Random matrices, permutations and processes from the integrable point of view

Pierre Van Moerbeke

(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.