Introduction to Random Walks on Groups

Explore the mathematical theory of random walks on groups through this comprehensive graduate-level course from the Fields Institute. Delve into the fundamental concepts, properties, and applications of random processes that evolve on group structures, examining how probability theory intersects with abstract algebra and geometric group theory. Study the behavior of random walks on various types of groups, including finite groups, free groups, and geometric groups, while investigating key topics such as mixing times, spectral properties, and convergence rates. Analyze the relationship between the algebraic structure of groups and the probabilistic behavior of random walks, covering essential techniques for understanding return probabilities, hitting times, and asymptotic behavior. Learn advanced mathematical tools and methods used to study these stochastic processes, including harmonic analysis, representation theory, and geometric techniques. Examine applications to areas such as computer science, statistical mechanics, and geometric group theory through detailed mathematical exposition and rigorous proofs across 21 comprehensive lectures totaling over 6 hours of instruction.