Random Walks and Invariant Random Subgroups

In November 2024, I gave a talk at the Seoul National University Dynamics Seminar, supervised by Prof. Seonhee Lim. I introduced basic concepts related to random walks on groups, the notion of invariant random subgroups, and generalizations of the tools for random walks on groups to random coset spaces.

Abstract

Upon first introduced by Kesten in 1959, the study of random walks on groups has proven to be a valuable method for investigating group structures, giving rise to many intriguing questions. Meanwhile, invariant random subgroups have emerged as central objects in the study of group actions, serving as natural generalizations of normal subgroups and lattices. In the first part of this talk, we will introduce fundamental concepts related to random walks on groups, including the Poisson–Furstenberg boundary and Avez asymptotic entropy. In the second part, we will present the notion of invariant random subgroups and discuss related results. Finally, we will address generalizations of former results on random walks on groups, extending them to random coset spaces, which are quotients by invariant random subgroups.

Slides

These are the slides for my talk.

References

[1] Abért, M., Invariant Random Subgroups and Their Applications, Slides from a presentation at IHP, 2014. (Online)

[2] Benjamini, I. and G. Kozma, Nonamenable liouville graphs, arXiv preprint, 2010. arXiv: 1010.3365

[3] Bowen, L., Random walks on random coset spaces with applications to Furstenberg entropy, Inventiones Mathematicae 196 (2014), 485-510.

[4] Furman, A., Random walks on groups and random transformations, Handbook of Dynamical Systems Volume 1A. Elsevier Science, 2002. 931-1014.

[5] Gelander, T., Lecture notes on Invariant Random Subgroups and Lattices in rank on and higher rank, arXiv preprint, 2015. arXiv: 1503.08402

[6] Kaimanovich, V. A. and A. M. Vershik, Random walks on discrete groups: boundary and entropy, The Annals of Probability 11 (1983), 457-490.

[7] Lalley, S. P., Random Walks on Infinite Discrete Groups, Electronic Edition, 2018. (Online)

[8] Lyons, R. and Y. Peres, Probability on Trees and Networks. Cambridge University Press, 2017.

[9] Tamuz, O., Lecture Notes on Random Walks, Electronic Edition, 2022. (Online)

[10] Yadin, A., Harmonic Functions and Random Walks on Groups. Cambridge University Press, 2024.

[11] Zheng, T., Asymptotic behaviors of random walks on countable groups, Proceedings of the International Congress of Mathematicians, 2022.

[12] Zheng, T., Random Walks on Random Coset Spaces, 13th Cornell Probability Summer School, 2024.