Date of Award

2018

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Graduate Group

Mathematics

First Advisor

Robin Pemantle

Abstract

The frog model refers to a system of interacting random walks on a rooted graph. It begins with a single active particle (i.e. frog) at the root, and some distribution of inactive particles among the non-root vertices. Active particles perform discrete-time-nearest neighbor random walks on the graph and activate passive particles upon landing on them. Once activated, the trajectories of distinct particles are independent. In this thesis, we examine the frog model in several different environments, and in each case, work towards identifying conditions under which the model is recurrent, transient, or neither, in terms of the number of distinct frogs that return to the root. We begin by looking at a continuous analog of the model on $\R$ in chapter 2, following which I analyze several different models on $\Z$ in chapters 2 and 3. I then conclude by examining the frog model on trees in chapter 4. The strategy used for analyzing the model on $\R$ primarily revolves around looking at a closely related birth-death process. Somewhat similar techniques are then used for the model on $\Z$. For the frog model on trees I exploit some of the self-similarity properties of the model in order to construct an operator which is used to analyze its long term behaviour, as it relates to questions of recurrence vs. transience.

Additional Files

JR_Frog.py (1 kB)

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