First passage percolation (FPP) is a model of shortest distances through a random medium. In dimension 2, FPP is conjectured to be in the KPZ universality class, and passage times are expected to have variances scaling as n^{2/3}. To date, however, the best known upper bounds are n/log n by Benjamini, Kalai and Schramm. In this talk we will study models FPP which have an additional rotational symmetry. We will discuss methods to establish the chaotic nature of the optimal path which roughly states that after resampling a small fraction of the environment, the new optimal path has a vanishing overlap with the original. By applying this on every length scale we get new upper bounds that improve upon the Benjamini, Kalai and Schramm in the rotationally invariant case.

Joint work with Riddhipratim Basu and Vladas Sidoravicius.