Abstract
We study large scale properties of the uniform spanning tree in ℤd. It is shown that inside the n-cube of ℤd, there are o(n d) vertices with large scale degree 2. In 2D the number of vertices with large scale degree 3 is uniformly bounded and there are no vertices with large scale degree bigger than 3. Also, it is shown that the number of spanning clusters for the uniform spanning tree in a square is tight as the square grows, and that this is not the case in dimensions 4 and higher. A similar transition is a known conjecture for critical Bernoulli percolation.
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© 1999 Birkhäuser Boston
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Benjamini, I. (1999). Large Scale Degrees and the Number of Spanning Clusters for the Uniform Spanning Tree. In: Bramson, M., Durrett, R. (eds) Perplexing Problems in Probability. Progress in Probability, vol 44. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2168-5_10
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DOI: https://doi.org/10.1007/978-1-4612-2168-5_10
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-7442-1
Online ISBN: 978-1-4612-2168-5
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