Lower bounds for fluctuations in first-passage percolation for general distributions

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 parallel to x parallel to. This result was found in the '90s and there has not been any improvement since. In this paper, we address 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 parallel to x parallel to)(1/2) , and similar statements for FPP in thin cylinders. Such statements have been proved for special edge-weight distributions, and here we obtain such bounds for general edge-weight distributions. The methods involve inducing a fluctuation in the number of edges in a box whose weights are of "hi-mode" (large).