Minimally stable site (MSS) clusters play a dominant role in shaping avalanches in the self-organized critical (SOC) systems. The manipulation of MSS clusters through local smoothings (diffusion) alter the MSS landscape, suppressing rare avalanches and postponing them until they manifest as spanning avalanches. By leveraging the Inverse Ising problem, we uncover a duality between diffusive sandpiles and equilibrium statistical physics. Our analysis reveals an emergent magnetic instability in the dual Ising model, coinciding with the formation of spanning avalanches and marking a transition to a correlated percolation regime. At this point, the MSS loop soups exhibit fractal self-similarity and power-law distributions, while the effective pairwise interactions in the dual 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 anti-correlations of MSSs in standard SOC systems transform into positive correlations when diffusion is introduced. These findings bridge self-organized criticality, percolation theory, and equilibrium phase transitions, shedding new light on emergent criticality and large-scale correlations in non-equilibrium systems.
Makani et al. (Mon,) studied this question.