阈态与Poghosyan，Poghosyan，Priezzhev和Ruelle的一个猜想
Threshold state and a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle
摘要：
我们证明了固定能量沙堆的阈值状态和Dhar的阿贝尔沙堆的定态之间的精确关系：在初始条件趋于负无穷大的极限下，前者是通过根据爆发大小（雪崩统计量）对后者进行大小偏置而获得的。自2000年以来，这两种状态是否相关以及如何相关的问题一直存在争议。在我们的结果中的大小偏置作为马尔可夫更新定理的一个实例而出现，并且意味着即使在负无穷大极限中，阈值分布和平稳分布也不相等。我们证明，尽管如此，在这个极限中的阈值状态的沙子的总量收敛于分布的沙子的总量在静止状态，证实了Poghosyan，Poghosyan，Priezzhev和Ruelle的猜想。
英文摘要：
We prove a precise relationship between the threshold state of the fixed-energy sandpile and the stationary state of Dhar's abelian sandpile: In the limit as the initial condition tends to negative infinity, the former is obtained by size-biasing the latter according to burst size, an avalanche statistic. The question of whether and how these two states are related has been a subject of some controversy since 2000. The size-biasing in our result arises as an instance of a Markov renewal theorem, and implies that the threshold and stationary distributions are not equal even in the negative infinity limit. We prove that nevertheless in this limit the total amount of sand in the threshold state converges in distribution to the total amount of sand in the stationary state, confirming a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle.
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