  Consider the first passage percolation model on ${\bf Z}^d$ for $d\geq 2$. 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 $T({\bf 0}, v)$ the
passage time from the origin to $v$ for $v\in {\bf R}^d$ and $$B(t)=\{v\in {\bf
R}^d: T({\bf 0}, v)\leq t\}{and} G(t)=\{v\in {\bf R}^d: ET({\bf 0}, v)\leq
t\}.$$ It is well known that if $p < p_c$, there exists a compact shape
$B_d\subset {\bf R}^d$ such that for all $\epsilon >0$ $$t B_d(1-\epsilon)
\subset {B(t)} \subset tB_d(1+\epsilon){and} G(t)(1-{\epsilon}) \subset {B(t)}
\subset G(t)(1+{\epsilon}) {eventually w.p.1.}$$ We denote the fluctuations of
$B(t)$ from $tB_d$ and $G(t)$ by &&F(B(t), tB_d)=\inf \{l:tB_d(1-{l\over
t})\subset B(t)\subset tB_d(1+{l\over t})\} && F(B(t),
G(t))=\inf\{l:G(t)(1-{l\over t})\subset B(t)\subset G(t)(1+{l\over t})\}.
  The means of the fluctuations $E[F(B(t), tB_d]$ and $E[F(B(t), G(t))]$ have
been conjectured ranging from divergence to non-divergence for large $d\geq 2$
by physicists. In this paper, we show that for all $d\geq 2$ 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.
