Chip-firing references

### Parallel chip-firing:

F. Bagnoli, F. Cecconi, A. Flammini, and A. Vespignani, Short-period attractors and non-ergodic behavior in the deterministic fixed-energy sandpile model, Europhysics Letters, 2003. (physics paper with the original discovery of the devil's staircase; no theorems!)

M. A. Kiwi, R. Ndoundam, M. Tchuente and E. Goles, No polynomial bound for the period of the parallel chip firing game on graphs, Theoretical Comp. Sci., 1994. (counterexample to Bitar's conjecture on short period lengths)

E. Prisner, Parallel chip-firing on digraphs, Complex Systems, 1994. (contains several useful lemmas, including the fact that $a(\sigma) + a(2d-1-\sigma) = 1$.)

L. Levine, Parallel chip-firing on the complete graph, Ergodic Theory & Dynamical Systems, 2011. (proves devil's staircase on the complete graph, explains the connection to iteration of a circle map)

A. Fey, L. Levine and D. Wilson, The approach to criticality in sandpiles, Physical Review E 82, 2010. (Section 5 contains a staircase result plus proof for the Flower graph)

Special case of cycle graphs:

J. Jeffs and S. Seager, Chip-firing game on n-cycles, Graphs and Combinatorics, 1995.

L. Dall'Asta, Exact solution of the one-dimensional deterministic Fixed-Energy Sandpile, Physical Review Letters, 2006.

### Abelian networks:

Chip-firing and rotor-routing are special cases of a more general construction called an abelian network. You can find the definition on page 6 of Lionel's grant proposal. Browsing through the proposal might be helpful if you want to see where your project fits into the bigger picture of research in this area.

The original idea for abelian networks is due to Deepak Dhar in the following paper:

D. Dhar, Theoretical studies of self-organized criticality, Physica A, 2006. (search for the text "abelian distributed processors")

An older version is here:

### Applications outside math

Physicists are interested in chip-firing ("sandpiles") as a model of what they call "self-organized criticality". Per Bak - one of the inventors of the abelian sandpile model - wrote a popular book on this idea called How Nature Works. (I haven't read it, so I'd be curious of any reviews.)

Parallel chip-firing is a natural model of systems that synchronize themselves. All kinds of systems do this: everything from pendulum clocks to fireflies to neurons, pacemaker cells, and moons and planets in the solar system. Steven Strogatz wrote a great popular book on this called Sync.

page revision: 9, last edited: 06 Jul 2011 09:32