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Geometry of Uniform Spanning Forest Components in High Dimensions

Part of: Graph theory Geometric probability and stochastic geometry Other generalizations

Published online by Cambridge University Press:  07 January 2019

Martin T. Barlow  and
Antal A. Járai
Martin T. Barlow
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada Email: barlow@math.ubc.ca
Antal A. Járai
Affiliation:
Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK Email: a.jarai@bath.ac.uk
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Abstract

We study the geometry of the component of the origin in the uniform spanning forest of $\mathbb{Z}^{d}$ and give bounds on the size of balls in the intrinsic metric.

Keywords

uniform spanning forestloop-erased random walk

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 05C05: Trees 31C20: Discrete potential theory and numerical methods
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 6 , December 2019 , pp. 1297 - 1321
Copyright
© Canadian Mathematical Society 2018 

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Footnotes

Research partially supported by NSERC (Canada).

References

Aizenman, M. and Newman, C. M., Tree graph inequalities and critical behavior in percolation models . J. Stat. Phys. 36(1984), nos. 1/2, 107–143. https://doi.org/10.1007/BF01015729.Google Scholar
Barlow, M. T. and Masson, R., Exponential tail bounds for loop-erased random walk in two dimensions . Ann. Probab. 38(2010), no. 6, 2379–2417. https://doi.org/10.1214/10-AOP539.Google Scholar
Barlow, M. T. and Masson, R., Spectral dimension and random walks on the two dimensional uniform spanning tree . Comm. Math. Phys. 305(2011), 23–57. https://doi.org/10.1007/s00220-011-1251-8.Google Scholar
Benjamini, I., Lyons, R., Peres, Y., and Schramm, O., Uniform spanning forests . Ann. Probab. 29(2001), 1–65.Google Scholar
Bhupatiraju, S., Hanson, J., and Járai, A. A., Inequalities for critical exponents in d-dimensional sandpiles . Electron. J. Probab. 22(2017), paper no. 85, 1–51. https://doi.org/10.1214/17-EJP111.Google Scholar
Lawler, Gregory F., A self-avoiding random walk . Duke Math. J. 47(1980), no. 3, 655–693. https://doi.org/10.1215/S0012-7094-80-04741-9.Google Scholar
Lawler, Gregory F., Intersections of random walks . Probability and its Applications . Birkhäuser Boston, Boston, MA, 1991.Google Scholar
Lawler, Gregory F., The logarithmic correction for loop-erased walk in four dimensions . In: Proceedings of the Conference in Honor of Jean-Pierre Kahane . J. Fourier Anal. Appl. (1995) Special Issue, 347–361.Google Scholar
Lawler, Gregory F., Loop-erased random walk . In: Perplexing problems in probability . Progress in probability, 44. Birkhäuser Boston, Boston, MA, 1999.Google Scholar
Lawler, Gregory F. and Limic, Vlada, Random walk: a modern introduction . Cambridge University Press, 2009.Google Scholar
Lyons, R., Morris, B. J., and Schramm, O., Ends in uniform spanning forests . Electron. J. Probab. 13(2008), no. 58, 1702–1725. https://doi.org/10.1214/EJP.v13-566.Google Scholar
Lyons, R. and Peres, Y., Probability on trees and networks . Cambridge Series in Statistical and Probabilistic Mathematics, 42. Cambridge University Press, New York, 2016.Google Scholar
Masson, Robert, The growth exponent for planar loop-erased random walk . Electron. J. Probab. 14(2009), no. 36, 1012–1073. https://doi.org/10.1214/EJP.v14-651.Google Scholar
Pemantle, R., Choosing a spanning tree for the integer lattice uniformly . Ann. Probab. 19(1991), no. 4, 1559–1574. https://doi.org/10.1214/aop/1176990223.Google Scholar
Wilson, D. B., Generating spanning trees more quickly than the cover time . Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing . ACM, New York, 1996, pp. 296–303.Google Scholar