Cluster distribution in mean-field percolation: Scaling and universality

Abstract

The partition function of the finite q-state Potts model in the limit q→1 is shown to yield a closed form for the distribution of clusters in the immediate vicinity of the percolation transition. Various important properties of the transition are manifest, including scaling behavior and the emergence of the spanning cluster.

Percolation on two- and three-dimensional lattices

“…Therefore, approximate solutions are necessary, and much effort has been dedicated in this direction. From the theoretical point of view, one can utilize mean-field [8,9] and renormalization group [10,11,12,13] techniques, among others. In particular, computer simulations constitute a powerful tool in this area, since their application to percolation is simpler than for many other problems in statistical physics [14].…”

Phase transition of DNA-linked gold nanoparticles

“…The absorption of DNA-modified, 10 nm, 20 nm, and 40 nm gold nanoparticles are plotted versus reduced temperatures (T R = T /T m ). Above temperature T m (low connectivity), the curves appear insensitive to details, indicative of universal scaling at the percolation transition [13]. The curves can be fitted with an equation that describes percolation phenomena,…”

Clustering and percolation in lithium borate glasses

“…The particular value of the power-law exponent is in agreement with the predictions of random percolation theory, 40,41 whereas the pronounced peak at the large-size region is attributed to the existence of percolating clusters. 42 This is also inferred from the distributions of the percolating NBO clusters, which have been calculated independently and are superimposed in the same figure for xϭ0.5 and 0.6. It is clear that there is a perfect coincidence between the corresponding peaks in the NBO cluster size distribution and the percolating NBO cluster size distribution.…”