This thesis focuses a mathematical model from statistical mechanics called the Arboreal gas. The Arboreal gas on a graph $G$ is Bernoulli bond percolation on $G$ with the conditioning that there are no ``loops". This model is related to other models such as the random cluster measure. We mainly study the Arboreal gas and a related model on the $d$-ary wired tree which is simply the $d$-ary wired tree with the leaves identified as a single vertex. Our first result is finding a distribution on the infinite $d$-ary tree that is the weak limit in height $n$ of the Arboreal gas on the $d$-ary wired tree of height $n$. We then study a similar model on the infinite $d$-ary wired tree which is Bernoulli bond percolation with the conditioning that there is at most one loop. In this model, we only have a partial result which proves that the ratio of the partition function of the one loop model in the wired tree of height $n$ and the Arboreal gas model in the wired tree of height $n$ goes to $0$ as $n \rightarrow \infty$. This allows us to prove certain key quantities of this model is actually the same as analogues of that quantity in the Arboreal gas on the $d$-ary wired tree, under an additional assumption. / Graduate / 10000-01-01
| Identifer | oai:union.ndltd.org:uvic.ca/oai:dspace.library.uvic.ca:1828/14189 |
| Date | 02 September 2022 |
| Creators | Xiao, Ben |
| Contributors | Ray, Gourab |
| Source Sets | University of Victoria |
| Language | English, English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
| Rights | Available to the World Wide Web |
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