Abstract
Considering the wired uniform spanning forest on a nonunimodular transitive graph, we show that almost surely each tree of the wired uniform spanning forest is light. More generally we study the tilted volumes for the trees in the wired uniform spanning forest.
Regarding the free uniform spanning forest, we consider several families of nonunimodular transitive graphs. We show that the free uniform spanning forest is the same as the wired one on Diestel–Leader graphs. For grandparent graphs, we show that the free uniform spanning forest is connected and has branching number bigger than one. We also show that each tree of the free uniform spanning forest is heavy and has branching number bigger than one on a free product of a nonunimodular transitive graph with one edge when the free uniform spanning forest is not the same as the wired.
Funding Statement
Pengfei Tang was partially supported by the Natural Science Foundation under grant DMS-1612363.
Acknowledgments
We thank Russell Lyons for many helpful discussions and suggestions. We thank Ádám Timár and Gábor Pete for sharing their recent work [22] and pointing out the branching number part of Remark 5.12. We thank the referees for their helpful comments and suggestions. Especially one referee pointed out a serious mistake in the previous version and proposed a useful rescue, namely the tree-graph inequality method. This also leads to sharper bounds on certain estimates in Section 4. The study of in Proposition C.1 was also suggested by the referee. Question 5.11 was suggested by another referee.
Citation
Pengfei Tang. "Weights of uniform spanning forests on nonunimodular transitive graphs." Electron. J. Probab. 26 1 - 62, 2021. https://doi.org/10.1214/21-EJP709
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