Dynamical first-passage percolation
In first-passage percolation (FPP), we place i.i.d. nonnegative
weights on the edges of the cubic lattice Z^d and study the induced weighted
graph metric T = T(x,y). Letting F be the common distribution function of the
weights, it is known that if F(0) is less than the threshold p_c for Bernoulli
percolation, then T(x,y) grows like a linear function of the distance |x-y|. In
2015, Ahlberg introduced a dynamical model of first-passage percolation, in
which the weights are resampled according to Poisson clocks, and considered the
growth of T(x,y) as time varies. He showed that when F(0) < p_c, the model has
no ``exceptional times'' at which the order of the growth is anomalously large
or small. I will discuss recent work with J. Hanson, D. Harper, and W.-K. Lam,
in which we study this question in two dimensions in the critical regime, where
F(0) = p_c, and T(x,y) typically grows sublinearly. We find that the existence
of exceptional times depends on the behavior of F(x) for small positive x, and
we characterize the dimension of the exceptional sets for all but a small class
of such F.