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 pcpc with an inverse power, λλ, 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/ν1/ν, with νν the correlation length exponent, we demonstrate an infinite cluster with scale dimension given by DH=2−βλDH=2−βλ. Further, we investigate the critical case λc=1/νλc=1/ν and show that iterated logarithmic corrections will tip the balance between the possibility and impossibility of an infinite cluster.