Course Objectives:

The limit laws for the sums x₁+...+x_n of independent identically distributed random variables x_i are fundamental in mathematics and its applications. For instance, the Law of Large Numbers predicts that (x₁+...+x_n)/n converges to the expectated value of x_i, and the deviation from this limit is described by the Central Limit Theorem.

In this course we investigate the analogous noncommutative problem, namely, the asymptotic behaviour of the products X_n...X₁ of d-dimensional random matrices X_i. It turns out that to handle noncommuting products, one needs to develop quite different techniques, and a number of remarkable new phenomena emerge. This theory has been developed by Furstenberg, Guivarc'h, Kesten, Le Page, Raugi, and others. It is not only beautiful on its own right, but now it also plays important role in several areas in analysis and geometry related to recurrence properties, harmonic functions, and Schrodinger operators.

We plan to cover the following topics:

• Lyapunov exponents of random walks and Oseledets' ergodic theorem,
• amenability and speed of escape of random walks,
• spectral gap property and its applications,
• simplicity of the Lyapounov spectrum,
• stationary measures,
• boundary theory and harmonic functions,
• noncommutative central limit theorem and large deviation results.

In this course we only assume basic knowledge of Probability and Analysis.

Time: Tue. 3-5pm; See TCC Timetable

Homework Problems:

• Problem Set

Course Assessment: The course will be assessed by problem sheets that will be posted here.

References:

• P. Bougerol and J. Lacroix, Random Products of Matrices with Applications to Schrodinger Operators. Birkhauser 1985.
• E. Breuillard, Random walks on Lie groups.
• J. Cohen, H. Kesten and C. Newman, Random matrices and their applications. Contemporary Mathematics, 50. American Mathematical Society, Providence, RI, 1986.
• A. Furman, Random walks on groups and random transformations, in Handbook of Dynamical Systems, Vol. 1 A, Amsterdam: North-Holland, 2002, pp. 931-1014.
• Y. Guivarc'h, Limit theorems for random walks and products of random matrices. Probability measures on groups: recent directions and trends, 255-330, Tata Inst. Fund. Res., Mumbai, 2006.
• A. Karlsson, Ergodic theorems for noncommuting random products
• F. Ledrappier, Quelques propriétés des exposants caractéristiques. École d'été de probabilités de Saint-Flour, XII-1982, 305-396, Lecture Notes in Math., 1097, Springer, Berlin, 1984.