Shape curvatures and transversal fluctuations in the first passage percolation model

Abstract

We consider the first passage percolation model on the square lattice. In this model, $\{t(e):eanedgeof{\mathbf{Z}}^2\}$ is an independent identically distributed family with a common distribution $F$. We denote by $T(0,v)$ the passage time from the origin to $v$ for $v\in {\mathbf{R}}^2$ and $B(t)=\{v\in {\mathbf{R}}^d:T(0,v)\le t\}.$ It is well known that if $F(0)<p_c$, there exists a compact shape ${\mathbf{B}}_F\subset {\mathbf{R}}^2$ such that for all $\epsilon >0$, $t{\mathbf{B}}_F(1-\epsilon )\subset B(t)\subset t{\mathbf{B}}_F(1+\epsilon )$, 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 $\xi (u)\le 1-\max \{{\kappa }^{-}(u)/2,{\kappa }^{+}(u)/2\}$ for all shape boundary points $u$. To pursue a curvature on ${\mathbf{B}}_F$, 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 $r>0$, $\xi ((r,{\theta }_p^{\pm }))=0.5$ and $\xi ((r,\theta ))=1$ for all ${\theta }_p^{-}<\theta <{\theta }_p^{+}$ and $r>0$.

29 pages and 5 figures

References in corpus (1)

• Singularity points for first passage percolation