TU Lecture, WS 2007/08

Random matrices and related problems

Content: In this lecture we will explore some aspects of random matrix theory. We will then see how the underlying mathematical structure can be applied to various models in mathematics and physics. These models span from directed percolation, random domino tiling, equilibrium and non-equilibrium statistical mechanics.

Program: October 15: It will be an introduction to the lecture and a rough plan of the content will be also presented.
October 22 - December 17: We introduce the relevant mathematical objects / methods using as a basis model one ensemble of random matrices.
January 7 - February 11: The rest of the semester will be devoted to a few applications. We will apply the mathematical structure we learned to a few models (non random matrix models).

Schedule:WS 2007/08, Monday 10-12, MA 645

Scanned lectures notes: (first versions, are not missprints free: take them with a grain of salt)

A single file of the lecture notes
References used in the preparation of the lecture notes
Annexe to introduction: some pictures
The Classical Gaussian Ensembles of Random Matrices
Annexe: a variational formula
Gaussian Unitary Ensemble (GUE): determinantal correlation functions
Annexe: Cauchy-Binet and Christoffel-Darboux formulas
Universal scalings for GUE: Sine and Airy kernel
Annexe: Logarithmic fluctuations of sine kernel
Genaral point processes and determinantal point processes
Fredholm determinant, part 1
Annexe of functional analysis
Fredholm determinant, part 2
Tracy-Widom distribution and Painlevé II
Dnamics of point process: extended point processes
Non-intersecting Brownian bridges and the Airy process
Dyson's Brownian Motion and LGV theorem
Application to the 3D-Ising corner
Application to the PNG droplet, part 1
Application to the PNG droplet, part 2