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

Abstract

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.