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Spectral Dimension and Random Walks on the Two Dimensional Uniform Spanning Tree

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Abstract

We study the simple random walk on the uniform spanning tree on \({\mathbb {Z}^2}\) . We obtain estimates for the transition probabilities of the random walk, the distance of the walk from its starting point after n steps, and exit times of both Euclidean balls and balls in the intrinsic graph metric. In particular, we prove that the spectral dimension of the uniform spanning tree on \({\mathbb {Z}^2}\) is 16/13 almost surely.

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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, #121-1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada

    Martin T. Barlow & Robert Masson

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  1. Martin T. Barlow
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  2. Robert Masson
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Correspondence to Martin T. Barlow.

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Communicated by F.L. Toninelli

Research partially supported by NSERC (Canada) and by the Peter Wall Institute of Advanced Studies (UBC).

Research partially supported by NSERC (Canada).

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Barlow, M.T., Masson, R. Spectral Dimension and Random Walks on the Two Dimensional Uniform Spanning Tree. Commun. Math. Phys. 305, 23–57 (2011). https://doi.org/10.1007/s00220-011-1251-8

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  • DOI: https://doi.org/10.1007/s00220-011-1251-8

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