First passage percolation: Scaling and critical exponents

Abstract

Motivated by a percolation analysis of neural conduction, we write a scaling form for the expected length of the shortest path between two sites in the infinite cluster. $\ensuremath{\psi}$ is the fractal dimension of this path over distances small compared to a correlation length. Over long distances, path "tortuosity" and effective conduction velocity scale with a new critical exponent $\ensuremath{\theta}$. The scaling argument provides the first analytic expression for an effective velocity in a "first passage" percolation problem.