The divergence of fluctuations for shape in first passage percolation

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, nu) the passage time from the origin to nu for nu epsilon R(d) and [GRAPHICS] It is well known that if p < p(c), there exists a compact shape B(d) subset of R(d) such that for all epsilon > 0, [GRAPHICS] We denote the fluctuations of B(t) from tB(d) and G(t) by [GRAPHICS] 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.