Georgia Institute of Technology College of Sciences

In first-passage percolation (FPP), one assigns i.i.d. weights to the edges of the cubic lattice Z^d and analyzes the induced weighted graph metric. If T(x,y) is the distance between vertices x and y, then a primary question in the model is: what is the order of the fluctuations of T(0,x)? It is expected that the variance of T(0,x) grows like the norm of x to a power strictly less than 1, but the best lower bounds available are (only in two dimensions) of order \log |x|. This result was found in the '90s and there has not been any improvement since. In this talk, we discuss the problem of getting stronger fluctuation bounds: to show that T(0,x) is with high probability not contained in an interval of size o(\log |x|)^{1/2}, and similar statements for FPP in thin cylinders. Such a statement has been proved for special edge-weight distributions by Pemantle-Peres ('95) and Chatterjee ('17). In work with J. Hanson, C. Houdré, and C. Xu, we extend these bounds to general edge-weight distributions. I will explain some of the methods we use, including an old and elementary "small ball" probability result for functions on the hypercube.