In this course, we will present modern topics at the interface between probability and geometric group theory. We will define two random processes on Cayley graphs: the simple random walk and percolation, and discuss their respective behaviors depending on the geometric properties of the underlying group.
Objective
Present in an original framework important tools in the study of - random walks: spectral gap, harmonic functions, entropy,... - percolation: uniqueness of the infinite cluster, mass-transport principle,...
Content
In this course, we will present modern topics at the interface between probability and geometric group theory. To every group with a finite generating set, one can associate a graph, called Cayley graph. (For example, the d-dimensional grid is a Cayley graph associated to the group Z^d.) Then, we will define two random processes on Cayley graphs: the simple random walk and percolation, and discuss their respective behaviors depending on the geometric properties of the underlying group. The focus will be on the random processes and their properties, and we will use very few notions of geometric group theory.
Literature
Probability on trees and network (R. Lyons, Y. Peres)
Prerequisites / Notice
- Probability Theory - No prerequisite on group theory, all the background will be introduced in class.
Performance assessment
Performance assessment information (valid until the course unit is held again)