Critical behavior of the two-dimensional first passage time

We study the two-dimensional first passage problem in which bonds have zero and unit passage times with probability p and 1-p, respectively. We prove that as the zero-time bonds approach the percolation threshold pc, the first passage time exhibits the same critical behavior as the correlation function of the underlying percolation problem. In particular, if the correlation length obeys ξ(p) ∼|p-pc|^-v, then the first passage time constant satisfies μ(p)∼|p-pc|^v. At pc, where it has been asserted that the first passage time from 0 to x scales as |x| to a power ψ with 0<ψ<1, we show that the passage times grow like log |x|, i.e., the fluid spreads exponentially rapidly. © 1986 Plenum Publishing Corporation.

Duke Scholars

Author Richard Timothy Durrett Mathematics