2018/10/18 by Damron, Michael, Hanson, Jack, Houdré, Christian +1 · 1 citation
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1810.08270
In first-passage percolation (FPP), one assigns i.i.d.~weights to the edges of the cubic lattice ℤd 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 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 ‖x‖)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).