High-dimensional probability in ergodic theory and representation theory

This project begins with the interaction between (i) convergence and limits for labeled bounded-degree graphs and (ii) the This project begins with the interaction between (i) convergence and limits for labeled bounded-degree graphs and (ii) the ergodic theory of possible limit processes over infinite graphs, regarded as stationary processes for infinite groups of symmetries. It will deliberately focus on foundational issues that cannot be addressed using the most standard tools: models that are at ‘low temperature’, so mean-field and related methods do not provide good approximations; and limiting graphs that are not ‘amenable’, meaning that they do not allow approximations by decomposing into finite pieces with only ‘small’ interactions between them. The first goal of this project is to uncover new approaches that draw on notions of convergence for sparse graphs, conjectures about the behaviour of high-dimensional graphical models in statistical physics, and the ergodic theoretic study of limit processes via sofic entropy and related concepts.To shed new light beyond the reach of established techniques, the project will take inspiration from an analogous emerging theory where some of the same challenges can be addressed. This theory studies unitary matrices (rather than labeled graphs) and limiting unitary representations (rather than infinitary stochastic processes). The PI has recently been developing an analog of sofic entropy in this new setting. Several open problems about graph convergence admit analogs about tuples of unitary matrices, and the PI has already solvedsome of these using ideas from random matrix theory or operator algebras. The second goal of this proposal is to grow this new chapter in random matrix theory, where many further questions are now within reach. The third is to transplant the newly-discovered methods for tuples of matrices back to bounded-degree graphs to make progress towards open problems about graph convergence and processes.