Abstract
We show that the diameter of a uniformly drawn spanning tree of a connected graph on n vertices which satisfies certain high-dimensionality conditions typically grows like \(\Theta (\sqrt{n})\). In particular this result applies to expanders, finite tori \(\mathbb {Z}_m^d\) of dimension \(d \ge 5\), the hypercube \(\{0,1\}^m\), and small perturbations thereof.
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Notes
The spectral gap of a graph is the difference between 1 and the second largest eigenvalue of the transition matrix of the simple random walk on the graph.
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Acknowledgements
The authors wish to thank Tom Hutchcroft and Yinon Spinka for useful discussions and for their helpful comments on the paper. This research is supported by ERC starting Grant 676970 RANDGEOM and by ISF Grant 1207/15.
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Michaeli, P., Nachmias, A. & Shalev, M. The diameter of uniform spanning trees in high dimensions. Probab. Theory Relat. Fields 179, 261–294 (2021). https://doi.org/10.1007/s00440-020-00999-2
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DOI: https://doi.org/10.1007/s00440-020-00999-2