We define the $Q$ factor in the percolation problem as the quotient of the size of the largest cluster and the average size of all clusters. As the occupation probability $p$ is increased, the $Q$ factor for the system size $L$ grows systematically to its maximum value $Q_{\text{max}}\left(L\right)$ at a specific value $p_{\text{max}}\left(L\right)$ and then gradually decays. Our numerical study of site percolation problems on the square, triangular, and simple cubic lattices exhibits that the asymptotic values of $p_{\text{max}}$, though close, are distinct from the corresponding percolation thresholds of these lattices. We also show, using scaling analysis, that at $p_{\text{max}}$ the value of $Q_{\text{max}}\left(L\right)$ diverges as $L^d$ ($d$ denoting the dimension of the lattice) as the system size approaches its asymptotic limit. We further extend this idea to nonequilibrium systems such as the sandpile model of self-organized criticality. Here the $Q(\rho ,L)$ factor is the quotient of the size of the largest avalanche and the cumulative average of the sizes of all the avalanches, with $\rho$ the drop density of the driving mechanism. This study was prompted by some observations in sociophysics.

• Received 12 February 2024
• Revised 15 May 2024
• Accepted 8 July 2024

DOI:https://doi.org/10.1103/PhysRevE.110.014131