Classical dynamics studies how systems change in time. Ergodic theory, a subfield of dynamics, focuses on the statistical behavior of dynamical systems. Applications of ergodic theory are widespread: from traffic modeling to aerospace engineering and population dynamics. It is natural and of practical importance to generalize the role of time in a dynamical system to more complicated groups of symmetries. This generalized notion of dynamics leads to applications in statistical physics, number theory and geometry. However, new tools are needed when the group of symmetries is non-amenable which means that boundary phenomena are too significant to be safely ignored. One such tool is the weak local (or Benjamini-Schramm) limit. These limits formalize the asymptotic local behavior of large, possibly random, mathematical objects. This project is concerned with fundamental questions: when do these limits exist (for natural families of low-dimensional geometric objects) and given an infinite mathematical object (such as a manifold or network), can it be identified as the weak local limit of finite objects? This research project has three main objectives. 1) In recent work with M. Chapman, A. Lubotzky and T. Vidick, the primary investigator (PI) settled the Aldous-Lyons Conjecture in the negative: there exist non-sofic unimodular random graphs. These are random rooted graphs which cannot be approximated by finite graphs in the Benjamini-Schramm sense despite satisfying the Mass