Chip-firing and energy minimization on M-matrices

We consider the class of z-superstable configurations. We prove that for any M-matrix, the z-superstable configurations coincide with the energy minimizing configurations. Moreover, we prove that the z-superstable configurations are in simple duality with critical configurations. Thus for all avalanche-finite systems (including all directed graphs with a global sink) there exist unique critical, energy minimizing and z-superstable configurations. The critical configurations are in simple duality with energy minimizers which coincide with z-superstable configurations.