Scaling of cluster heterogeneity in percolation transitions

We investigate a critical scaling law for the cluster heterogeneity H in site and bond percolations in d-dimensional lattices with d=2,",6. The cluster heterogeneity is defined as the number of distinct cluster sizes. As an occupation probability p increases, the cluster size distribution evolves from a monodisperse distribution to a polydisperse one in the subcritical phase, and back to a monodisperse one in the supercritical phase. We show analytically that H diverges algebraically, approaching the percolation critical point p _c as H∼|p-p_c^|-1 ^/σ with the critical exponent σ associated with the characteristic cluster size. Interestingly, its finite-size-scaling behavior is governed by a new exponent ν_H=(1+d_f/d)ν, where d_f is the fractal dimension of the critical percolating cluster and ν is the correlation length exponent. The corresponding scaling variable defines a singular path to the critical point. All results are confirmed by numerical simulations.