Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp01j3860b04r
 Title: The Phase Shifting Coalescing-Branching Random Walk Authors: Blitz, Jackson Advisors: Nestoridi, Evita Department: Mathematics Class Year: 2021 Abstract: The Coalescing-Branching random (CoBra) walk is shown to exhibit a phase transition on an infinite $$n$$-ary homogenous tree, denoted $$T_n$$, as the branching factor $$k$$ is varied. The CoBra walk on $$T_n$$ does not infect the root infinitely often for $$k = O(\sqrt{n})$$. The CoBra walk on $$T_n$$ has a positive probability of infecting the root infinitely often for $$k = \Omega(\sqrt{n})$$. We provide leading lemmas about the $$\varepsilon$$-biased walk and inverse-degree biased walks which can help find the regions of recurrence for the CoBra walk on other infinite graphs. Further exposition on two general bounds of the 2-CoBra walk cover time of general finite graphs is provided. Miscellaneous results about the expected infections of a given CoBra particle and a few results about the branching random walk are also proven. Additionally, we provide CoBra walk simulation code on $$T_n$$ and $$\mathbb{Z}^d$$. We finally conjecture the 2-CoBra walk on the integer lattice $$\mathbb{Z}^d$$ is recurrent for any dimension $$d$$. URI: http://arks.princeton.edu/ark:/88435/dsp01j3860b04r Type of Material: Princeton University Senior Theses Language: en Appears in Collections: Mathematics, 1934-2021