Mathematical Physics Seminar

In First-Passage Percolation, one considers a random medium formed by associating random nonnegative weights to each edge in Z^d. The passage time between two points at large euclidean distance is then defined to be the distance in the random (pseudo)-metric formed by summing the weights along lattice paths between these two points. The first-order behavior of the passage time from the origin to a point at distance N is described by a Law of Large Numbers-type result, the sub additive ergodic theorem. The fluctuations about this first-order limit are supposed to be of much smaller order than N, and in dimensions d=2, there is a precise prediction coming from physics for the limiting distribution of the centered, rescaled passaged time.

Benjamini, Kalai, and Schramm found a way to show that the variance is sub linear, bounded by N/log N, when the edge-weights are Bernoulli. They used an inequality of Talagrand, based on discrete Fourier analysis. Later, Benaim and Rossignol showed how to use log-Sobolev inequalities to extend this to another special class of distributions. With Michael Damron and Jack Hanson, we showed, using the discrete log-Sobolev inequality, that the variance is always bounded by N/log N, regardless of what the distribution is. I will give some background on FPP, explain how functional inequalities can be used to estimate the variance, and give an idea of what we did.