Shape curvatures and transversal fluctuations in the first passage percolation model

Details Shape curvatures and transversal fluctuations in the first passage percolation model Shape curvatures and transversal fluctuations in the first passage percolation model

2007-01-24

arxiv.org/abs/math/0701689v1

Mathematics

Probability

29 pages and 5 figures

Scientific paper

We consider the first passage percolation model on the square lattice. In this model, $\{t(e): e{an edge of}{\bf Z}^2 \}$ is an independent identically distributed family with a common distribution $F$. We denote by $T({\bf 0}, v)$ the passage time from the origin to $v$ for $v\in {\bf R}^2$ and $B(t)=\{v\in {\bf R}^d: T({\bf 0}, v)\leq t\}.$ It is well known that if $F(0) < p_c$, there exists a compact shape ${\bf B}_F\subset {\bf R}^2$ such that for all $\epsilon >0$, $t {\bf B}_F(1-\epsilon) \subset {B(t)} \subset t{\bf B}_F(1+\epsilon)$, eventually with a probability 1. For each shape boundary point $u$, we denote its right- and left-curvature exponents by $\kappa^+(u)$ and $\kappa^-(u)$. In addition, for each vector $u$, we denote the transversal fluctuation exponent by $\xi(u)$. In this paper, we can show that $\xi(u) \leq 1-\max\{\kappa^-(u)/2, \kappa^+(u)/2\}$ for all shape boundary points $u$. To pursue a curvature on ${\bf B}_F$, we consider passage times with a special distribution infsupp$(F)=l$ and $F(l)=p > \vec{p}_c$, where $l$ is a positive number and $\vec{p}_c$ is a critical point for the oriented percolation model. With this distribution, it is known that there is a flat segment on the shape boundary between angles $0< \theta_p^- < \theta_p^+< 90^\circ$. In this paper, we show that the shape are strictly convex at the directions $\theta_p^\pm$. Moreover, we also show that for all $r>0$, $\xi((r, \theta^\pm_p)) = 0.5$ and $\xi((r, \theta)) =1$ for all $\theta_p^- <\theta< \theta_p^+$ and $r>0$.

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