Threshold state and a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle

Understand

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.