Phase boundaries and desynchronization in the single-defect Olami-Feder-Christensen model: Toward understanding earthquake power laws

We investigate the emergence of self-organized criticality (SOC) in the single-defect Olami-Feder-Christensen (OFC) model by quantifying the ordered phase and deriving theoretical phase boundaries. Using both square and honeycomb lattices, we analytically obtain expressions for the ordered-SOC transition boundary and the characteristic period, validating them against numerical simulations. The results show that while synchronization alone governs the transition to the ordered phase, the onset of SOC requires desynchronization triggered by supercritical toppling energies exceeding a certain threshold. These findings characterize the threshold-dependent nature of desynchronization under minimal inhomogeneity, providing a fundamental perspective that clarifies the mechanisms observed in the original OFC model.