In the site percolation problem, when the occupation probability $p \rightarrow p_c$, where $p_c$ is the critical probability. The characteristic length diverges, and assuming the usual scaling ansatz $$n(s,p) = f\left(s/s_\xi \right)s^{-\tau},$$ the cluster size distribution becomes a pure power law with no characteristic length, i.e. $f\left(s/s_\xi \right) \rightarrow C$ with $, C\in{\rm I\!R}$. The physical situation is completely scale-free.

Intuitively I would expect that since the problem becomes a statistical fractal, where all sizes are equivalent, then clusters of all sizes should be equally likely to be found. Why is it that in reality the cluster size distribution $n(s) \sim 1/s^\tau$ with $\tau > 1$, i.e. the probability of finding larger clusters is smaller? and if the problem is scale-free, what is the physical relevance of parameter $s$?


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In order for the system to be scale free, you don't need all the sizes to be equiprobable (that would be a degenerate case), it's only necessary that their relative probabilities stay constant.

That means that, for a given size $s_0$ and a scaling factor $k$, consecutive sizes appear equally less often. In equations: If

$$ \frac{n(s_1)}{n(s_0)} =\frac{n(s_2)}{n(s_1)} = C, $$

where $\;n(s_1)=kn(s_0)\;$ and $\;n(s_2)=kn(s_1)$, then, if we write $C=k^{-\tau}$, it must be that $$ n(s_2) = k^{-\tau}n(s_1) = (k^2)^{-\tau}n(s_0) $$ and, in general,

$$ n(s_N) = n(k^Ns_0) = (k^N)^{-\tau}n(s_0). $$

This statement is equivalent to equation (given in the question) $n(s) \propto s^{-\tau}$.

As for the parameter $s$, even if the system doesn't have a characteristic length, it doesn't mean everything is length independent. Particularly, when dealing with finite approximations of the thermodynamical limit, $s$ is important to quantify the limitations of the approximation.


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