Divisible Sandpile on Sierpinski Gasket Graphs

Huss, Wilfried; Sava-Huss, Ecaterina

doi:10.1142/s0218348x19500324

Cited by 8 publications

(20 citation statements)

References 15 publications

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“…In this section we will establish a lower bound in Lemma 3.5 needed for our arguments in Section 4. We derive this from the results in [HSH19], which are briefly outlined here. First, we define the sandpile cluster and the odometer function.…”

Section: The Divisible Sandpilementioning

confidence: 99%

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On the fluctuations of Internal DLA on the Sierpinski gasket graph

Heizmann¹

2021

Preprint

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Internal diffusion limited aggregation (IDLA) is a random aggregation model on a graph G, whose clusters are formed by random walks started in the origin (some fixed vertex) and stopped upon visiting a previously unvisited site. On the Sierpinski gasket graph the asymptotic shape is known to be a ball in the usual graph metric. In this paper we establish bounds for the fluctuations of the cluster from its asymptotic shape.

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Section: The Divisible Sandpilementioning

confidence: 99%

“…The main improvement from the already existing result is based on the analysis of the sandpile model, which is a different aggregation model. We will use the exact calculation of the odometer function of the sandpile model from [HSH19] to develop an improved inner bound and the outer bound then follows analogously from [CHSHT].…”

Section: Introductionmentioning

confidence: 99%

On the fluctuations of Internal DLA on the Sierpinski gasket graph

Heizmann¹

2021

Preprint

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“…On SG an exact expression of h can be obtained when n = 2 k for k ∈ N. Indeed, in the proof of Lemma 3.1 below, we will encounter a related Dirichlet problem with ∆h = 1 replaced by ∆h = 1 − |B o (n)|δ o , whose solution is fully addressed in [HSH18]. We believe it is possible to find sharper estimates of h for all radii n using harmonic splines [SU00], see also the recent work [GKQS14].…”

Section: Sierpinski Gasket Graphsmentioning

confidence: 99%

“…The limit functions µ and u do not depend on the order of topplings. A precise analysis of the divisible sandpile model on SG has been done in [HSH18], where the authors obtain the limit shape for the sandpile cluster.…”

Section: The Inner Bound For the Idla Clustermentioning

confidence: 99%

Internal DLA on Sierpinski gasket graphs

Chen,

Huss,

Sava-Huss

et al. 2017

Preprint

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Internal diffusion-limited aggregation (IDLA) is a stochastic growth model on a graph G which describes the formation of a random set of vertices growing from the origin (some fixed vertex) of G. Particles start at the origin and perform simple random walks; each particle moves until it lands on a site which was not previously visited by other particles. This random set of occupied sites in G is called the IDLA cluster. In this paper we consider IDLA on Sierpinski gasket graphs, and show that the IDLA cluster fills balls (in the graph metric) with probability 1.

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“…In this subsection, G is the one-sided gasket graph SG, and the "origin" o is the corner vertex of SG; see again Figure 1. Propositions 1.1 and 1.2, which are the main theorems of [15] and [33], respectively, were proved on the double-sided SG. It is straightforward to modify the proof to work on the one-sided SG, which results in no change in the statement.…”

mentioning

confidence: 94%

Laplacian growth & sandpiles on the Sierpinski gasket: limit shape universality and exact solutions

Chen,

Kudler-Flam

2018

Preprint

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We establish quantitative spherical shape theorems for rotor-router aggregation and abelian sandpile growth on the graphical Sierpinski gasket (SG) when particles are launched from the corner vertex.In particular, the abelian sandpile growth problem is exactly solved via a recursive construction of self-similar sandpile tiles. We show that sandpile growth and patterns exhibit a (2•3 n )-periodicity as a function of the initial mass. Moreover, the cluster explodes-increments by more than 1 in radius-at periodic intervals, a phenomenon not seen on Z d or trees. We explicitly characterize all the radial jumps, and use the renewal theorem to prove the scaling limit of the cluster radius, which satisfies a power law modulated by log-periodic oscillations. In the course of our proofs we also establish structural identities of the sandpile groups of subgraphs of SG with two different boundary conditions, notably the corresponding identity elements conjectured by Fairchild, Haim, Setra, Strichartz, and Westura.Our main theorems, in conjunction with recent results of Chen, Huss, Sava-Huss, and Teplyaev, establish SG as a positive example of a state space which exhibits "limit shape universality," in the sense of Levine and Peres, among the four Laplacian growth models: divisible sandpiles, abelian sandpiles, rotor-router aggregation, and internal diffusion-limited aggregation (IDLA). We conclude the paper with conjectures about radial fluctuations in IDLA on SG, possible extensions of limit shape universality to other state spaces, and related open problems.

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Divisible Sandpile on Sierpinski Gasket Graphs

Cited by 8 publications

References 15 publications

On the fluctuations of Internal DLA on the Sierpinski gasket graph

On the fluctuations of Internal DLA on the Sierpinski gasket graph

Internal DLA on Sierpinski gasket graphs

Laplacian growth & sandpiles on the Sierpinski gasket: limit shape universality and exact solutions

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