  For directed Bernoulli last passage percolation with i.i.d.~weights on
vertices over a $n\times n$ grid and for $n$ large enough, the geodesics are
shown to be concentrated in a cylinder, centered on the main diagonal and of
width of order $n^{(2\kappa+2)/(2\kappa+3)}\sqrt{\ln n}$, where
$1\le\kappa<\infty$ is the curvature power of the shape function at $(1,1)$.
The methodology of proof is robust enough to also apply to directed Bernoulli
first passage site percolation, and further to longest common subsequences in
random words.
