Shape fluctuations are different in different directions

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, σ 2 (T ((0, 0), (n, 0))) ≥ C log n for all n ≥ 1. Note that if F satisfies an additional special condition, infsupp(F ) = r > 0 and F (r) > ~ pc, 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.