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Arm Events in Two-Dimensional Invasion Percolation

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Abstract

We compare the probabilities of “arm events” in two-dimensional invasion percolation to those in critical percolation. Arm events are defined by the existence of a prescribed “color sequence” of invaded and non-invaded connections from the origin to distance n. We find that, for sequences of a particular form, arm probabilities in invasion percolation and critical percolation are comparable, uniformly in n, while they differ by a power of n for all others. A corollary of our results is the existence, on the triangular lattice, of arm exponents for invasion percolation, for any color sequence with at least two open (invaded) entries.

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Acknowledgements

The research of M. D. is supported by NSF Grant DMS-0901534 and an NSF CAREER Grant. The research of J. H. is supported by NSF Grant DMS-1612921. The research of P. S. is supported by the Center for Mathematical Sciences and Applications at Harvard University, and NSF Grant DMS-1811093.

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Authors and Affiliations

  1. Georgia Tech, Atlanta, USA

    Michael Damron

  2. City College, CUNY, New York, USA

    Jack Hanson

  3. Cornell University, Ithaca, New York, USA

    Philippe Sosoe

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  1. Michael Damron
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  2. Jack Hanson
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  3. Philippe Sosoe
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Correspondence to Michael Damron.

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Damron, M., Hanson, J. & Sosoe, P. Arm Events in Two-Dimensional Invasion Percolation. J Stat Phys 173, 1321–1352 (2018). https://doi.org/10.1007/s10955-018-2149-z

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  • DOI: https://doi.org/10.1007/s10955-018-2149-z

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