Probability Seminar

Jack HansonThe City College of New York
The boundary in first-passage percolation

Monday, November 27, 2017 - 4:00pm
Malott 406

First-passage percolation can be thought of as inducing a growth process on the graph Z^d. The "infected region" B(t) at time t is known to have diameter linearly growing in t. We describe work showing that, if the i.i.d. edge weights generating the model satisfy a weak moment condition, the cardinality of the boundary of B(t) is of order t^{d-1} for most times. For heavy-tailed variables, the boundary is larger, due to the presence of small holes, but the exterior boundary is still of order t^{d-1}. Under a further (but standard) unproven "curvature" assumption, we show that the boundary has cardinality at most (log t)^C t^{d-1} for all large t.