  We consider a model of first passage percolation (FPP) where the
nearest-neighbor edges of the standard two-dimensional Euclidean lattice are
equipped with random variables. These variables are i.i.d.\, nonnegative,
continuous, and have a finite moment generating function in a neighborhood of
$0$. We derive consequences about transverse increments of passage times,
assuming the model satisfies certain properties. Approximately, the assumed
properties are the following: We assume that the standard deviation of the
passage time on scale $r$ is of some order $\sigma(r)$, and $\left\{\sigma(r),
r > 0\right\}$ grows approximately as a power of $r$. Also, the tails of the
passage time distributions for distance $r$ satisfy an exponential bound on a
scale $\sigma(r)$ uniformly over $r$. In addition, the boundary of the limit
shape in a neighborhood of some fixed direction $\theta$ has a uniform
quadratic curvature. By transverse increment we mean the difference of passage
times from the origin to a pair of points which are located as follows: they
are approximately in the same direction, say $\theta$, from the origin; the
direction of one of them from the other is the direction of the tangent of the
boundary of the limit shape at the point on the limit shape in the direction
$\theta$. The main consequence derived is the following. If $\sigma(r)$ varies
as $r^\chi$ for some $\chi>0$, and $\xi$ is such that $\chi=2\xi-1$, then the
fluctuation of the transverse increment of passage time between a pair of
points situated at distance $r$ from each other is of the order of
$r^{\chi/\xi}$.
