Title: Lower bounds for fluctuations in first-passage percolation
Speaker: Michael Damron (Georgia Tech)
Abstract: 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 ongoing work with J. Hanson, C. Houdré, and C. Xu, we aim to extend these bounds to general edge-weight distributions. I will explain some of the methods we are using, including an old and elementary ``small ball'' probability result for functions on the hypercube.
