Large scale properties of the IIIC for 2D percolation

We reinvestigate the 2D problem of the inhomogeneous incipient infinite cluster where, in an independent percolation model, the density decays to p c with an inverse power, λ , of the distance to the origin. Assuming the existence of critical exponents (as is known in the case of the triangular site lattice) if the power is less than 1 / ν , with ν the correlation length exponent, we demonstrate an infinite cluster with scale dimension given by D H = 2 − β λ . Further, we investigate the critical case λ c = 1 / ν and show that iterated logarithmic corrections will tip the balance between the possibility and impossibility of an infinite cluster.