Shape curvatures and transversal fluctuations in the first passage percolation model

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 $r>0$.