Confluence in labeled chip-firing

Abstract

In 2016, Hopkins, McConville, and Propp proved that labeled chip-firing on a line always leaves the chips in sorted order provided that the initial number of chips is even. We present a novel proof of this result. We then apply our methods to resolve a number of related conjectures concerning the confluence of labeled chip-firing systems.

Introduction

This paper is concerned with a labeled variant of the chip-firing process as defined by Hopkins, McConville, and Propp [7]. In unlabeled chip-firing, a collection of indistinguishable chips are placed at the nodes of a graph. If a node has at least as many chips as it has neighbors, it can “fire” by sending one chip to each of its neighbors. The process terminates if no site has enough chips to fire. For background on chip-firing in general, see [8].

The labeled chip-firing process is a variation of the chip-firing process on the infinite path graph, where the chips are made to be distinguishable. Each chip is given an integer label, and additional rules govern which chip moves in each direction during a firing move. Again, if a node has at least as many chips as it has neighbors, it can fire. The difference in the labeled case is that two distinct chips at the node are chosen and the chip of larger label moves to the right while the chip of smaller label moves to the left, see Example 2.1.

In [7], it was shown that, for an even number of chips, labeled chip-firing terminates in a unique configuration regardless of the order in which nodes fire and regardless of the choice of chips made at each node. Moreover, in the unique terminal configuration, the chips are in sorted order. The property that the final configuration is unique regardless of the intermediate steps is known as global confluence and is a fundamental property of unlabeled chip-firing. In unlabeled chip-firing global confluence is proved via local confluence. Local confluence states that for any two available fires, there is a common configuration that can be reached after either of them in only one additional fire. In the case of unlabeled chip-firing, any two available fires may be performed in either order without changing the resulting configuration. Local confluence combined with Newman's Lemma on abstract rewriting systems [9] gives a global confluence property for unlabeled chip-firing, in which any terminating chip-firing process must have a unique final configuration.

In labeled chip-firing, local confluence does not hold. The sorting result for labeled chip-firing from [7] is particularly notable for proving global confluence even though local confluence, and thus Newman's Lemma does not apply. Without this tool, the labeled case proved significantly more challenging to establish.

Here we give a novel, more general and more illuminating proof of global confluence for labeled chip-firing and related systems. Our proof is based on the analysis of a firing order poset. Fig. 1, Fig. 2 visually demonstrate the structure of confluent versus non-confluent labeled chip-firing processes that arise if we have an even or odd number of chips initially. The existence of the diamond shape at the bottom of the Haase diagram in the even case is crucial for confluence.

There have been attempts to generalize the results of [7] including several conjectures from [6] and [7] in which labeled chip-firing is extended to modified versions of the 1-dimensional grid graph. Additionally, Galashin et al. [4], [5] treat the labeled chip-firing problem as chip-firing on a Type A root system, and then generalize the problem to apply to other types of root systems and more general classes of firing moves. In [3], global confluence is proved without local confluence for higher dimensional forms of chip-firing. Our methods allow us to prove many of the above cases via a unified methodology.

In Section 2, we present the proof that labeled chip-firing sorts. In Section 3, we apply these methods to prove a series of related conjectures on sorting via chip-firing on modified versions of the one-dimensional grid graph. In Section 4, we discuss how these methods can shed light on the case where the number of chips is odd.