The Infinite Volume Limit of Dissipative Abelian Sandpiles

We give a non-trivial upper bound for the critical density when stabilizing i.i.d. distributed sandpiles on the lattice Z 2 . We also determine the asymptotic spectral gap, asymptotic mixing time and prove a cutoff phenomenon for the recurrent state abelian sandpile model on the torus pZ{mZq 2 . The techniques use analysis of the space of functions on Z 2 which are harmonic modulo 1. In the course of our arguments, we characterize the harmonic modulo 1 functions in ℓ p pZ 2 q as linear combinations of certain discrete derivatives of Green's functions, extending a result of Schmidt and Verbitskiy [35].2010 Mathematics Subject Classification. Primary 82C20, 60B15, 60J10. Key words and phrases. Abelian sandpile model, random walk on a group, spectral gap, cutoff phenomenon, critical density, harmonic modulo 1 functions.

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“…(Possibly Lemma 14 could help resolve this question.) Some further studies of sandpile dynamics on Z d are [31,25,6].…”

Section: 2mentioning

confidence: 99%

Sandpiles on the Square Lattice

Hough

Jerison

Levine

2019

Commun. Math. Phys.

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“…In the scaling limit, the subtracted cluster variables give rise to the fields h S (z), whose mixed correlators are given by the terms displayed in equation (52). Interestingly, it has been observed [33] that these fields have a realization in terms of the symplectic free fermions, which is discussed at the end of Section 4.…”

Section: Minimal Height Cluster Correlationsmentioning

confidence: 90%

“…In this case, it has been argued that indeed criticality is broken, resulting in an exponential decay of the correlations [33,50,51]. A mathematically rigorous proof that all correlations decay exponentially has been provided in Ref [52,53]. Presumably, a nonzero density of dissipative sites could be sufficient to break criticality, but to our knowledge, this possibility has not been investigated.…”

Section: The Massive Sandpile Modelmentioning

confidence: 99%

Sandpile Models in the Large

Ruelle

2021

Front. Phys.

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This contribution is a review of the deep and powerful connection between the large-scale properties of critical systems and their description in terms of a field theory. Although largely applicable to many other models, the details of this connection are illustrated in the class of two-dimensional Abelian sandpile models. Bulk and boundary height variables, spanning tree–related observables, boundary conditions, and dissipation are all discussed in this context and found to have a proper match in the field theoretic description.

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Infinite Volume Limit for the Stationary Distribution of Abelian Sandpile Models

Athreya

Járai

2004

Commun. Math. Phys.

104

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C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a PNA Probability, Networks and Algorithms Probability, Networks and AlgorithmsInfinite volume limit for the stationary distribution of Abelian sandpile models Siva R. Athreya, Antal A. Járai REPORT PNA-E0304 DECEMBER 8, 2003CWI is the National Research Institute for Mathematics and Computer Science. It is sponsored by the Netherlands Organization for Scientific Research (NWO). CWI is a founding member of ERCIM, the European Research Consortium for Informatics and Mathematics.CWI's research has a theme-oriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms.

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The Infinite Volume Limit of Dissipative Abelian Sandpiles

Cited by 11 publications

References 15 publications

Sandpiles on the Square Lattice

Sandpiles on the Square Lattice

Sandpile Models in the Large

Infinite Volume Limit for the Stationary Distribution of Abelian Sandpile Models

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