Percolation characteristic functions

The cluster-size distribution

It is evident, that for p << 1, in the low-concentration regime, all clusters are small and isolated. When p increases, clusters merge together and grow. At p = pc infinite spanning cluster appears. It makes random thin web and covers the hole space. Finite clusters are confined in pores of infinite cluster. For bigger p it absorbs finite clusters and becomes more dense. Mean size of its pores and closed clusters decreases.
subcritical supercritical
ns is the number of clusters of size s divided by the total number of sites in the system. It is shown in Fig.1,2 for different p values. In the subcritical p < pc and critical |p - pc| << 1 regions it decreases for small s in accordance with the power low. For s bigger then critical sc value, it decreases exponentially.

Averaged ns values are plotted in the right part of the applet for s = 1,2...200. ns grows to the right and cluster size grows to the bottom. P,S and other parameters are shown in the Status bar. Click mouse to get a new iteration and to average ns one more (+<Alt> to make 100 averagings). Press "Enter" to set a new p value. "Clear" button cleans averaging. "Print" sends data to the Java console (see also 640x640 lattice).

Characteristic functions P and S(p)

functions W,P,S(p) are shown in Fig.3. They are obtained by the upper applet for L = 640 and 100 averagings.
W(p) is the probabilities of appearance of spanning cluster.
P(p) is the probability that an occupied site belongs to the spanning cluster
    P(p) = (number of sites in the spanning cluster) / (total number of occudied sites)
The quantity discribes the "density" of the spanning cluster. Because sns is the part of occupied sites in clusters of size s, then the mean cluster size S(p) is
    S(p) = ∑s s2ns / ∑s sns = ∑i si2 / ∑i si ,
where si is the size of i-th cluster. It is the site weighted average. S(p) goes to infinity S(p) ~ |p - pc| in the critical region |p - pc| << 1 . But the simpe average s sns / ∑s ns is finite at p → pc and the main part of clusters has size s ~ 1. Divergence of S(p) at p → pc is the result of increasing of critical clusters size and number.
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updated 15 Nov 2001