## 6 Step 2: Computation of Effective Conductivity Through the Structural Sums
However, RSA protocols can be used for all concentrations [20], and near a fixed concentration, they generate different formulas. This is due to the fact that the location of the disks in the RSA protocol [20; 34] depends on \(f\). In particular, the disks satisfy the Poisson distribution for infinitesimally small \(f\). If \(f=\frac{\pi}{\sqrt{12}}\), the RSA admits only one event, the perfect hexagonal array, with zero probability. The Formula (40) was constructed for all generated \(0\leq f\leq\frac{\pi}{\sqrt{12}}\). At the same time, the RSA protocols used during the derivation of Equations (43)-(46) yield different formulas. This observation enables us to introduce the different RSA protocols that were not elaborated enough in previous publications. The RSA protocol used in [20] leads to quality fits for all concentrations \(f\). However, the higher-order terms for the corresponding effective conductivity are found to be unstable. Denote the \(RSA(f)\) protocol used near a fixed concentration \(f\). The stability of the Formulas (43)-(46) during the simulations is different and depends on \(f\), as shown in Tables 1-3 and Figures 4-6. One can see that the shape of cell \(Q\), be it square or hexagonal, practically does not influence the structural sums. The strongest dependence is on the basic concentration of the stable protocol \(RSA(f)\). The \(RSA(f)\) protocol becomes less stable with \(f\) decreasing. Thus, the deviation of \(e_{2}\) near \(\pi\) for \(f=0.126\) as shown in Figure 4 is higher than the deviation of \(e_{2}\) for \(f=0.377\) shown in Figures 5b and 6b. \[\lambda_{e}^{RSA}=1+2f+2f^{2}+5.13057f^{3}+5.9969f^{4}+\Delta_{RSA}(f) \tag{39}\]1 Fifty values of structural sum \(e_{2}\) on the complex plane simulated by RSA for the square array with \(f=0.126\). The dashed red lines correspond to the mean value \(3.1085+\mathrm{i}0.0199\) from Table 1 near the point \(e_{2}=\pi\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Structural**} & \multicolumn{3}{c}{**Real Part**} & \multicolumn{3}{c}{**Imaginary Part**} \\ \cline{2-7}  & **Mean** & **Median** & **St. Dev.** & **Mean** & **Median** & **St. Dev.** \\ \hline \(e_{2}\) & 3.1085 & 3.0754 & 0.3021 & 0.0199 & 0.0016 & 0.3594 \\ \hline \(e_{22}\) & 25.5157 & 25.7152 & 2.9871 & 0 & 0 & 0 \\ \hline \(e_{33}\) & \(-60.2746\) & \(-58.2463\) & 8.7796 & 0 & 0 & 0 \\ \hline \(e_{44}\) & 296.702 & 305.671 & 55.7525 & 0 & 0 & 0 \\ \hline \(e_{222}\) & 126.848 & 128.309 & 29.7531 & \(-0.8130\) & \(-6.4829\) & 27.8381 \\ \hline \(e_{332}\) & \(-186.629\) & \(-183.168\) & 61.6119 & 4.5258 & 15.9828 & 62.4439 \\ \hline \(e_{2222}\) & 1375.21 & 1385.35 & 316.249 & 0 & 0 & 0 \\ \hline \end{tabular}
\end{table} 
Table 1: Structural sums simulated by RSA for the square array 50 times with \(f=0.126\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Structural**} & \multicolumn{3}{c}{**Real Part**} & \multicolumn{3}{c}{**Imaginary Part**} \\ \cline{2-7}  & **Mean** & **Median** & **St. Dev.** & **Mean** & **Median** & **St. Dev.** \\ \hline \(e_{2}\) & 3.1203 & 3.1168 & 0.0496 & 0.0045 & \(-0.0032\) & 0.0592 \\ \hline \(e_{22}\) & 12.1183 & 12.1194 & 0.3369 & 0 & 0 & 0 \\ \hline \(e_{33}\) & \(-5.5869\) & \(-5.5707\) & 0.4074 & 0 & 0 & 0 \\ \hline \(e_{44}\) & 13.2102 & 13.1334 & 0.755 & 0 & 0 & 0 \\ \hline \(e_{222}\) & 45.2035 & 45.234 & 1.982 & \(-0.0761\) & \(-0.1515\) & 1.0108 \\ \hline \(e_{332}\) & \(-17.4618\) & \(-17.5099\) & 1.3918 & 0.0066 & 0.0298 & 0.6047 \\ \hline \(e_{2222}\) & 191.953 & 192.084 & 11.0702 & 0 & 0 & 0 \\ \hline \end{tabular}
\end{table} 
Table 2: Structural sum simulated by RSA for the square array 300 times with \(f=0.377\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Structural**} & \multicolumn{3}{c}{**Real Part**} & \multicolumn{3}{c}{**Imaginary Part**} \\ \cline{2-7}  & **Mean** & **Median** & **St. Dev. ** & **Mean** & **Median** & **St. Dev.