Shape fluctuations are different in different directions

Details Shape fluctuations are different in different directions Shape fluctuations are different in different directions

2005-12-23

arxiv.org/abs/math/0512537v3

Annals of Probability 2008, Vol. 36, No. 1, 331-362

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/009117907000000213 the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117907000000213

We consider the first passage percolation model on $\mathbf{Z}^2$. 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 $F(0)0$ and $F(r)>\vec{p}_c$, it is known that there exists $M$ such that for all $n$, $\sigma^2(T((0,0),(n,n)))\leq M$. 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.

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