Minimally stable site (MSS) clusters play a dominant role in shaping avalanches in the self-organized critical (SOC) systems. A local smoothing (diffusion) mechanism reshapes the MSS landscape, delaying rare avalanches until the emergence of spanning avalanches, thereby inducing self-organized bistability. We find bistability-induced self-similarity in MSS clusters, which is suppressed in the thermodynamic limit. The inverse Ising problem technique is employed to track the dynamical transition to this point at which the spanning avalanches form. An emergent magnetic instability in the dual Ising model is observed, marking a transition to a correlated percolation regime. At the bistability-induced self-similarity point, the effective pairwise interactions in the dual Ising system vanish—signaling a magnetic transition characterized by abrupt changes in magnetization and spin susceptibility. Crucially, we show that diffusion fundamentally reshapes avalanche dynamics: the spatial anticorrelations of MSSs in standard SOC systems transform into positive correlations when diffusion is introduced.

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• Magnetic order
• Percolation
• Disordered systems
• Critical exponents
• Ising model
• Self-similarity