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

@article{Levine2014ThresholdSA, title={Threshold State and a Conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle}, author={Lionel Levine}, journal={Communications in Mathematical Physics}, year={2014}, volume={335}, pages={1003-1017} }

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…

## 15 Citations

Sandpiles on the Square Lattice

- MathematicsCommunications in Mathematical Physics
- 2019

AbstractWe give a non-trivial upper bound for the critical density when stabilizing i.i.d. distributed sandpiles on the lattice $${\mathbb{Z}^2}$$Z2 . We also determine the asymptotic spectral gap,…

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The abelian sandpile model defines a Markov chain whose states are integer-valued functions on the vertices of a simple connected graph
G
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Non-fixation for Conservative Stochastic Dynamics on the Line

- Mathematics
- 2015

We consider activated random walk (ARW), a model which generalizes the stochastic sandpile, one of the canonical examples of self organized criticality. Informally ARW is a particle system on…

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We consider Activated Random Walk (ARW), a particle system with mass conservation, on the cycle $\mathbb{Z}/n\mathbb{Z}$. One starts with a mass density $\mu>0$ of initially active particles, each of…

Abelian networks IV. Dynamics of nonhalting networks

- MathematicsMemoirs of the American Mathematical Society
- 2022

An intrinsic definition of the torsion group of a finite irreducible (halting or nonhalting) abelian network is given, and it is shown that it coincides with the critical group of Bond and Levine (2016) if the network is halting.

Absorbing-state transition for Stochastic Sandpiles and Activated Random Walks

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- 2014

We study the dynamics of two conservative lattice gas models on the infinite d-dimensional hypercubic lattice: the Activated Random Walks (ARW) and the Stochastic Sandpiles Model (SSM), introduced in…

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- 2021

Activated Random Walk (ARW) is an interacting particle system on the d-dimensional lattice Z. On a finite subset V ⊂ Z it defines a Markov chain on {0, 1} . We prove that when V is a Euclidean ball…

CoEulerian graphs

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- 2015

We suggest a measure of “Eulerianness” of a finite directed graph and define a class of “coEulerian” graphs. These are the graphs whose Laplacian lattice is as large as possible. As an application,…

Abelian Networks I. Foundations and Examples

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- 2016

In Deepak Dhar's model of abelian distributed processors, automata occupy the vertices of a graph and communicate via the edges. We show that two simple axioms ensure that the final output does not…

The Avalanche Polynomial of a Graph

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The (multivariate) avalanche polynomial that enumerates the toppling sequences of all principal avalanches is introduced and is characterized for trees, cycles, wheels, and complete graphs.

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