Michael Damron

Critical first-passage percolation in two dimensions

In 2d first-passage percolation (FPP), we place nonnegative i.i.d. weights (t_e) on the edges of Z^2 and study the induced weighted graph pseudometric T = T(x,y). If we denote by p = P(t_e = 0), then there is a transition in the large-scale behavior of the model as p varies from 0 to 1. When p < 1/2, T(0,x) grows linearly in x, and when p > 1/2, it is stochastically bounded. The critical case, where p = 1/2, is more subtle, and the sublinear growth of T(0,x) depends on the behavior of the distribution function of t_e near zero. I will discuss my work over the past few years that (a) determines the exact rate of growth of T(0,x), (b) determines the ``time constant'' for the site-FPP model on the triangular lattice and, more recently (c) studies the growth of T(0,x) in a dynamical version of the model, where weights are resampled according to independent exponential clocks. These are joint works with J. Hanson, D. Harper, W.-K. Lam, P. Tang, and X. Wang.