From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models

Asselah, Amine; Gaudillière, Alexandre

doi:10.1214/12-aop762

Cited by 42 publications

(111 citation statements)

References 13 publications

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“…Although external DLA has had more attention in the physics literature, there has been much more mathematical progress on internal DLA, beginning with works by Diaconis and Fulton and by Lawler, Bramson, and Griffeath from the early 1990's [DF91,LBG92]. More recently, the second author was part of an IDLA paper series with Levine and Jerison that describes the size and nature of internal DLA fluctuations in great detail and relates these fluctuations to a variant of the GFF [JLS12, JLS10, JLS11], see also [AG13a,AG13b].…”

Section: Diffusion Limited Aggregation (Dla)mentioning

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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Section: Diffusion Limited Aggregation (Dla)mentioning

confidence: 99%

Quantum Loewner evolution

Miller¹,

Sheffield⋆²

2016

Duke Math. J.

124

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“…We discuss a possible approach to proving matching lower bounds, as well as several other open questions, in Section 3. Using methods rather different from ours, Asselah and Gaudillière [AG10b] have recently announced an independently obtained upper bound of order log 2 r in all dimensions d ≥ 2. In the third part, entitled Internal DLA and the Gaussian free field [JLS11], we show that the fluctuations from circularity have a weak limit which is a variant of the Gaussian free field.…”

Section: Introductionmentioning

confidence: 98%

Logarithmic fluctuations for internal DLA

Jerison

Levine

Sheffield⋆

2011

J. Amer. Math. Soc.

135

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Let each of n particles starting at the origin in Z 2 perform simple random walk until reaching a site with no other particles. Lawler, Bramson, and Griffeath proved that the resulting random set A(n) of n occupied sites is (with high probability) close to a disk B r of radius r = n/π. We show that the discrepancy between A(n) and the disk is at most logarithmic in the radius: i.e., there is an absolute constant C such that with probability 1, B r−C log r ⊂ A(πr 2 ) ⊂ B r+C log r for all sufficiently large r.

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“…The research in the field of fractal theory is a hotspot. It is good at the growth model such as Diffusion Limited Aggregation model (DAL) [5][6][7][8], Lindenmayer system (L-system) [9][10][11][12]. DLA is the process whereby particles undergoing a random walk due to Brownian motion cluster together to form aggregates of such particles.…”

Section: Figure 1 Graphical Design Of Batikmentioning

confidence: 99%