Georgia Institute of Technology College of Sciences

In first-passage percolation (FPP), one places random non-negative weights on the edges of a graph and considers the induced weighted graph metric. Of particular interest is the case where the graph is Z^d, the standard d-dimensional cubic lattice, and many of the questions involve a comparison between the asymptotics of the random metric and the standard Euclidean one. In this talk, I will survey some of my recent work on the order of fluctuations of the metric, focusing on (a) lower bounds for the expected distance and (b) our recent sublinear bound for the variance for edge-weight distributions that have 2+log moments, with corresponding concentration results. This second work addresses a question posed by Benjamini-Kalai-Schramm in their celebrated 2003 paper, where such a bound was proved for only Bernoulli weights using hypercontractivity. Our techniques draw heavily on entropy methods from concentration of measure.