Random forests on trees

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dc.contributor.authorXiao, Ben
dc.contributor.supervisorRay, Gourab
dc.date.accessioned2022-09-03T00:10:36Z
dc.date.copyright2022en_US
dc.date.issued2022-09-02
dc.degree.departmentDepartment of Mathematics and Statisticsen_US
dc.degree.levelMaster of Science M.Sc.en_US
dc.description.abstractThis 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.en_US
dc.description.embargo10000-01-01
dc.description.scholarlevelGraduateen_US
dc.identifier.bibliographicCitationGourab Ray and Ben Xiao. Forests on wired regular trees. ALEA, to appear, 2022. arXiv:2108.04287en_US
dc.identifier.urihttp://hdl.handle.net/1828/14189
dc.languageEnglisheng
dc.language.isoenen_US
dc.rightsAvailable to the World Wide Weben_US
dc.subjectprobabilityen_US
dc.subjectpercolationen_US
dc.subjectarboreal gasen_US
dc.subjecttreesen_US
dc.titleRandom forests on treesen_US
dc.typeThesisen_US

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