Q factor: A measure of competition between the topper and the average in percolation and in self-organized criticality

• Published in: Physical review. E 2024-07, Vol.110 (1-1), p.014131, Article 014131
• Main Authors: Ghosh, Asim, Manna, S S, Chakrabarti, Bikas K
• Format: Article
• Language: English
• ISSN: 2470-0045; 2470-0053; 2470-0053
• DOI: 10.1103/PhysRevE.110.014131

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_{max}(L) at a specific value p_{max}(L) 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_{max}, though close, are distinct from the corresponding percolation thresholds of these lattices. We also show, using scaling analysis, that at p_{max} the value of Q_{max}(L) 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(ρ,L) factor is the quotient of the size of the largest avalanche and the cumulative average of the sizes of all the avalanches, with ρ the drop density of the driving mechanism. This study was prompted by some observations in sociophysics.