This dissertation proves several results for first passage percolation on graphs of polynomial growth.The class of limit shapes for first passage percolation with stationary weights on Cayley graphs of virtually nilpotent groups is characterized. Then strict monotonicity theorems for independent first passage percolation on graphs of polynomial growth and quasi-trees are given. Specifically, for such graphs, when we compare the expected passage time metrics with respect to two different weight measures, strict stochastic domination of weight measures implies (an analogue of) strict inequality of the associated ``time constants'' as long as the dominating measure satisfies an appropriate subcriticality condition. This is proven by showing that in the subcritical regime, long geodesics ``use all possible weights linearly often in expectation,'' which is a result of independent interest. Moreover, a similar strict monotonicity theorem with respect to variability of measures holds for such graphs if and only if the graphs satisfy a geometric condition we call admitting detours. Lastly, we show that for Cayley graphs of virtually nilpotent groups, in the supercritical regime, there is a nontrivial ``percolation cone'' where strict monotonicity with respect to stochastic domination fails; that is, the subcriticality assumption in our strict monotonicity theorems is necessary.

Creator

• Gorski, Christian Stanley Edward

DOI

• https://doi.org/10.21985/n2-abd6-sb70

Subject

• Mathematics

Language

• en

Alternate Identifier

• http://dissertations.umi.com/northwestern:16688
• etdadmin_upload_1009568

Keyword

• coarse geometry
• first-passage percolation
• Cayley graphs

Date created

• 2023-01-01

Resource type

• Dissertation

Rights statement

• In Copyright