Mathematics – Combinatorics
Scientific paper
2008-11-17
Mathematics
Combinatorics
22 pages, 2 figures. Rewrote sections 2 and 3 in terms of "generalized chip configurations" to address referee comments. To ap
Scientific paper
10.1017/S0143385710000088
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 equals the Poincare rotation number of the circle map. The stairs of the devil's staircase correspond to periodic chip-firing states of small period.
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