Probability On Groups: Random Walks and Limit Theorems

• Main Author: He, Jimmy
• Format: Dissertation
• Language: English
• Published: ProQuest Dissertations & Theses 01-01-2021
• Subjects: Dynamical systems; Eigenvalues; Inequality; Markov analysis; Mathematics; Operations research; Probability; Psychology; Random variables; Theorems
• ISBN: 9798544204671

This thesis studies three problems in probability on groups and related structures. A variety of techniques are used, with the common thread connecting the various works being the interplay between the algebraic structure of the group and the probability within the problem being studied. The first problem concerns the mixing time of random walk on finite fields given by applying an almost rational bijection between steps of a random walk. This work is contained in [64]. It is shown that the mixing time is much faster than the random walk alone, using Cheeger’s inequality and the Weil bounds.The second problem concerns a bi-invariant random walk on the symmetric space of symplectic forms over a finite field. This work is contained in [63] and [65]. Cutoff is established using spectral theory. An elementary argument is given for the eigenvalue computation, along with a proof through constructing a characteristic map relating the spherical functions to Macdonald symmetric functions. Other applications of this characteristic map are also given.The final problem concerns the distribution of descents in the symmetric group under the Mallows measure. This work is contained in [66]. A joint central limit theorem is established for descents of both the permutation and its inverse, as well as stronger results for their sum. The key tool is Stein’s method with size-bias coupling.