First passage percolation in many dimensions

Abstract

We consider first passage percolation on a d-dimensional hypercubical lattice. The passage time for each bond is an independent exponentially distributed random variable with mean l. For large separations, the limiting ratio of expected minimum passage time between two points along an axis and their separation is a constant μ d , and we show that lim d→∞ μ d /ln d = 1 2 .