## 6 Step 2: Computation of Effective Conductivity Through the Structural Sums
The RWs method is particularly effective for modeling systems with high concentrations of inclusions, ensuring uniformity and stability over extended domains. The RSA protocol is a sequential process that could be applied at moderately high concentrations up to \(f=0.5773\) in two-dimensional composites. In this approach, inclusions are placed incrementally. A random point within \(Q\) is selected as the center of the initial inclusion. Each subsequent center is chosen statistically uniformly, excluding regions occupied by previous inclusions to prevent overlap. The probability distribution for each new center depends on the positions of earlier inclusions. This process continues until the desired number of inclusions is placed. There is also a special case where the inclusion conductivity is much higher than the conductivity (\(\varrho=1\)) of the host [20]. Below, the expansion for such high-contrast materials is presented in the truncated numerical form by use of the RSA protocol, see Formula (9.1.2) from [20; 34]:  
\[\lambda_{e}^{RSA}=1+2f+2f^{2}+5.13057f^{3}+5.9969f^{4}+\Delta_{RSA}(f) \tag{39}\]  
and by random walk, see Formula (9.2.37) from [20; 34]:  
\[\lambda_{e}^{RW}=1+2f+2f^{2}+5.00392f^{3}+6.3495f^{4}+\Delta_{RW}(f). \tag{40}\]  
The remainders \(\Delta_{RSA}(f)\) and \(\Delta_{RW}(f)\) have highly irregular forms and do not contribute to the critical properties. For instance, the coefficients on \(f^{k}\) (\(k=5,6,7,8\)) in the term \(\Delta_{RSA}(f)\) are of order \(10^{-10}\), see the estimation [34] (Formula (9.1.3)) of other terms. The mean values of the above coefficients do vanish, but their moduli are greater than two. Random walks in the RW protocol start with the perfect hexagonal arrangement that has the maximum packing array with concentration \(f_{c}=\frac{\pi}{\sqrt{12}}\approx 0.9069\). Therefore, this protocol admits the percolation regime of touching disks and [34]:  
\[\lambda_{e}\simeq B\;(f_{c}-f)^{-s},\quad\text{as}\,f\to f_{c}. \tag{41}\]  
The critical index \(s\approx 1.3\), while the critical amplitude \(B\) was estimated as \(B\approx 1.71\)[34]. We now consider another, the so-called RSA protocol near the concentration \(f=0.126\). More precisely, we generate 100 disks of radius \(r=0.02\) distributed without overlaps in the unit square \(Q\) using the RSA method and calculate the effective conductivity by Equations (33) and (34). This computational experiment is repeated 50 times. The averaged result of computations has the form  
\[\lambda_{sq}^{RSA}(0.126)=1+2f+2f^{2}+5.21269f^{3}+16.1938f^{4}+62.5597f^{5}+ O(f^{6}). \tag{42}\]  
It should be noted that the coefficients of \(\lambda_{e}^{RSA}(0.126)\) and \(\lambda_{e}^{RSA}\) in \(f^{3}\) differ, and the coefficients in \(f^{4}\) differ significantly. Moreover, the term of the order of \(f^{5}\) is highly irregular in \(\lambda_{e}^{RSA}\). At the same time, this term in \(\lambda_{e}^{RSA}(0.126)\) is more stable. After performing 50 independent simulations with 300 disks by RSA for the square and hexagonal arrays, the mean polynomials for the effective conductivity are obtained. In particular,  
\[\lambda_{sq}^{RSA}(\varrho,0.377)=1+2f\varrho+2f^{2}\varrho^{2}+2.48277f^{3} \varrho^{3}+0.720739f^{4}\varrho^{3}+0.813692f^{5}\varrho^{3} \tag{43}\]  
\[+2.96542f^{4}\varrho^{4}+1.44148f^{5}\varrho^{4}+4.01997f^{5}\varrho^{5}+O(f^{6 }).\]Substituting \(\varrho=1\) into Equation (43) yields  
\[\lambda_{sq}^{RSA}(0.377)=1+2f+2f^{2}+2.48277f^{3}+3.68616f^{4}+6.27514f^{5}+O(f^ {6}). \tag{44}\]  
Almost the same results hold for the hexagonal array:  
\[\lambda_{hex}^{RSA}(\varrho,0.377)=1+2f\varrho+2f^{2}\varrho^{2}+2.48155f^{3} \varrho^{3}+0.725247f^{4}\varrho^{3}+0.793752f^{5}\varrho^{3} \tag{45}\]  
\[+2.96295f^{4}\varrho^{4}+1.45049f^{5}\varrho^{4}+4.00576f^{5}\varrho^{5}+O(f^ {6}).\]  
Substituting \(\varrho=1\) into Equation (45) yields  
\[\lambda_{e}^{RSA}=1+2f+2f^{2}+5.13057f^{3}+5.9969f^{4}+\Delta_{RSA}(f) \tag{39}\]0  
The RSA protocols for the square and hexagonal arrays give very close results.