# From Color-Avoiding to Color-Favored Percolation in Diluted Lattices
## 6 Multilayer Model  
We can have an alternative view of color-avoiding percolation by mapping it onto a multilayer directed process. Let us consider two (or more) layers, defined by an index \(c\), \(c=1,\ldots,C\), so that now the equations concerns quantities such as \(s_{i}^{(c)}(t)\) and random numbers \(r_{i}^{(c)}(t)\). We can assume that each layer corresponds to a given color. Let us first discuss the color-favored situation, in which the percolation clusters are composed by sites carrying at least one color. Equation (4) becomes  
\[s_{i}^{(c)}(t)=[r_{i}^{(c)}(t)<\tau(S_{i-1}(t-1)+S_{i+1}(t-1))], \tag{12}\]  
where for color-favored percolation,  
\[S_{i}(t)=\bigvee_{c}s_{i}^{(c)}(t).\]  
Sites with more than one color act as gateways among layers. For color-avoiding percolation and not-exclusive colors, Equation (4) becomes  
\[s_{i}^{(c)}(t)=[r_{i}^{(c)}(t)<\tau(S_{i-1}^{(c)}(t-1)+S_{i+1}^{(c)}(t-1))], \tag{13}\]  
where now \(S_{i}^{(c)}(t)\) denotes the presence of color \(c\) and the absence of other colors \(c^{\prime}\
eq c\) at site \(i\) and time \(t\),  
\[S_{i}^{(c)}(t)=s_{i}^{(c)}(t)\bigwedge_{c^{\prime}\
eq c}\overline{s_{i}^{(c^ {\prime})}(t)},\]  
where the overbar denotes the negation. For exclusive colors, \(S_{i}^{(c)}(t)\) is simply \(s_{i}^{(c)}(t)\), since the presence of one color \(c\) implies the absence of other colors \(c^{\prime}\) in that site. In the following, we shall use the terminology of color-favored percolation, i.e., we shall say that the system percolates when there is activity in the phase space. Up to now we described the situation in which colors do not interact, so that the color-avoiding percolation phase just corresponds to the absorbing (\(\rho^{(c)}=0\)) state of the standard DK model of Figure 1.  
What happens when colors interfere? To reduce the number of parameters we assume that the model is symmetric among colors. For two colors (\(c=1,2\)) we have now that the transition probability for layer 1 is \(\tau(n^{(1)},n^{(1)})\), (and symmetrically for layer 2 is \(\tau(n^{(2)},n^{(0)})\)). We again assume that the empty state is absorbing, so that \(\tau(0,0)=0\). If we assume that the presence of a color is sufficient to stop percolation on that layer, we have the same model as before except for the case in which there is no color on one layer but there is one on the other: \(\tau(1,0)=\tau(1,1)=\tau(1,2)=p\) and \(\tau(2,0)=\tau(2,1)=\tau(2,2)=q\). We are left with two free parameters: \(\tau(0,1)\) and \(\tau(0,2)\), which express the \"interference\" for which a color on a layer can affect the percolation on the other layer. We studied two cases: \(\tau(0,1)=\tau(0,2)=\varepsilon\) and \(\tau(0,1)=\varepsilon\), \(\tau(0,2)=0\). The first case (a) is similar to site percolation, while the other (b) reminds the \"disruptive interference\" of the DK model. For small enough values of \(\varepsilon\) nothing new happens, since in the absorbing phase the \"activity\" on both layers goes to zero, and in the \"active\" phase, the influence of the interference term \(\varepsilon\) is small. The only influence of \(\varepsilon>0\) is that of shifting the border of the active phase (the simplest mean-field analysis predicts that the threshold \(p_{c}=1/2\) for the DK model now becomes \(p_{c}=1/2-\varepsilon\)). However, when \(\varepsilon\) is about \(0.7-0.75\), one can see that the phase boundary becomes more complex: an active phase appears near the corner \(p=0,q=0\) for case (a), and this extend to the whole side \(p=0\) for case (b). What happens here is that the \"percolation paths\" jump from one layer to the other. Indeed, the value \(\varepsilon\simeq 0.7\) is near the site percolation threshold for the standard DK model, so that even an isolated infected site on a layer can propagate to the other layer, although is almost-immediately disappearing on the original layer due to the small values of \(p\) and \(q\).