Concentration of Geodesics in Directed Bernoulli Percolation

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 $n^{(2\kappa +2)\mathrm{/}(2\kappa +3)}\sqrt{\ln n}$, where $1\le \kappa <\mathrm{\infty }$ 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.