The divergence of fluctuations for shape in first passage percolation |
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Authors: | Yu Zhang |
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Institution: | (1) Department of Mathematics, University of Colorado, Colorado Springs, CO 80933, USA |
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Abstract: | We consider the first passage percolation model on Z
d
for d ≥ 2. In this model, we assign independently to each edge the value zero with probability p and the value one with probability 1−p. We denote by T(0, ν) the passage time from the origin to ν for ν ∈ R
d
and
It is well known that if p < p
c
, there exists a compact shape B
d
⊂ R
d
such that for all
> 0,
t
B
d
(1 −
) ⊂ B(t) ⊂ tB
d
(1 +
) and G(t)(1 −
) ⊂ B(t) ⊂ G(t)(1 +
) eventually w.p.1. We denote the fluctuations of B(t) from tB
d
and G(t) by
In this paper, we show that for all d ≥ 2 with a high probability, the fluctuations F(B(t), G(t)) and F(B(t), tB
d
) diverge with a rate of at least C log t for some constant C. The proof of this argument depends on the linearity between the number of pivotal edges of all minimizing paths and the
paths themselves. This linearity is also independently interesting.
Research supported by NSF grant DMS-0405150 |
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Keywords: | 60K 35 |
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