Internal Diffusion Limited Aggregation

Lawler, Gregory F.; Bramson, Maury; Griffeath, David

doi:10.1214/aop/1176989542

Cited by 144 publications

(237 citation statements)

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“…Let s(x) = inf{f (x)|f is continuous, superharmonic and f ≥ γ} (20) be the least superharmonic majorant of γ, and let…”

Section: Superharmonic Potentialsmentioning

confidence: 99%

“…The Diaconis-Fulton smash sum generalizes the model of internal diffusion limited aggregation ("internal DLA") studied in [20], and in fact was part of the original motivation for that paper. In classical internal DLA, we start with n particles at the origin o ∈ Z d and let each perform simple random walk until it reaches an unoccupied site.…”

mentioning

confidence: 94%

“…We will use the term internal DLA to refer to the same process run from an arbitrary starting configuration of particles. In this broader sense of the term, both the Diaconis-Fulton sum and the model studied in [20] are particular cases of internal DLA.…”

mentioning

confidence: 99%

“…Our shape theorem for multiple point sources, which is deduced from Theorem 1.3 using the main results of [20] and [21], is the following. Theorem 1.4.…”

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confidence: 99%

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Scaling limits for internal aggregation models with multiple sources

Levine

Peres

2010

JAMA

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We study the scaling limits of three different aggregation models on Z d : internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in R d . In particular, internal DLA has a deterministic scaling limit. We find that the scaling limits are quadrature domains, which have arisen independently in many fields such as potential theory and fluid dynamics. Our results apply both to the case of multiple point sources and to the DiaconisFulton smash sum of domains.

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“…Let s(x) = inf{f (x)|f is continuous, superharmonic and f ≥ γ} (20) be the least superharmonic majorant of γ, and let…”

Section: Superharmonic Potentialsmentioning

confidence: 99%

mentioning

confidence: 94%

mentioning

confidence: 99%

“…Our shape theorem for multiple point sources, which is deduced from Theorem 1.3 using the main results of [20] and [21], is the following. Theorem 1.4.…”

mentioning

confidence: 99%

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Scaling limits for internal aggregation models with multiple sources

Levine

Peres

2010

JAMA

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“…Rotor sequence (←, ↑, →, ↓) (←, ↑, ↓, →) (←, →, ↑, ↓) average ∆(n) for 2 · 10 6 < n ≤ 3 · 10 6 1.920 ± 0.004 1.782 ± 0.003 1.781 ± 0.003 maximal ∆(n) for n ≤ 3 · 10 6 2.541 ± 0.051 2.351 ± 0.053 2.364 ± 0.067…”

Section: Aggregating Modelmentioning

confidence: 99%

Deterministic Random Walks on the Two-Dimensional Grid

Doerr

Friedrich

2006

Algorithms and Computation

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Jim Propp's rotor router model is a deterministic analogue of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbors in a fixed order. We analyze the difference between Propp machine and random walk on the infinite two-dimensional grid. It is known that, apart from a technicality, independent of the starting configuration, at each time, the number of chips on each vertex in the Propp model deviates from the expected number of chips in the random walk model by at most a constant. We show that this constant is approximately 7.8, if all vertices serve their neighbors in clockwise or counterclockwise order and 7.3 otherwise. This result in particular shows that the order in which the neighbors are served makes a difference. Our analysis also reveals a number of further unexpected properties of the two-dimensional Propp machine.

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Design and assessment of a novel static mixer

Al-Atabi

2010

Can J Chem Eng

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This paper examines the performance of a novel static mixer comprising a circular tube fitted with eight alternating equi-spaced semicircular rigid insert (baffles) as the mixer elements. Experiments were carried out to obtain the coefficient of variance (CoV) for the mixing of two streams of water and brine for Reynolds number between 60 and 700. Decreasing the baffles clearance ratio significantly reduces the CoV but at a cost of an increase in the pressure drop across the static mixer. The presence of the mixing elements (baffles) promotes a non-laminar, turbulent-like flow which considerably enhances the mixing. The static mixer described here represents a cost effective, easy to manufacture, low maintenance, and flexible alternative to the more sophisticated static mixers currently in use.Ce document examine le rendement d'un nouveau mélangeur statique composé d'un tube circulaire comportant huit garnitures rigides demicirculaireséquidistantes alternantes (déflecteurs) commeéléments du mélangeur. Des expériences ontété réalisées pour obtenir le coefficient de variation (CV) pour le mélange de deux jets d'eau et de saumure pour un nombre de Reynolds entre 60 et 700. Diminuer le rapport de dégagement des déflecteurs réduit considérablement le CV, mais il en coûte une augmentation de la perte de charge dans tout le mélangeur statique. La présence deséléments de mélange (déflecteurs) favorise unécoulement non laminaire semblableà de la turbulence qui accroît considérablement le mélange. Le mélangeur statique décrit ici représente une solution de rechange souple,économique, facileà fabriquer et présentant peu d'entretien aux mélangeurs statiques plusélaborés utilisésà l'heure actuelle.

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Internal Diffusion Limited Aggregation

Cited by 144 publications

References 0 publications

Scaling limits for internal aggregation models with multiple sources

Scaling limits for internal aggregation models with multiple sources

Deterministic Random Walks on the Two-Dimensional Grid

Design and assessment of a novel static mixer

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