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Published October 2019 | Submitted + Published
Journal Article Open

The component graph of the uniform spanning forest: transitions in dimensions 9,10,11, ...


We prove that the uniform spanning forests of Z^d and Z^ℓ have qualitatively different connectivity properties whenever ℓ > d ≥ 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.

Additional Information

© The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Received: 30 June 2017 / Revised: 4 October 2018 / Published online: 23 November 2018. This work was carried out while T.H. was an intern at Microsoft Research, Redmond. T.H. thanks Mathav Murugan for many useful discussions on heat kernel estimates. We thank Omer Angel for his comments on an earlier draft of this manuscript, and thank the anonymous referee for many helpful comments and corrections.

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Published - Hutchcroft-Peres2019_Article_TheComponentGraphOfTheUniformS.pdf

Submitted - 1702.05780.pdf


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August 19, 2023
October 23, 2023