Large Scale Properties of the IIIC for 2D Percolation

TL;DR

This paper explores the properties of the incipient infinite cluster in 2D percolation, revealing how the decay rate of the density influences the existence and scale of infinite clusters, including critical cases with logarithmic corrections.

Contribution

It demonstrates the scale dimension of the infinite cluster under power-law decay and analyzes the critical case with logarithmic corrections, extending understanding of 2D percolation.

Findings

Infinite cluster with scale dimension D_H=2-eta\lambda for decay rate \\lambda<1/\\nu

Critical case \\lambda=1/\\nu involves logarithmic corrections

Conditions for the existence of infinite clusters based on decay parameters

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