# 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.
Type de document :
Pré-publication, Document de travail
MAP5 2017-27. 17 pages. 2017
Domaine :

https://hal.archives-ouvertes.fr/hal-01670417
Soumis le : jeudi 21 décembre 2017 - 13:32:17
Dernière modification le : jeudi 11 janvier 2018 - 06:19:45

### Identifiants

• HAL Id : hal-01670417, version 1
• ARXIV : 1712.06529

### Citation

Frank Redig, Wioletta M. Ruszel, Ellen Saada. Non-criticality criteria for Abelian sandpile models with sources and sinks. MAP5 2017-27. 17 pages. 2017. 〈hal-01670417〉

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