Surface structure and anisotropy of Eden clusters

Freche, Peter; Stauffer, Dietrich; Stanley, H. Eugene

doi:10.1088/0305-4470/18/18/009

Cited by 68 publications

(25 citation statements)

References 14 publications

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“…At large radius, r > 1000, for circular clusters grown over a square lattice, a crossover comes out. Several authors [12,13] this increase in the exponent is the lattice anisotropy effect and the origin of the reference system [14]. In this figure, we show two insets that represent cluster interfaces (r vs. θ) in two different regions: Left inset (r < 1000) does not show any characteristic length scale.…”

Section: Circular Eden Modelmentioning

confidence: 89%

New dynamic scaling in increasing systems

Pastor

Galeano

2007

Open Physics

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Abstract:We report a new dynamic scaling ansatz for systems whose system size is increasing with time. We apply this new hypothesis in the Eden model in two geometries. In strip geometry, we impose the system to increase with a power law, L ∼ h a . In increasing linear clusters, if a < 1/z, where z is the dynamic exponent, the correlation length reaches the whole system, and we find two regimes: the first, where the interface fluctuations initially grow with an exponent β = 0.3, and the second, where a crossover comes out and fluctuations evolve asis not a crossover and fluctuations keep on growing in a unique regimen with the same exponent β. In particular, in circular geometry, a = 1, we find this kind of regime and in consequence, a unique regime holds.

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Section: Circular Eden Modelmentioning

confidence: 89%

New dynamic scaling in increasing systems

Pastor

Galeano

2007

Open Physics

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“…With a very quick glance at Figure 1.5, one might guess that the limiting shape of the Eden model (whose existence is guaranteed by the Cox and Durrett theorem mentioned above) is circular; but early and subsequent computer experiments suggest that though the deterministic limit shape is "roundish" it is not exactly circular (e.g., [FSS85,BH91]). …”

Section: 5mentioning

confidence: 99%

Quantum Loewner evolution

Miller¹,

Sheffield⋆²

2016

Duke Math. J.

124

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What is the scaling limit of diffusion limited aggregation (DLA) in the plane? This is an old and famously difficult question. One can generalize the question in two ways: first, one may consider the dielectric breakdown model η-DBM, a generalization of DLA in which particle locations are sampled from the η-th power of harmonic measure, instead of harmonic measure itself. Second, instead of restricting attention to deterministic lattices, one may consider η-DBM on random graphs known or believed to converge in law to a Liouville quantum gravity (LQG) surface with parameter γ ∈ [0, 2].In this generality, we propose a scaling limit candidate called quantum Loewner evolution, QLE(γ 2 , η). QLE is defined in terms of the radial Loewner equation like radial SLE, except that it is driven by a measure valued diffusion ν t derived from LQG rather than a multiple of a standard Brownian motion. We formalize the dynamics of ν t using an SPDE. For each γ ∈ (0, 2], there are two or three special values of η for which we establish the existence of a solution to these dynamics and explicitly describe the stationary law of ν t .We also explain discrete versions of our construction that relate DLA to loop-erased random walk and the Eden model to percolation. A certain "reshuffling" trick (in which concentric annular regions are rotated randomly, like slot machine reels) facilitates explicit calculation.We propose QLE(2, 1) as a scaling limit for DLA on a random spanningtree-decorated planar map, and QLE(8/3, 0) as a scaling limit for the Eden model on a random triangulation. We propose using QLE(8/3, 0) to endow pure LQG with a distance function, by interpreting the region explored by a branching variant of QLE(8/3, 0), up to a fixed time, as a metric ball in a random metric space.

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“…Starting with a seed particle at the origin, at each time step one adds a new particle on a randomly chosen site on the perimeter of the cluster [1,22]. This however turned out to induce anisotropy due to the lattice structure [1,23] unless a specific growth rule is introduced, such as the growth probability dependent on the number of the occupied nearest-neighbors [24,25]. Off-lattice versions of the Eden model have therefore been considered occasionally in the literature [25][26][27][28], but in most cases time t is measured by the global radius h of the radial cluster, which is not correct at finite times because the two quantities are in general connected by [25,27,30], a random number for the angular position of the new particle is generated within [π/6, φ − π/6], where φ is defined in the sketch and π/6 is due to the finite radius of the new particle.…”

Section: Modelmentioning

confidence: 99%

Statistics of circular interface fluctuations in an off-lattice Eden model

Takeuchi¹

2012

J. Stat. Mech.

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Scale-invariant fluctuations of growing interfaces are studied for circular clusters of an off-lattice variant of the Eden model, which belongs to the (1 + 1)-dimensional Kardar-Parisi-Zhang (KPZ) universality class. Statistical properties of the height (radius) fluctuations are numerically determined and compared with the recent theoretical developments as well as the author's experimental result on growing interfaces in turbulent liquid crystal [K. A. Takeuchi and M. Sano, arXiv:1203.2530]. We focus in particular on analytically unsolved properties such as the temporal correlation function and the persistence probability in space and time. Good agreement with the experiment is found in characteristic quantities for them, which implies that the geometry-dependent universality of the KPZ class holds here as well, but otherwise a few dissimilarities are also found. Finite-time corrections in the cumulants of the distribution are also studied and shown to decay, for the mean, as t −2/3 within the time window of the simulations, instead of t −1/3 which arises as the typical leading term in the previously known cases.

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Surface structure and anisotropy of Eden clusters

Cited by 68 publications

References 14 publications

New dynamic scaling in increasing systems

New dynamic scaling in increasing systems

Quantum Loewner evolution

Statistics of circular interface fluctuations in an off-lattice Eden model

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