The component graph of the uniform spanning forest: transitions in dimensions 9 , 10 , 11 , …
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Publication Date
2019-10Journal Title
Probability Theory and Related Fields
ISSN
0178-8051
Publisher
Springer Science and Business Media LLC
Volume
175
Issue
1-2
Pages
141-208
Type
Article
This Version
VoR
Metadata
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Hutchcroft, T., & Peres, Y. (2019). The component graph of the uniform spanning forest: transitions in dimensions 9 , 10 , 11 , …. Probability Theory and Related Fields, 175 (1-2), 141-208. https://doi.org/10.1007/s00440-018-0884-3
Abstract
We prove that the uniform spanning forests of $\mathbb{Z}^d$ and
$\mathbb{Z}^{\ell}$ have qualitatively different connectivity properties
whenever $\ell >d \geq 4$. In particular, we consider the graph formed by
contracting each tree of the uniform spanning forest down to a single vertex,
which we call the component graph. We introduce the notion of ubiquitous
subgraphs and show that the set of ubiquitous subgraphs of the component graph
changes whenever the dimension changes and is above $8$. To separate dimensions
$5,6,7,$ and $8$, we prove a similar result concerning ubiquitous
subhypergraphs in the component hypergraph. Our result sharpens a theorem of
Benjamini, Kesten, Peres, and Schramm, who proved that the diameter of the
component graph increases by one every time the dimension increases by four.
Sponsorship
Microsoft Research
Identifiers
External DOI: https://doi.org/10.1007/s00440-018-0884-3
This record's URL: https://www.repository.cam.ac.uk/handle/1810/287212
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