Concentration of Geodesics in Directed Bernoulli Percolation

Abstract

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.

23 pages, 2 figures. The proof is simplified in this version and some bibliographic and typographic corrections are included

References in corpus (1)

• A Central Limit Theorem for the Length of the Longest Common Subsequences in Random Words

Cited by in corpus (2)

• Lower bounds for fluctuations in first-passage percolation for general distributions
• Power Lower Bounds for the Central Moments of the Last Passage Time for Directed Percolation in a Thin Rectangle