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January 2008 Shape fluctuations are different in different directions
Yu Zhang
Ann. Probab. 36(1): 331-362 (January 2008). DOI: 10.1214/009117907000000213

Abstract

We consider the first passage percolation model on Z2. In this model, we assign independently to each edge e a passage time t(e) with a common distribution F. Let T(u, v) be the passage time from u to v. In this paper, we show that, whenever F(0)<pc, σ2(T((0, 0), (n, 0)))≥C log n for all n≥1. Note that if F satisfies an additional special condition, inf supp (F)=r>0 and F(r)>p⃗c, it is known that there exists M such that for all n, σ2(T((0, 0), (n, n)))≤M. These results tell us that shape fluctuations not only depend on distribution F, but also on direction. When showing this result, we find the following interesting geometrical property. With the special distribution above, any long piece with r-edges in an optimal path from (0, 0) to (n, 0) has to be very circuitous.

Citation

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Yu Zhang. "Shape fluctuations are different in different directions." Ann. Probab. 36 (1) 331 - 362, January 2008. https://doi.org/10.1214/009117907000000213

Information

Published: January 2008
First available in Project Euclid: 28 November 2007

zbMATH: 1130.60093
MathSciNet: MR2370607
Digital Object Identifier: 10.1214/009117907000000213

Subjects:
Primary: 60K35

Rights: Copyright © 2008 Institute of Mathematical Statistics

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Vol.36 • No. 1 • January 2008
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