Fluctuations for internal DLA on the comb

Asselah, Amine; Rahmani, Houda

doi:10.1214/14-aihp629

Cited by 12 publications

(13 citation statements)

References 13 publications

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“…Similar results were also obtained independently in [2,1]. The model has been also studied on the comb lattice in [3], and similar limiting shape theorems have been obtained.…”

Section: Introductionsupporting

confidence: 78%

Border aggregation model

Thacker

Volkov

2018

Ann. Appl. Probab.

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Start with a graph with a subset of vertices called the border. A particle released from the origin performs a random walk on the graph until it comes to the immediate neighbourhood of the border, at which point it joins this subset thus increasing the border by one point. Then a new particle is released from the origin and the process repeats until the origin becomes a part of the border itself. We are interested in the total number ξ of particles to be released by this final moment.We show that this model covers OK Corral model as well as the erosion model, and obtain distributions and bounds for ξ in cases where the graph is star graph, regular tree, and a d−dimensional lattice.

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“…Similar results were also obtained independently in [2,1]. The model has been also studied on the comb lattice in [3], and similar limiting shape theorems have been obtained.…”

Section: Introductionsupporting

confidence: 78%

Border aggregation model

Thacker

Volkov

2018

Ann. Appl. Probab.

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“…• "4 2 3 − " and "5 1 3 − ," followed by "4 2 3 ." • "4 4 9 " and "5 1 3 ," which are proved in tandem.…”

Section: 4mentioning

confidence: 72%

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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“…Theorem 7. 4 The directed infinite-volume IDLA forest F ∞ satisfies the following properties: 4). This result is one of the original motivation of this paper.…”

Section: The Directed Infinite-volume Idla Forestmentioning

confidence: 99%

“…Recently, many variants of this problem have been considered. In particular, IDLA on discrete groups with polynomial or exponential growth have been studied in [5,10], on non-amenable graphs in [18], with multiple sources in [27], on supercritical percolation clusters in [15,33], on comb lattices in [4,19], on cylinder graphs in [23,28,34], constructed with drifted random walks in [29] or with uniform starting points in [7]. A random infinite tree T ∞ can be associated with the sequence of IDLA aggregates (A n ) n≥0 defined above in a very natural way.…”

Section: Introductionmentioning

confidence: 99%

IDLA aggregates with infinitely many sources and the directed IDLA forest

Chenavier,

Coupier,

Rousselle

2020

Preprint

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We investigate three types of Internal Diffusion Limited Aggregation (IDLA) models. These models are based on simple random walks on Z 2 with infinitely many sources that are the points of the vertical axis I (∞) = {0} × Z. Various properties are provided, such as stationarity, mixing, stabilization and shape theorems. Our results allow us to define a new directed (w.r.t. the horizontal direction) random forest spanning Z 2 , based on an IDLA protocol, which is invariant in distribution w.r.t. vertical translations.

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Fluctuations for internal DLA on the comb

Cited by 12 publications

References 13 publications

Border aggregation model

Border aggregation model

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

IDLA aggregates with infinitely many sources and the directed IDLA forest

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