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 \(p_ c\), 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 \(\xi (p)\sim | p-p_ c|^ v\), then the first pasage time constant satisfies \(\mu (p)\sim | p-p_ c|^ v\). At \(p_ c\), where it has been asserted that the first passage time from 0 to x scales as \(| x|\) to a power \(\psi\) with \(0<\psi <1\), we show that the passage times grow like log\(| x|\), i.e., the fluid spreads exponentially rapidly.
scientific article; zbMATH DE number 4024482
scientific article