Fluctuations of transverse increments in two-dimensional first passage percolation

Abstract

We consider a model of first passage percolation (FPP) where the nearest-neighbor edges of the standard two-dimensional Euclidean lattice are equipped with random variables. These variables are i.i.d. nonnegative, continuous, and have a finite moment generating function in a neighborhood of 0. We derive consequences about transverse increments of passage times, assuming the model satisfies certain properties. Approximately, the assumed properties are the following: We assume that the standard deviation of the passage time on scale r is of some order $\mathit{\sigma }(r)$, and $\left\{\mathit{\sigma }(r),r>0\right\}$ grows approximately as a power of r. Also, the tails of the passage time distributions for distance r satisfy an exponential bound on a scale $\mathit{\sigma }(r)$ uniformly over r. In addition, the boundary of the limit shape in a neighborhood of some fixed direction θ has a uniform quadratic curvature. By transverse increment we mean the difference between passage times from the origin to a pair of points which are approximately at the direction θ and the direction between the pair of points is the direction of the tangent to the boundary of the limit shape at the direction θ. The main consequence derived is the following. If $\mathit{\sigma }(r)$ varies as $r^{\mathit{\chi }}$ for some $\mathit{\chi }>0$, and ξ is such that $\mathit{\chi }=2\mathrm{\xi }-1$, then the fluctuation of the transverse increment of passage time between a pair of points situated at distance r from each other is of the order of $r^{\mathit{\chi }∕\mathrm{\xi }}$.