## Thesis Defense |

**Random Entanglement and History-Dependent Random Sequences**

**Date:**

**Monday, August 8th**

**Time:**1:00 pm

**Place:**4272 Chamberlin Hall

**Speaker:**Gage Bonner, Physics PhD Graduate Student

**Abstract:**This thesis is comprised of two main parts, each concerned with a different stochastic process. In the first section, we consider a two-dimensional reflecting Brownian motion in a bounded region divided into two halves by a wall with three or more small windows. In the limit of small windows, we appeal to the narrow escape problem to construct a Markov chain on the fundamental groupoid of the region. Taking the transition probabilities between windows as inputs, our Markov chain can be cast as a random walk on a regular language. We obtain a law of large numbers as well as a central limit theorem for this process in which the constants appearing in the limit theorems are expressed in terms of a coupled system of quadratic equations. Our result requires the solution of a simpler problem than those seen previously in the literature, and requires less assumptions.

In the second section, we consider several history-dependent sequences which have attracted recent interest. We primarily study the Ulam-Kac adder, a sequence for which very little is known explicitly except for its first two moments, which have been computed in some generality. Our main contribution is to compute the asymptotic behavior of all moments and obtain bounds on their size. We also provide a semi-analytic formulation of the moment problem which allows them to be computed directly. We discuss many combinatorial interpretations of this sequence and others which, in particular, lead to a novel formula for the first passage times of a related sequence. Connections of these sequences to other areas of mathematics and physics are explored.

**Host:**Jean-Luc Thiffeault and Lisa Everett