Stationary random systems via an asymptotic approach

Large random networks naturally arise as models for real-life networks, as various characteristics can be effectively captured and analyzed in such models. In the study of random graphs, stationarity can be viewed as a stochastic homogeneity assumption; while in geometry, homogeneity is associated with a transitive symmetry group. In this project we aim to investigate stationary random networks through the lens of random walks on groups, entropy and boundary theory, and graph limits, while utilizing the growing theory of probability measures on the Chabauty space of subgroups. This project offers collaboration opportunities, mentorship and training for postdoctoral scholars and graduate students. The proposed project primarily focuses on three research directions, with entropy theory of random walks being the underlying fundamental tool used to approach all these threads. The first part focuses on the Furstenberg entropy realization problem for classical matrix groups, with an aim to understand the interplay between rigidity and flexibility in the structure of stationary random systems. The second part focuses on the longstanding amenability problem of a class of groups (called iterated monodromy groups) which originate from conformal dynamics. The conformal geometry of the limit space, which can be viewed as limits of finite level Schreier graphs, plays a crucial role in the recent solution in the strongly recurrent regime. In this part the goal is to shed light on an important critical case where the limit space is the full two-dimensional sphere. In the third part we propose a new procedure called random walk cocycle sampling limit and explore its potential applications. The procedure is inspired by the local-convergence graph limits initiated by Benjamini-Schramm and it naturally produces stationary random systems that carry rich information about random walk transition probabilities. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.