The divergence of fluctuations for the shape on first passage percolation

Details The divergence of fluctuations for the shape on first passage percolation The divergence of fluctuations for the shape on first passage percolation

2005-01-06

arxiv.org/abs/math/0501095v1

Mathematics

Probability

Scientific paper

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.

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