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Probability and Phase Transition pp 261–264Cite as

Planar First-Passage Percolation Times are not Tight

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Part of the book series: NATO ASI Series ((ASIC,volume 420))

Abstract

Let the edges of ℤ2 be assigned independent, identically distributed passage times that are exponentials of mean one, and let T(0, n) denote the resulting first-passage time from the origin to the point (0, n). We show that T(0, n) is not tight around its median. A fractional power lower bound for the dispersion of T(0, n) may be obtained by combining this method with that of Newman and Piza (1993).

Supported in part by NSF grant DMS 93-00191, by a Sloan Foundation Fellowship, and by a Presidential Faculty Fellowship.

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References

  • Alexander, K. (1992). Fluctuations in the boundary of the wet region for first-passage percolation in two and three dimensions. Preprint.

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  • Cox, J. T. and Durrett, R. (1981). Some limit theorems for percolation processes with necessary and sufficient conditions. Annals of Probability 9, 583–603.

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  • Kesten, H. (1986). Aspects of first passage percolation. Lecture Notes in Mathematics 1180, 125–264, Springer, Berlin.

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  • Kesten, H. (1992). On the speed of convergence in first passage percolation. Annals of Applied Probability, to appear.

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  • Newman, C. and Piza, M. (1993). Divergence of shape fluctuations in two dimensions. Preprint.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin, Madison, WI, 53706, USA

    R. Pemantle

  2. Department of Statistics, University of California, Berkeley, CA, 94720, USA

    Y. Peres

Authors
  1. R. Pemantle
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  2. Y. Peres
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Editor information

Editors and Affiliations

  1. Statistical Laboratory, University of Cambridge, Cambridge, UK

    Geoffrey Grimmett

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© 1994 Springer Science+Business Media Dordrecht

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Pemantle, R., Peres, Y. (1994). Planar First-Passage Percolation Times are not Tight. In: Grimmett, G. (eds) Probability and Phase Transition. NATO ASI Series, vol 420. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8326-8_16

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  • DOI: https://doi.org/10.1007/978-94-015-8326-8_16

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4370-2

  • Online ISBN: 978-94-015-8326-8

  • eBook Packages: Springer Book Archive

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