Threshold State and a Conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle

  title={Threshold State and a Conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle},
  author={Lionel Levine},
  journal={Communications in Mathematical Physics},
  • Lionel Levine
  • Published 13 February 2014
  • Mathematics
  • Communications in Mathematical Physics
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 s0 tends to $${-\infty}$$-∞ , 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… 
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