The travel time to infinity in percolation

Stochastics Seminar
Thursday, September 7, 2017 - 3:05pm
1 hour (actually 50 minutes)
Skiles 006
Georgia Institute of Technology

On the two-dimensional square lattice, assign i.i.d. nonnegative weights to the
edges with common distribution F. For which F is there an infinite self-avoiding path with
finite total weight? This question arises in first-passage percolation, the study of the
random metric space Z^2 with the induced random graph metric coming from the above edge-weights. It has long been known that there is no such infinite path when F(0)<1/2
(there are only finite paths of zero-weight edges), and there is one when F(0)>1/2 (there
is an infinite path of zero-weight edges). The critical case, F(0)=1/2, is considerably
more difficult due to the presence of finite paths of zero-weight edges on all scales. I will
discuss work with W.-K. Lam and X. Wang in which we give necessary and sufficient
conditions on F for the existence of an infinite finite-weight path. The methods involve
comparing the model to another one, invasion percolation, and showing that geodesics
in first-passage percolation have the same first order travel time as optimal paths in an
embedded invasion cluster.