Chip-firing and energy minimization on M-matrices
  We consider chip-firing dynamics defined by arbitrary M-matrices. M-matrices generalize graph Laplacians and were shown by Gabrielov to yield avalanche finite systems. Building on the work of Baker and Shokrieh, we extend the concept of energy minimizing chip configurations. Given an M-matrix, we show that there exists a unique energy minimizing configuration in each equivalence class defined by the matrix.   We define the class of $z$-superstable configurations which satisfy a strictly stronger stability requirement than superstable configurations (equivalently $G$-parking functions or reduced divisors). 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. 
Johnny Guzm\'an and Caroline Klivans
math.CO