Large Scale Properties of the IIIC for 2D Percolation

Abstract

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 νthe correlation length exponent, we demonstrate an infinite cluster with scale dimension given by D_H=2-βλ. Further, we investigate the critical case λ_c=1/νand show that iterated logarithmic corrections will tip the balance between the possibility and impossibility of an infinite cluster.

19 pages, 2 figures