An Infinite Number of Effectively Infinite Clusters in Critical Percolation

An infinite number of effectively infinite clusters are predicted at the percolation threshold, if “effectively infinite” means that a cluster's mass increases with a positive power of the lattice size L. All these cluster masses increase as LD with the fractal dimension D = d − β/v, while the mass of the rth largest cluster for fixed L decreases as 1/rλ, with λ = D/d in d dimensions. These predictions are confirmed by computer simulations for the square lattice, where D = 91/48 and λ = 91/96.