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We propose a generalization of the graphical chip-firing model allowing for the redistribution dynamics to be governed by any invertible integer matrix while maintaining the long term critical, superstable, and energy minimizing behavior of the classical model.
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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.
We study a particular chip-firing process on an infinite path graph. At any time when there are at least $a+b$ chips at a vertex, $a$ chips fire to the left and $b$ chips fire to the right. We describe the final state of this process when we start with $n$ chips at the origin.
Motivated by the notion of chip-firing on the dual graph of a planar graph, we consider `integral flow chip-firing on an arbitrary graph $G$. The chip-firing rule is governed by ${mathcal L}^*(G)$, the dual Laplacian of $G$ determined by choosing a basis for the lattice of integral flows on $G$. We show that any graph admits such a basis so that ${mathcal L}^*(G)$ is an $M$-matrix, leading to a firing rule on these basis elements that is avalanche finite. This follows from a more general result on bases of integral lattices that may be of independent interest. Our results provide a notion of $z$-superstable flow configurations that are in bijection with the set of spanning trees of $G$. We show that for planar graphs, as well as for the graphs $K_5$ and $K_{3,3}$, one can find such a flow M-basis that consists of cycles of the underlying graph. We consider the question for arbitrary graphs and address some open questions.
In this paper, we study the dynamics of sand grains falling in sand piles. Usually sand piles are characterized by a decreasing integer partition and grain moves are described in terms of transitions between such partitions. We study here four main transition rules. The more classical one, introduced by Brylawski (1973) induces a lattice structure $L_B (n)$ (called dominance ordering) between decreasing partitions of a given integer n. We prove that a more restrictive transition rule, called SPM rule, induces a natural partition of L_B (n) in suborders, each one associated to a fixed point for SPM rule. In the second part, we extend the SPM rule in a natural way and obtain a model called Chip Firing Game (Goles and Kiwi, 1993). We prove that this new model has interesting properties: the induced order is a lattice, a natural greedoid can be associated to the model and it also defines a strongly convergent game. In the last section, we generalize the SPM rule in another way and obtain other lattice structure parametrized by some t: L(n,t), which form for -n+2 <= t <= n a decreasing sequence of lattices. For each t, we characterize the fixed point of L(n,t) and give the value of its maximal sized chains lenght. We also note that L(n,-n+2) is the lattice of all compositions of n.
We show a collection of scripts, called $G$-strongly positive scripts, which is used to recognize critical configurations of a chip firing game (CFG) on a multi-digraph with a global sink. To decrease the time of the process of recognition caused by the stabilization we present an algorithm to find the minimum G-strongly positive script. From that we prove the non-stability of configurations obtained from a critical configuration by firing inversely any non-empty multi-subset of vertices. This result is a generalization of a very recent one by Aval emph{et.al} which is applied for CFG on undirected graphs. Last, we give a combinatorial proof for the duality between critical and super-stable configurations.
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