PC Seminar: Superconcentration, chaos and multiple valleys in first passage percolation

Abstract:We consider a dynamical version of first-passage percolation on the d-dimensional integer lattice with i.i.d. edge weights, where edge weights are resampled independently in time. Let T(n) denote the passage time from the origin to the site n steps along the first coordinate axis at time t=0, and let \mu(t) denote the expected overlap between the time minimizing paths at time 0 and t>0. We show that a subdiffusive behaviour of T(n) is equivalent to a chaotic behavior of the time minimizing paths, manifested in that \mu(t)=o(n). Known bounds for Var(T(n)) thus imply that indeed \mu(t)=o(n) for t>0. As a consequence we show that there are many almost disjoint paths with almost optimal passage time. This gives evidence that earlier work by Sourav Chatterjee for certain Gaussian disordered systems reflects a more general principle.  

This is a seminar in our seminar series on Probability and Combinatorics (PC).