# From Color-Avoiding to Color-Favored Percolation in Diluted Lattices
## 3 Self-organized Criticality and the Fragment Method  
In systems showing a continuous phase transition, like the percolation ones, order parameters such as the asymptotic density of active sites \(\rho(p)\) show a power-law behavior in the vicinity of the critical value \(p_{c}\):  
\[\rho(p)\propto(p-p_{c})^{\beta},\]  
which is a mark of the self-similar (scale-free) character of the transition. While in \"standard\" critical systems this power-law behavior can be observed only via a fine triggering of parameters, the so-called auto-critical systems self-organizes into a critical state, which is therefore denoted as \"self-organized criticality\", to underline their ability to \"go freely\" to the critical point. Known examples of auto-critical systems have been used for modeling earthquakes [27] and evolution [28], and the invasion percolation model [29]. Some classic percolation problems can be reformulated as SOC system [30], and the evolution of certain quantities can be used to obtain information on the criticality of the systems, such as the critical values of their control parameters. We shall illustrate this idea for the simple directed site percolation problem with \(k=2\). Let us introduce the quantity \(p_{i}(t)\), which is the minimum value of \(p\) for which the giant percolation cluster includes site \(i\) at time \(t\). The quantity \(s_{i}(t)\) can therefore be written as \([p>p_{i}(t)]\), and Equation (3) as  
\[[p>p_{i}(t)]=[p>r_{i}(t)]\wedge\big{(}[p>p_{i-1}(t-1)]\vee[p>p_{i+1}(t-1)] \big{)}, \tag{5}\]  
The expression \([p>a]\vee[p>b]\) can be rewritten as \([p>\min(a,b)]\) and \([p>a]\wedge[p>b]\) is equivalent to \([p>\max(a,b)]\). Equation (5) becomes  
\[[p>p_{i}(t)]=\Big{[}p>\max\big{(}r_{i}(t),\min(p_{i-1}(t-1),p_{i+1}(t-1)\big{)} \Big{]}.\]  
We can therefore extract the evolution rule for the \(p_{i}(t)\):  
\[p_{i}(t)=\max\big{(}r_{i}(t),\min(p_{i-1}(t-1),p_{i+1}(t-1)\big{)},\]  
which has a sort of magic: automatically (or self-critically) the \(p_{i}\)'s evolve so that all of them are above the critical value \(p_{c}\), for the given lattice and the set of random number used, since, by definition, the critical percolation probability is the smallest value of \(p\) for which there is a giant percolation cluster, spanning the whole system. Thus, \(p_{c}\) can be approximated by  
\[p_{c}=\underset{0<i\leq N}{\text{MIN}}p_{i}(t),\]  
This estimation is just an approximation, since one should consider the thermodynamic (\(\lim_{t\to\infty}\lim_{N\to\infty}\)) limit. Moreover, the previous result depends on the realization of the disorder, i.e., the set of random numbers used for the simulation, and one should average over different realizations. In practice, the variance of the critical values of \(p_{c}\) obtained in different realizationsalways drops to zero when increasing the system size, showing the presence of strong self-averaging in these systems [31]. This method is powerful and elegant, but it can be used only if the rule can be expressed using the AND and OR operation. It is possible to think to sites as segments initially marking all values of \(p\), from zero to one. Since site \(i\) can be wet only for \(p>p_{i}\), the associated segment can be painted with white (dry) from zero to \(p_{i}\), and black (wet) above it. At beginning, all segments are black, meaning that initially all are wet. The AND operation is similar to taking the intersection of the black parts of the segments, and the OR to the union of them. Since these operations keep the black part contiguous, one has only to keep track of the lower boundary, which is exactly \(p_{i}(t)\)[18]. Clearly, if the rule includes also other operations like NOT or XOR, the black part of the segment is no more compact. We can however approximate the evolution of a system by iterating segments (now called fragments) sampled at many values of \(p\), using a multi-bit technique.