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.
Categories: math.PR