Is there a lower critical dimension for chemical distance?

Estimates of the fractal dimension phi for 'chemical distance' (shortest-path distance) between points on a percolation cluster are inferred from computations of the first-passage velocity nu (p) on square (d=2) and simple cubic (d=3) bond lattices with bonds randomly assigned time delay b with probability p, otherwise time delay a>>b. The computations, implemented on strips in a manner analogous to the transfer matrix for conductivity, yield estimates of phi based on a new scaling law, nu (p c ) approximately a -1 (a/b) 1 phi /, where p c is the percolation threshold for b-bonds. For d=2, the authors obtain phi =1.021+or-0.005, which is significantly lower than previous estimates. For d=3, they obtain phi =1.26+or-0.06, in agreement with the Havlin-Nossal proposal phi =d-(1+ beta )/ nu . Their results do not exclude the possibility that phi =1 for d=2, indicating that the chemical distance may be non-fractal below some lower critical dimension between two and three.