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 $H\sim |p-p_c|^{-1/σ}$ 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.

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