The 1D activated random walk mixes rapidly to a stationary state whose avalanche sizes obey a power law and whose limiting density equals the critical density of the fixed-energy version.
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Activated random walk on the complete graph has a Gumbel scaling limit for sleeping particles and hyperuniform stationary law when the sink probability satisfies exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2}.
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Activated random walk exhibits self-organized criticality
The 1D activated random walk mixes rapidly to a stationary state whose avalanche sizes obey a power law and whose limiting density equals the critical density of the fixed-energy version.
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Scaling limit and density conjecture for activated random walk on the complete graph
Activated random walk on the complete graph has a Gumbel scaling limit for sleeping particles and hyperuniform stationary law when the sink probability satisfies exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2}.