Shape fluctuations are different in different directions

We consider the first passage percolation model on $\mathbf{Z}^2$. In this model, we assign independently to each edge $e$ a passage time $t(e)$ with a common distribution $F$. Let $T(u,v)$ be the passage time from $u$ to $v$. In this paper, we show that, whenever $F(0)<p_c$, $\sigma^2(T((0,0),(n,0)))\geq C\log n$ for all $n\geq1$. Note that if $F$ satisfies an additional special condition, $\inf\operatorname {supp}(F)=r>0$ and $F(r)>\vec{p}_c$, it is known that there exists $M$ such that for all $n$, $\sigma^2(T((0,0),(n,n)))\leq M$. These results tell us that shape fluctuations not only depend on distribution $F$, but also on direction. When showing this result, we find the following interesting geometrical property. With the special distribution above, any long piece with $r$-edges in an optimal path from $(0,0)$ to $(n,0)$ has to be very circuitous.