The paper proves that criticality of the 1D dissipative abelian sandpile with arbitrary traps is equivalent to divergence of an explicit recursive sequence, giving a three-regime classification with thresholds at ((3+√5)/2)^k and a^{2^k}.
The Abelian sandpile and related models
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Criticality of the abelian sandpile in dimension 1
The paper proves that criticality of the 1D dissipative abelian sandpile with arbitrary traps is equivalent to divergence of an explicit recursive sequence, giving a three-regime classification with thresholds at ((3+√5)/2)^k and a^{2^k}.