Chip-Firing: Dynamics, Divisors, and Generalizations

• Chip-firing is a discrete dynamical system involving local chip redistributions on graphs, encapsulating properties like termination, confluence, and abelian behavior.
• The framework employs graph Laplacians and divisor theory to analyze stabilization and criticality, yielding explicit bounds and unique canonical configurations.
• Generalizations extend chip-firing to labeled, signed, flow, and higher-dimensional variants, with significant applications in computational algorithms and combinatorial optimization.

Chip-firing is a discrete dynamical system in which a configuration of chips evolves by local redistribution moves called firings. In the classical graphical model, a vertex may fire when it has at least as many chips as its degree, sending one chip along each incident edge; in divisor language the same operation is a Laplacian move. The term also covers labeled chip-firing, root-system firing, diffusion-type comparison dynamics, rational and quantized variants, higher-dimensional flow-firing, and matrix-governed generalizations, all organized around termination, confluence, periodicity, criticality, and canonical representatives (Goldberg, 2014, Guzman et al., 2015).

1. Classical graphical model and abelian behavior

On a connected graph $G$ on $n$ vertices with degrees $d_i$, a chip configuration is a vector $\mathbf a=(a_1,\dots,a_n)$. A firing at vertex $i$ is legal when $a_i\ge d_i$; it decreases $a_i$ by $d_i$ and increases each neighbor by $1$. If $x_i$ denotes the number of times vertex $n$0 fires, the duration of a terminating game is $n$1, and the initial and final configurations satisfy

$n$2

where $n$3 is the graph Laplacian. A theorem of Björner–Lovász–Shor states that if the game terminates, then the total number of firings and the final position are independent of the order of legal moves. The same line of analysis yields duration bounds from the Moore–Penrose pseudo-inverse $n$4: if $n$5 is a vertex that never fires, then $n$6, which leads to the implicit estimate

$n$7

and recovers the classical bound

$n$8

For strongly regular graphs this improves the scale from $n$9 to $d_i$0 (Goldberg, 2014).

The same abelian principle underlies many later formulations. In the standard unlabeled setting, confluence means that whenever stabilization occurs, the final stable state is independent of the order of legal firings. Much of the subject consists of identifying new state spaces and firing rules for which this order-independence survives, fails, or is replaced by weaker canonicality properties.

2. Divisors, reduction, and graph invariants

In divisor language on a finite connected undirected multigraph without loops, a divisor is an integer-valued function on vertices,

$d_i$1

with degree

$d_i$2

Firing a vertex $d_i$3 sends one chip along each incident edge, and a firing script $d_i$4 acts by

$d_i$5

This yields linear equivalence of divisors and the associated groups $d_i$6, $d_i$7, and $d_i$8. The basic decision problem is winnability: whether a divisor is linearly equivalent to an effective divisor. Dhar’s burning algorithm gives the canonical reduction procedure. Relative to a chosen vertex $d_i$9, a divisor is $\mathbf a=(a_1,\dots,a_n)$0-reduced if all non-$\mathbf a=(a_1,\dots,a_n)$1 vertices are nonnegative and no nonempty subset of $\mathbf a=(a_1,\dots,a_n)$2 can legally fire; a divisor $\mathbf a=(a_1,\dots,a_n)$3 is winnable iff its unique $\mathbf a=(a_1,\dots,a_n)$4-reduced representative $\mathbf a=(a_1,\dots,a_n)$5 satisfies $\mathbf a=(a_1,\dots,a_n)$6 (Mavani et al., 1 Aug 2025).

These notions support the chip-firing definition of gonality. In the Gonality Game, one player places $\mathbf a=(a_1,\dots,a_n)$7 chips, an adversary places a single $\mathbf a=(a_1,\dots,a_n)$8 chip, and the question is whether chip-firing can eliminate all debt. The smallest such $\mathbf a=(a_1,\dots,a_n)$9 is the gonality $i$0. Rank is defined in the same divisor-theoretic framework: $i$1 iff the Dollar Game is winnable for $i$2, and $i$3 iff $i$4 wins the Gonality Game. The graph Riemann–Roch theorem takes the form

$i$5

where the canonical divisor is $i$6 and $i$7. On the Platonic-solid graphs, the resulting gonality values are

$i$8

(Beougher et al., 2024).

3. Matrix, signed, and flow generalizations

A major generalization replaces the reduced graph Laplacian by an arbitrary invertible integer matrix $i$9, together with an $a_i\ge d_i$0-matrix $a_i\ge d_i$1 that determines which configurations are valid. Writing

$a_i\ge d_i$2

the admissible configurations are

$a_i\ge d_i$3

A firing subtracts $a_i\ge d_i$4 while remaining in $a_i\ge d_i$5. In this generalized setting, each equivalence class has a unique critical configuration, a unique superstable configuration, and a unique energy minimizer. The energy is

$a_i\ge d_i$6

and superstability is equivalent to energy minimization in the equivalence class. The number of critical or superstable representatives is $a_i\ge d_i$7 (Guzman et al., 2015).

Signed graphs modify the redistribution rule rather than the validity cone alone. If an edge is negative, then firing a vertex incident to that edge causes a loss of chips at both endpoints. The dynamics are encoded by the reduced signed Laplacian $a_i\ge d_i$8, and the critical group is

$a_i\ge d_i$9

Using the pair $a_i$0, valid configurations are the lattice points of a rational cone

$a_i$1

Each equivalence class contains exactly one critical configuration and exactly one $a_i$2-superstable configuration, both counted by $a_i$3. Vertex switching preserves the Smith normal form of $a_i$4, so switching-equivalent signed graphs have isomorphic critical groups (Cho et al., 2023).

A dual rather than signed extension places chips on a basis of the integral flow lattice

$a_i$5

For a chosen basis $a_i$6, the governing matrix is the dual Laplacian

$a_i$7

Every graph admits a basis for which $a_i$8 is an $a_i$9-matrix, making the resulting flow chip-firing avalanche finite. The associated $d_i$0-superstable flow configurations are in bijection with the spanning trees of $d_i$1. In planar graphs, and also for $d_i$2 and $d_i$3, one can choose such a flow $d_i$4-basis to consist of cycles of the underlying graph (Dochtermann et al., 2020).

4. Labeled chip-firing and root-system formulations

A labeled variant on the infinite path graph $d_i$5 assigns distinct labels $d_i$6 to chips. At an unstable vertex one chooses two chips $d_i$7, moves the lesser-labeled chip one step left, and moves the greater-labeled chip one step right. When all chips start at the origin and $d_i$8 is even, every legal sequence stabilizes in sorted order. The final stable configuration places the labels increasingly from left to right, so labeled chip-firing becomes a sorting mechanism. The same analysis shows that this is subtler than ordinary unlabeled chip-firing: odd starting multiplicity at the origin does not have unique stabilization (Hopkins et al., 2016).

This labeled line process has an exact reformulation in the language of root systems. In type $d_i$9, the positive roots are $1$0, and a labeled firing is exactly the move

$1$1

when the configuration vector $1$2 is orthogonal to that root. For an arbitrary crystallographic root system, central-firing is defined by

$1$3

After quotienting by the Weyl group, unlabeled central-firing is always terminating and confluent for every root system and every weight. In the simply-laced case it becomes a Dynkin-diagram number game defined on zero connected components (Galashin et al., 2017).

Interval-firing enlarges the firing condition from a single Coxeter hyperplane to an interval of affine hyperplanes. For $1$4, the symmetric process allows

$1$5

while the truncated process uses

$1$6

Both are confluent from every initial weight. Their stable points can be labeled consistently across all $1$7 by a piecewise-linear map $1$8, and the numbers of initial weights stabilizing to $1$9 are Ehrhart-like polynomials in $x_i$0: for the symmetric process this holds for every root system, and for the truncated process it holds for every simply laced root system. The coefficients are conjectured to be nonnegative integers (Galashin et al., 2017).

5. Infinite graphs, trees, and exact stabilization formulas

On the infinite path with parameters $x_i$1, a vertex fires whenever it has at least $x_i$2 chips, sending $x_i$3 chips left and $x_i$4 chips right. This game is finite and has a unique final state. Its analysis uses the Laurent polynomial

$x_i$5

with invariants

$x_i$6

The symmetric case $x_i$7 has an explicit closed form, and for coprime $x_i$8 the right side eventually settles into explicit periodic patterns governed by

$x_i$9

with

$n$00

for all sufficiently large $n$01. The left side then behaves like an elevated fractional-base system (Borodin et al., 2018).

On the infinite $n$02-ary tree with a self-loop at the root, starting from $n$03 chips at the root yields a rigid unlabeled stable configuration. If

$n$04

and

$n$05

then the occupied vertices form a perfect $n$06-ary tree of height $n$07, every vertex in the same layer has the same number of chips, and those layer counts are determined by the base-$n$08 digits of

$n$09

The same analysis gives explicit formulas for the number of fires on each layer, for the number of root fires, and for the total number of fires, together with self-similar difference sequences (Agrawal et al., 12 Jan 2025).

In the directed labeled $n$10-ary tree, starting with $n$11 labeled chips at the root and firing any vertex with at least $n$12 chips produces a stable configuration with exactly one chip on each vertex in layer $n$13. The number of possible stable configurations is recursive in $n$14-dimensional Catalan numbers; a distinguished stable permutation $n$15 is the radix-$n$16 digit-reversal permutation and maximizes the number of inversions among stable configurations. A complementary strategy-based analysis indexes stable permutations by $n$17, proves that distinct strategies yield distinct stable configurations, and expresses inversion and descent statistics through the Lehmer code of $n$18 (Inagaki et al., 2024, Inagaki et al., 12 Mar 2025).

Chip-firing on the Hasse diagram of $n$19 with $n$20 chips at the origin introduces an intermediate firing configuration $n$21, defined as the number of chips at $n$22 after all rows with smaller index than $n$23 have finished firing but before row $n$24 begins. The stable configuration is the parity shadow of $n$25: $n$26 holds one chip in the stable state exactly when $n$27 is odd. The table $n$28 is symmetric, has even diagonal entries $n$29, agrees with the scaled Pascal formula

$n$30

in the top triangle, and has a bottom triangle governed by explicit minimal rows $n$31 (Inagaki et al., 14 Jan 2026).

6. Diffusion, rational deformations, asymptotics, and computation

Not all chip-firing variants are threshold-based. In the diffusion game, every vertex simultaneously sends one chip to each neighbor with fewer chips, so the dynamics are comparison-driven and negative chip counts may occur. Eventual periodicity is proved for paths, cycles, wheels, complete graphs, and complete bipartite graphs. A configuration is tight—eventually fixed or eventually periodic with period length $n$32—iff some iterate has property plus. Perturbation Diffusion adds an initial forcing step from a subset $n$33, leading to the notions of $n$34-invoking and $n$35-invoking sets; a subset is $n$36-invoking exactly when it is Complementary Component Dominant, and on paths

$n$37

while the total number of $n$38-invoking subsets satisfies a Fibonacci-type recurrence (Duffy et al., 2016, Cox et al., 2020).

Parallel chip-firing admits a dense-graph limit through graphons. For a graphon $n$39 and chip configuration $n$40, the update uses

$n$41

when $n$42, and mass is conserved. Activity need not exist for every connected graphon, but it exists for every chip configuration when $n$43 has finite diameter, equivalently when $n$44 is connected and $n$45 almost everywhere. Under a smoothness condition, activity is continuous in cut distance and $n$46; for Erdős–Rényi graphs $n$47, suitable activity diagrams converge almost surely to a Devil’s staircase (Kiss et al., 2020).

Quantized rational chip-firing changes the arithmetic itself. In this model, each vertex in a firing set provisionally sends $n$48 chips to the sink and $n$49 chips to each non-sink neighbor outside the set, after which the total amount leaving or arriving at a vertex is rounded down. In the complete-graph case with $n$50, superstable configurations are rational parking functions, each equivalence class modulo firing and borrowing contains a unique $n$51-skeletal representative, and the quotient group is

$n$52

(Backman et al., 16 Mar 2026).

Higher-dimensional flow-firing moves integer flow around faces rather than redistributing chips on vertices. For the pulse configuration $n$53 on the two-dimensional grid complex, the three-regime theorem states: if $n$54, firing terminates uniquely in the Aztec diamond; if $n$55, termination is not unique but can end in the Aztec diamond; and if $n$56, termination is not unique, while

$n$57

prevents termination in the Aztec diamond altogether (Brauner et al., 2023).

A further deterministic analogue of stochastic dynamics is the hunger game, a greedy chip-firing-style process that mimics recurrent and absorbing Markov chains. For finite recurrent chains it concentrates around the stationary distribution with discrepancy falling off like $n$58, and for finite absorbing chains it approximates hitting measures and expected hitting times with the same $n$59 discrepancy scale. When transition probabilities are rational, the process is eventually periodic (Li et al., 2021).

Computation has become part of the field’s infrastructure. The Python package chipfiring models finite connected undirected multigraphs without loops through classes such as CFGraph, CFDivisor, and CFLaplacian, and implements Dhar’s algorithm, q-reduction, linear-equivalence testing, winnability, rank, and data import/export for graphs, divisors, orientations, and firing scripts (Mavani et al., 1 Aug 2025).