Joint series with Mark Adler)

**Abstract: **

Lecture 1. (Monday June 20, P. van Moerbeke)
** An introduction to random matrices **

The Gaussian Hermitian ensemble (GUE) can be described
by Hermitian matrices whose entries (real and imaginary)
are all independent identically distributed Gaussian
random variables. The Laguerre ensemble, another random
matrix ensemble, is related to an unbiased estimator of
the covariance matrix of p independent and identically
distributed Gaussian complex random variables, whereas a
maximum likelihood estimator for the correlation matrix
of two gaussian populations relate to the Jacobi
Hermitian ensemble.

Lecture 2. (Monday June 20, M. Adler)
** Recursion relations for Unitary integrals.**

Recursion relations for unitary integrals are discussed with emphasis on
examples from combinatorics. The integral system called the Toeplitz lattice
hierarchy is used along with its Virasoro symmetry operator, which together
conspire to yield recursion relations. These relations are then shown to have
the "Painleve property" of singularity confinement with maximal degrees of
freedom, no matter how many steps in the recursion relation. The latter
property depends on the "Painleve property" of the Toeplitz lattice and thus
is inherited.

Lecture 3. (Tuesday June 21, P. van Moerbeke)
**
The length of longest increasing sequences
in random permutations and words: some basic facts.
**

The length of the longest increasing sequences in random
permutations is expressed in terms of the width of the
Young diagram, via the Robinson-Shensted-Knuth
correspondence. Uniform measure on permutations
translates into Plancherel measure on Young diagrams.
Some related generating functions are expressed as
integrals over the Unitary group. Some percolation
problems will also be discussed.

Lecture 4. (Tuesday June 21, M. Adler)
**Random matrices and solitons.**

Random matrix ensembles are introduced along with their determinantal nature
and various universal limiting behaviors are discussed via limiting Fredholm
determinants. PDE's for these determinants are gotten using KP theory. In
particular, the probabilistic Fredholm determinants are seen to be continuous
solitons. Then KP deformation theory via vertex-operators and the Virasoro
algebra can be applied to generate the PDE's for a variety of cases in a
uniform fashion.

Lecture 5. (Wednesday June 22, P. van Moerbeke)
**
A probability measure on partitions,
Toeplitz and Fredholm determinants and the 2d-Toda lattice.
**

A general measure, which upon specialization leads to
Plancherel measure, has many interesting features:

(1) It is expressible in terms of a Unitary matrix integral,

(2) It can also be expressed as a Fredholm determinant
for a certain kernel,

(3) It satisfies the Toeplitz lattice,

(4) It satisfies Virasoro constraints.

Lecture 6. (Wednesday June 22, M. Adler)
**Coupled Random matrices and the 2d-Toda lattice.**

We derive a fundamental PDE for the probabilities for coupled random Hermtian
matrices. The essential method being that a natural deformation of the
probabilities leads to the 2-Toda hierarchy. The latter is well related to the
biorthogonal polynomials going with this problem, which generate the wave
functions of the hierarchy. The bilinear identities for the hierarchy leads to
the Fay identities, yielding Virasoro relations for the probabilities - which
have been deformed into tau functions. The PDE's then follow from a fundamental
new PDE for the 2-Toda hierarchy.

Lecture 7. (Thursday June 23, P. van Moerbeke)
**
Length of longest increasing sequences and
Painlevé equations.
**

The features in Lecture 5 will be used effectively to
show that the generating functions for the statistics of
the length of longest increasing subsequences satisfies
Painlevé equations. These ideas will be applied to
queuing problems, discrete polynuclear growth problems,
random walks, which are expressed as unitary matrix
integrals as well. Some other combinatorial features of
random permutations lead to integrals over the
Grassmannian space of p-dimensional planes in C^n.

Lecture 8. (Thursday June 23, M. Adler)
**The Dyson Brownian motion and the Airy process.**

Dyson showed that the spectrum of a nxn Hermitian matrix whose entries diffuse
according to n^2 independent Ornstein-Uhlenbeck processes, evolves as n
non-colliding Brownian particles, held together by harmonic forces. We explain
the result carefully and derive a PDE for this Dyson process. Then when n gets
large, the largest eigenvalue, with space and time properly rescaled, goes to
a non-Markovian continuous stationary process, called the Airy process , and
similarly in the bulk one finds the Sine process. Using the PDE for the Dyson
process we derive PDE's for the joint distributions for the Airy and Sine
processes. These then lead to asymptotic behavior for the distributions and
covariances at different times, t1 and t2, when t1-t2 become large.

Lecture 9. (Friday June 24, P. van Moerbeke):
** Distribution of the length of longest
increasing subsequences for very large permutations
**

For large permutations, the length of the longest
increasing sequences, properly rescaled, fluctuates
around 2\sqrt{n} according to the Tracy-Widom
distribution, when the size n of the permutations tends
to infinity

Lecture 10. (Friday June 24, M. Adler)
**The Pearcey distribution.**

Consider 2n-noncolliding Brownian motions, conditioned to emanate from x=0 at
t=0 , with half to end at n^(1/2) and the other half to end at -n^(1/2), all at
t=1. This is equivalent to a random Heritian matrix potential, but with a
specific magnetic field. At t=1/2, for n large, a bifurcation takes place and
the probability distribution goes from having support on one interval to
support on two intervals separating at x=0, as the Brownian motions finally
realize they are destined for two distinct futures at t=1. The limiting
behavior is characterized by a Fredholm determinant involving the Pearcey
kernel with a parameter . Using the 3-Toda equations we derive a PDE , first
for the probabilistic Fredholm determinant for a finite n problem, itself of
interest, and then after appropriate rescaling for the Fredholm determinant
involving the actual Pearcey kernel. The methods involve understanding the
multi-orthogonal polynomials (mops) inherent in this problem and how they
relate to the 3-Toda lattice, which has its own natural mops. Also the Virasoro
analysis is quite novel in this problem and finally the bilinear identities in
the 3-Toda lattice yield important PDE's, which finally lead to the desired
probabilistic PDE's.