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