Alexander E. Holroyd, Lionel Levine, Karola Meszaros, Yuval Peres, James Propp, David B. Wilson
We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems.
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Chip-firing and rotor-routing are two well-studied examples of abelian networks. We study the complexity of their respective reachability problems. We show that the rotor-routing reachability problem is decidable in polynomial time, and we give a simple characterization of when a chip-and-rotor configuration is reachable from another one. For chip-firing, it has been known that the reachability problem is in P if we have a class of graphs whose period length is polynomial (for example, Eulerian digraphs). Here we show that in the general case, chip-firing reachability is hard in the sense that…
We analyze the poset of moves in chip-firing, as defined by Klivans and Liscio. Answering a question of Propp, we show that the move poset forms the join-irreducibles of the poset of configurations. The proof involves a graph augmentation and an analysis of configurations in which only one firing move is available. We then use this framework to analyze the problem of chip-firing on a line, where the move poset is relevant to the problem of labeled chip-firing.
Jim Propp recently proposed a labeled version of chip-firing on a line and conjectured that this process is confluent from some initial configurations. This was proved by Hopkins-McConville-Propp. We reinterpret Propp's labeled chip-firing moves in terms of root systems: a "central-firing" move consists of replacing a weight $λ$ by $λ+α$ for any positive root $α$ that is orthogonal to $λ$. We show that central-firing is always confluent from any initial weight after modding out by the Weyl group, giving a generalization of unlabeled chip-firing on a line to other types. For simply-laced root s…
In 2016, Hopkins, McConville, and Propp proved that labeled chip-firing on a line always leaves the chips in sorted order if the 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.
Periods and atomic firing sequences of parallel chip-firing games on directed graphs
本文研究有向图上并行筹码点火博弈的周期与原子发射序列，证明无向图核心结果的有向图对应版本，并给出强连通有向图中顶点原子发射序列的完整刻画。
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.
Labeled Chip-firing on Binary Trees with $2^n-1$ Chips
研究二叉树上带 $2^n-1$ 个芯片的标记芯片发射，证明终止构型的排序性质及终局移动偏序集为模格。
We investigate a variant of the chip-firing process on the infinite path graph: rather than treating the chips as indistinguishable, we label them with positive integers. To fire an unstable vertex, i.e. a vertex with more than one chip, we choose any two chips at that vertex and move the lesser-labeled chip to the left and the greater-labeled chip to the right. This labeled version of the chip-firing process exhibits a remarkable confluence property, similar to but subtler than the confluence that prevails for unlabeled chip-firing: when all chips start at the origin and the number of chips i…
Focusing on gap lengths in the noisy violinist chip-firing problem
该研究将噪声小提琴家芯片激发问题的状态编码为间隙长度序列，并以此视角重新证明了平坦簇子最终状态的相关命题。
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.
Labeled Chip-Firing on Star Graphs
本文研究无限细分k-星图上标记点火游戏的稳定配置，证明其排序性质，并分析与标准杨表对应的特殊稳定配置。
On the minimum number of chips allowing an infinite Chip-firing game
本文针对强连通无环有向多重图，给出了计算导致无限筹码发射游戏所需最小初始筹码数的三个公式，推广了欧拉情形下的已知结果。
In this paper, we study the complexity of the chip-firing reachability problem. We show that for Eulerian digraphs, the reachability problem can be decided in strongly polynomial time, even if the digraph has multiple edges. We also show a special case when the reachability problem can be decided in polynomial time for general digraphs: if the target distribution is recurrent restricted to each strongly connected component. As a further positive result, we show that the chip-firing reachability problem is in co-NP for general digraphs. We also show that the chip-firing halting problem is in co…
Algorithms for Chip-Firing on Weighted Graphs
本文将筹码点火及相关算法推广到加权图，给出q-约化除子数量的上界，并提出寻找所有最大不可赢除子的方法。
Jim Propp recently introduced a variant of chip-firing on a line where the chips are given distinct integer labels. Hopkins, McConville, and Propp showed that this process is confluent from some (but not all) initial configurations of chips. We recast their set-up in terms of root systems: labeled chip-firing can be seen as a root-firing process which allows the moves $λ\to λ+ α$ for $α\in Φ^{+}$ whenever $\langleλ,α^\vee\rangle = 0$, where $Φ^{+}$ is the set of positive roots of a root system of Type A and $λ$ is a weight of this root system. We are thus motivated to study the exact same root…
On Chip-Firing on Undirected Binary Trees
本文研究无限二叉树上的芯片点火游戏，给出根节点放置2^ℓ - 1个带标签芯片时稳定配置数的上界，并确定任意正整数N个芯片时各顶点点火次数及总点火次数。
Chip Firing on Directed $k$-ary Trees
该研究针对无限根有向$k$叉树的芯片射击变体，在根置$k^\ell$个可区分芯片、顶点满$k$个芯片即向出邻接点各分散1个的规则下，明确了稳定构型的精确数量与性质。
Finite record sets of chip-firing games
本文研究 chip-firing 博弈的记录集，在记录集有限且仅发射两个顶点的假设下，发现了两个新性质以刻画图的记录集。
Chip-firing on the Lattice of Nonnegative Integer Points
本文研究了在非负整数点格子的有向图上进行芯片发射游戏，分析了中间配置的结构及稳定配置的性质。
Baker and Norine proved a Riemann--Roch theorem for divisors on undirected graphs. The notions of graph divisor theory are in duality with the notions of the chip-firing game of Björner, Lovász and Shor. We use this connection to prove Riemann--Roch-type results on directed graphs. We give a simple proof for a Riemann--Roch inequality on Eulerian directed graphs, improving a result of Amini and Manjunath. We also study possibilities and impossibilities of Riemann--Roch-type equalities in strongly connected digraphs and give examples. We intend to make the connections of this theory to graph th…
On z-Superstable and Critical Configurations of Chip Firing Pairs
本研究将图的超稳定与临界构型的对偶映射，从M-矩阵进一步自然推广到(L,M)-芯片发射对，并研究了该映射的相关性质，拓展了芯片发射理论的适用范围。
Cools, Draisma, Payne, and Robeva proved that generic metric graphs that are "paths of loops" are Brill-Noether general. We show that Brill-Noether generality does not hold for "trees of loops": the only trees of loops that are Brill-Noether general are paths of loops. We study various notions of generality and examine which of these graphs satisfy them.
In [5], Holroyd, Levine, Mészáros, Peres, Propp and Wilson characterize recurrent chip-and-rotor configurations for strongly connected digraphs. However, the number of steps needed to recur, and the number of orbits is left open for general digraphs. Recently, these questions were answered by Pham [6], using linear algebraic methods. We give new, purely combinatorial proofs for these formulas. We also relate rotor-router orbits to the chip-firing game: The number of recurrent rotor-router unicycle-orbits equals the order of the Picard group of the graph, defined in the sense of [1], and during…
Permutation-based Strategies for Labeled Chip-Firing on $k$-ary Trees
本文研究了在k叉树上基于排列的标记芯片点燃策略，探讨了稳定配置的排列性质及逆序数和下降数等特征。
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 undi…
Labeled Chip-Firing on Directed $k$-ary Trees and Where Chips Land
本文研究无限有根有向 $k$-ary 树上的有标记 chip-firing 变体，证明稳定配置中 chip 落点呈区间分布及其长度等性质，并描述每个 chip 可能落点以刻画可能的稳定配置。
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 o…
Chip-Firing on Infinite $k$-ary Trees
该研究针对根带自环的无限$k$叉树芯片发射过程，刻画稳定构型、计算各顶点及总发射次数，并分析根发射次数与总发射次数随芯片数$N$变化的序列性质。
The Middle Stair for Complete Bipartite Parallel Chip-Firing
该研究证明完全二部并行芯片点火的中间楼梯猜想，引入双参数共轭方法，得出特定构型下游戏最终周期为2的结论。
Non-Stabilizing Parallel Chip-Firing Games
本文研究了并行点火博弈的非稳定化条件，证明了在2|E|-|V|<|σ|<2|E|范围内博弈周期不为3或4，并扩展共轭配置方法证明了完全二部图上的相关猜想。
Labeled Chip-Firing on Undirected $k$-ary Trees
本文研究无向 $k$-叉树上的标记芯片发射，推广二叉树结果，提出稳定配置数的之字形上界及新下界。
The Chip-firing game is a discrete dynamical system played on a graph, in which chips move along edges according to a simple local rule. Properties of the underlying graph are of course useful to the understanding of the game, but since a conjecture of Biggs that was proved by Merino López, we also know that the study of the Chip-firing game can give insights on the graph. In particular, a strong relation between the partial Tutte polynomial $T_G(1,y)$ and the set of recurrent configurations of a Chip-firing game (with a distinguished sink vertex) has been established for undirected graphs. A …
Chip-firing and critical groups of signed graphs
该研究将可逆矩阵芯片发射理论应用于带符号图，定义其芯片发射规则、临界群及相关构型，给出性质验证方法并计算多类带符号图的临界群，推广了已有结果。
Kirchhoff's matrix-tree theorem states that the number of spanning trees of a graph G is equal to the value of the determinant of the reduced Laplacian of $G$. We outline an efficient bijective proof of this theorem, by studying a canonical finite abelian group attached to $G$ whose order is equal to the value of same matrix determinant. More specifically, we show how one can efficiently compute a bijection between the group elements and the spanning trees of the graph. The main ingredient for computing the bijection is an efficient algorithm for finding the unique $G$-parking function (reduce…
A Three-Regime Theorem for Flow-Firing
本文研究 2 维胞腔复形上的流重路由点火系统，借鉴前人工作，回答了该过程是否终止及终止是否唯一的问题，提出了完全判定系统终止性的三区域定理。
We investigate a special variant of chip-firing, in which we consider an infinite set of rooms on a number line, some of which are occupied by violinists. In a move, we take two violinists in adjacent rooms, and send one of them to the closest unoccupied room to the left and the other to the closest unoccupied room to the right. We classify the different possible final states from repeatedly performing this operation. We introduce numbers $R(N,\ell,x)$ that count labeled recursive rooted trees with $N$ vertices, $\ell$ leaves, and the smallest rooted path ending in $x$. We describe the propert…
In this expository article intended to be accessible to undergraduate students we introduce a finite abelian group that can be associated to any finite connected graph. This group can be defined in an elementary combinatorial way in terms of chip-firing operations, and has been an object of interest in combinatorics, algebraic geometry, statistical physics, and several other areas of mathematics. We will begin with basic definitions and examples and develop a number of properties that can be derived by looking at this group from different angles. Throughout, we will give exercises, some of whi…
We continue our studies of burn-off chip-firing games from [Discrete Math. Theor. Comput. Sci. 15 (2013), no. 1, 121-132; MR3040546] and [Australas. J. Combin. 68 (2017), no. 3, 330-345; MR3656659]. The latter article introduced randomness by choosing successive seeds uniformly from the vertex set of a graph $G$. The length of a game is the number of vertices that fire (by sending a chip to each neighbor and annihilating one chip) as an excited chip configuration passes to a relaxed state. This article determines the probability distribution of the game length in a long sequence of burn-off ga…
In earlier work in collaboration with Pavel Galashin and Thomas McConville we introduced a version of chip-firing for root systems. Our investigation of root system chip-firing led us to define certain polynomials analogous to Ehrhart polynomials of lattice polytopes, which we termed the symmetric and truncated Ehrhart-like polynomials. We conjectured that these polynomials have nonnegative integer coefficients. Here we affirm "half" of this positivity conjecture by providing a positive, combinatorial formula for the coefficients of the symmetric Ehrhart-like polynomials. This formula depends …
Chip-Firing and the Sandpile Group of the $R_{10}$ Matroid
本文通过复数在五边形顶点上描述了 $R_{10}$ 矩阵的 chip-firing 动力学，并提供了 162 个等价类的代表集。
Chip-Firing Games on Banana Trees
本文研究香蕉树上的芯片发射博弈，给出香蕉路径除子亏格的多项式算法及香蕉树 scramble number、screewidth 公式，并证明删边可任意改变亏格且亏格猜想对香蕉树成立。
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 …
We study the interplay between chip-firing games and potential theory on graphs, characterizing reduced divisors ($G$-parking functions) on graphs as the solution to an energy (or potential) minimization problem and providing an algorithm to efficiently compute reduced divisors. Applications include an "efficient bijective" proof of Kirchhoff's matrix-tree theorem and a new algorithm for finding random spanning trees. The running times of our algorithms are analyzed using potential theory, and we show that the bounds thus obtained generalize and improve upon several previous results in the lit…
The parallel chip-firing game is an automaton on graphs in which vertices "fire" chips to their neighbors when they have enough chips to do so. The game is always periodic, and we concern ourselves with the firing sequences of vertices. We introduce the concepts of motorized parallel chip-firing games and motor vertices, study the effects of motors connected to a tree and show that motorized games can be transformed into ordinary games if the motors' firing sequences occur in some ordinary game. We then characterize exactly which periodic firing patterns can occur in an ordinary game and state…
Parking functions and chip-firing on hypergraphs
本文将图上的停车函数与筹码发射概念扩展至超图，定义了超图停车函数，证明其极大序列由无环定向刻画，且可由$q$-根生成树计数，并揭示了其与超图筹码发射超稳定构型的等价关系。
Chip-firing games, Jacobians, and Prym varieties
该研究对度量图芯片射击游戏理论与热带Prym簇理论做自成体系的介绍，探讨二者和代数对应概念的关联，提出未来研究方向。
Let $Γ$ be a finite graph and let $Γ_n$ be the "$n$th cone over $Γ$" (i.e., the join of $Γ$ and the complete graph $K_n$). We study the asymptotic structure of the chip-firing group $\text{Pic}^0(Γ_n)$.
We study a variant of the chip-firing game called \emph{diffusion}. In diffusion on a graph, each vertex of the graph is initially labelled with an integer interpreted as the number of chips at that vertex, and at each subsequent step, each vertex simultaneously fires one chip to each of its neighbours with fewer chips. Since this firing rule may result in negative labels, diffusion, unlike the parallel chip-firing game, is not obviously periodic. In 2016, Duffy, Lidbetter, Messinger and Nowakowski nevertheless conjectured that diffusion is always eventually periodic, and moreover, that the pr…
Primitive periods in greedy chip-firing games
本文研究贪心芯片博弈的周期性质，证明饥饿博弈及单调得分推广中所有周期轨道的最小周期均为 T，并解决了 Li 和 Propp 的两个猜想，同时揭示了与主席分配问题的联系。
Quantized rational chip-firing
本文提出一种量化芯片 firing 模型，与有理格路和有理停车函数理论相关。通过定义 superstable、k-stable 和 k-skeletal 配置，建立 combinatorial 理论，并证明其与芯片配置的群结构等价。
Skeletal generalizations of Dyck paths, parking functions, and chip-firing games
该研究引入由Catalan数计数的k-骨架路径与对应生成树数的k-骨架停车函数，将其推广到非整数参数的梯形路径族，建立参数变化下的双射，关联芯片射击、组合表示论与热带几何。
Baker and Norine introduced a graph-theoretic analogue of the Riemann-Roch theory. A central notion in this theory is the rank of a divisor. In this paper we prove that computing the rank of a divisor on a graph is NP-hard. The determination of the rank of a divisor can be translated to a question about a chip-firing game on the same underlying graph. We prove the NP-hardness of this question by relating chip-firing on directed and undirected graphs.
The parallel chip-firing game is a periodic automaton on graphs in which vertices "fire" chips to their neighbors. In 1989, Bitar conjectured that the period of a parallel chip-firing game with n vertices is at most n. Though this conjecture was disproven in 1994 by Kiwi et. al., it has been proven for particular classes of graphs, specifically trees (Bitar and Goles, 1992) and the complete graph K_n (Levine, 2008). We prove Bitar's conjecture for complete bipartite graphs and characterize completely all possible periods for positions of the parallel chip-firing game on such graphs. Furthermor…
We study how parallel chip-firing on the complete graph K_n changes behavior as we vary the total number of chips. Surprisingly, the activity of the system, defined as the average number of firings per time step, does not increase smoothly in the number of chips; instead it remains constant over long intervals, punctuated by sudden jumps. In the large n limit we find a "devil's staircase" dependence of activity on the number of chips. The proof proceeds by reducing the chip-firing dynamics to iteration of a self-map of the circle S^1, in such a way that the activity of the chip-firing state eq…
Hereditary chip-firing models generalize the Abelian sandpile model and the cluster firing model to an exponential family of games induced by covers of the vertex set. This generalization retains some desirable properties, e.g. stabilization is independent of firings chosen and each chip-firing equivalence class contains a unique recurrent configuration. In this paper we present an explicit bijection between the recurrent configurations of a hereditary chip-firing model on a graph and its spanning trees.
A Variation on Chip-Firing: the diffusion game
本文提出并行芯片触发游戏的变体——扩散游戏，研究其周期性，证明了若干图类上的周期性，并提出了未来研究问题。
Aval et al. proved that starting from a critical configuration of a chip- firing game on an undirected graph, one can never achieve a stable configuration by reverse firing any non-empty subsets of its vertices. In this paper, we generalize the result to digraphs with a global sink where reverse firing subsets of vertices is replaced with reverse firing multi-subsets of vertices. Consequently, a combinatorial proof for the duality between critical configurations and superstable configurations on digraphs is given. Finally, by introducing the concept of energy vector assigned to each configurat…
We investigate Riemann-Roch theory for directed graphs. The Riemann-Roch criteria of Amini and Manjunath is generalized to all integer lattices orthogonal to some positive vector. Using generalized notions of a $v_0$-reduced divisor and Dhar's algorithm we investigate two chip-firing games coming from the rows and columns of the Laplacian of a strongly connected directed graph. We discuss how the "column" chip-firing game is related to directed $\vec{G}$-parking functions and the "row" chip-firing game is related to the sandpile model. We conclude with a discussion of arithmetical graphs, whic…
Formalizing chip-firing and Riemann--Roch for graphs in Lean 4
使用 Lean 4 定理证明器形式化了 Baker-Norine 的图 Riemann-Roch 定理，包括 q-约化除子的存在唯一性、Dhar 燃烧算法、无环定向与超稳定构型的双射以及 Clifford 定理。
In this paper we explore enumeration problems related to the number of reachable configurations in a chip-firing game on a finite connected graph G. We define an auxiliary notion of debt-reachability and prove that the number of debt-reachable configurations from an initial configuration with c chips on one vertex is a quasipolynomial in c. For the cycle graph C_n, we apply these results to compute a near explicit formula for the number of debt-reachable configurations. We then derive polynomial asymptotic bounds for the number of debt-reachable and reachable configurations, and finally provid…
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