Return Probabilities on Groups and Large Deviations for Permuton Processes

The topic of this thesis are random processes on finite and infinite groups. More specifically, we are concerned with random walks on finitely generated amenable groups and stochastic processes which arise as limits of trajectories of the interchange process on a line. In the first part of the thesis we construct a new class of finitely generated groups, called bubble groups. Analysis of the random walk on such groups shows that they are non-Liouville, but have return probability exponents close to $1/2$. Such behavior was previously unknown for random walks on groups. Our construction is based on permutational wreath products over tree-like Schreier graphs and the analysis of large deviations of inverted orbits on such graphs. In the second part of we analyze large deviations of the interchange process on a line, which can be thought of as a random walk in the group of all permutations, with adjacent transpositions as generators. This is done in the setting of random permuton processes, which provide a notion of a limit for a permutation-valued stochastic processes. More specifically, we provide bounds on the probability that the trajectory of the interchange process (as a permuton process) is close in distribution to a deterministic permuton process. As an application, we show that short paths joining the identity and the reverse permutation in the Cayley graph of $\mathcal{S}_{n}$ are typically close to the so-called sine curve process, which is the conjectured limit of random sorting networks. The analysis is done in the framework of interacting particle systems.Ph.D