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Chip-Firing And A Devil's Staircase

Abstract : The devil's staircase ― a continuous function on the unit interval $[0,1]$ which is not constant, yet is locally constant on an open dense set ― is the sort of exotic creature a combinatorialist might never expect to encounter in "real life.'' We show how a devil's staircase arises from the combinatorial problem of parallel chip-firing on the complete graph. This staircase helps explain a previously observed "mode locking'' phenomenon, as well as the surprising tendency of parallel chip-firing to find periodic states of small period.
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Lionel Levine. Chip-Firing And A Devil's Staircase. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. pp.573-584, ⟨10.46298/dmtcs.2693⟩. ⟨hal-01185385⟩



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