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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alexander, K. (1992). Fluctuations in the boundary of the wet region for first-passage percolation in two and three dimensions. Preprint.
Cox, J. T. and Durrett, R. (1981). Some limit theorems for percolation processes with necessary and sufficient conditions. Annals of Probability 9, 583–603.
Durrett, R. and Liggett, T. (1981). The shape of the limit set in Richardson’s growth model. Annals of Probability 9, 186–193.
Kesten, H. (1986). Aspects of first passage percolation. Lecture Notes in Mathematics 1180, 125–264, Springer, Berlin.
Kesten, H. (1992). On the speed of convergence in first passage percolation. Annals of Applied Probability, to appear.
Newman, C. and Piza, M. (1993). Divergence of shape fluctuations in two dimensions. Preprint.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
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
Download citation
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