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We prove a precise relationship between the threshold state of thefixed-energy sandpile and the stationary state of Dhar's abelian sandpile: Inthe limit as the initial condition tends to negative infinity, the former isobtained by size-biasing the latter according to burst size, an avalanchestatistic. The question of whether and how these two states are related hasbeen a subject of some controversy since 2000.  The size-biasing in our result arises as an instance of a Markov renewaltheorem, and implies that the threshold and stationary distributions are notequal even in the negative infinity limit. We prove that nevertheless in thislimit the total amount of sand in the threshold state converges in distributionto the total amount of sand in the stationary state, confirming a conjecture ofPoghosyan, Poghosyan, Priezzhev and Ruelle.

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