# Non-criticality criteria for Abelian sandpile models with sources and sinks

Abstract : We prove that the Abelian sandpile model on a random binary and binomial tree, as introduced in \cite{rrs}, is not critical for all branching probabilities $p<1$; by estimating the tail of the annealed survival time of a random walk on the binary tree with randomly placed traps, we obtain some more information about the exponential tail of the avalanche radius. Next we study the sandpile model on $\Zd$ with some additional dissipative sites: we provide examples and sufficient conditions for non-criticality; we also make a connection with the parabolic Anderson model. Finally we initiate the study of the sandpile model with both sources and sinks and give a sufficient condition for non-criticality in the presence of a finite number of sources, using a connection with the homogeneous pinning model.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-01670417
Submitted on : Thursday, December 21, 2017 - 1:32:17 PM
Last modification on : Thursday, November 7, 2019 - 3:44:14 PM

### Citation

Frank Redig, Wioletta M. Ruszel, Ellen Saada. Non-criticality criteria for Abelian sandpile models with sources and sinks. Journal of Mathematical Physics, American Institute of Physics (AIP), 2018, 59 (6), pp.1-16. ⟨10.1063/1.5022128⟩. ⟨hal-01670417⟩

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