Tails of the empirical distribution on a geodesic in first-passage percolation
First-passage percolation defines a random pseudo-metric on Z^d by attaching to each nearest-neighbor edge of the lattice a non-negative weight. Geodesics are paths which realize the distance between sites. This project considers the question of what the environment looks like on a geodesic through the lens of the empirical distribution on that geodesic when the weights are i.i.d.. We obtain upper and lower tail bounds for the upper and lower tails which quantify and limit the intuitive statement that the typical weight on a geodesic should be small compared to the marginal distribution of an edge weight.
Based on joint work-in-progress with Michael Damron, Wai-Kit Lam, and Xiao Shen which was started at the AMS MRC on Spatial Stochastic Models in 2019