The Apollonian structure of integer superharmonic matrices

Levine, Lionel; Pegden, Wesley; Smart, Charles K.

doi:10.4007/annals.2017.186.1.1

Cited by 31 publications

(48 citation statements)

References 33 publications

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“…(4) As proved in [47] (and illustrated in Figure 7), the maximal elements of Γ(Z 2 ) correspond to the circles in the Apollonian band packing of R 2 . Because the radius and the coordinates of the center of each such circle are rational numbers, each maximal element of Γ(Z 2 ) is a matrix with rational entries.…”

Section: Open Problemsmentioning

confidence: 81%

“…An explicit description of Γ(Z 2 ) appears in [47] (see Figure 7), and explicit fractal solutions of the sandpile PDE…”

Section: Multiple Sources; Quadrature Domainsmentioning

confidence: 99%

“…(i) According to the main theorem of[47], the set of allowed Hessians Γ(Z 2 ) is the union of slope 1 cones based at the circles of an Apollonian circle packing in the plane of 2 × 2 real symmetric matrices of trace 2. (ii) The same set viewed from above: Color of point (a, b) indicates the largest c such that c−a b b c+a ∈ Γ(Z 2 ) The rectangle shown, (a, b) ∈ [0, 2] × [0, 4], extends periodically to the entire plane.…”

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confidence: 99%

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Laplacian growth, sandpiles, and scaling limits

Levine

Peres

2017

Bull. Amer. Math. Soc.

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Laplacian growth is the study of interfaces that move in proportion to harmonic measure. Physically, it arises in fluid flow and electrical problems involving a moving boundary. We survey progress over the last decade on discrete models of (internal) Laplacian growth, including the abelian sandpile, internal DLA, rotor aggregation, and the scaling limits of these models on the lattice Z d as the mesh size goes to zero. These models provide a window into the tools of discrete potential theory: harmonic functions, martingales, obstacle problems, quadrature domains, Green functions, smoothing. We also present one new result: rotor aggregation in Z d has O(log r) fluctuations around a Euclidean ball, improving a previous power-law bound. We highlight several open questions, including whether these fluctuations are O(1).

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Section: Open Problemsmentioning

confidence: 81%

“…An explicit description of Γ(Z 2 ) appears in [47] (see Figure 7), and explicit fractal solutions of the sandpile PDE…”

Section: Multiple Sources; Quadrature Domainsmentioning

confidence: 99%

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confidence: 99%

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Laplacian growth, sandpiles, and scaling limits

Levine

Peres

2017

Bull. Amer. Math. Soc.

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“…• allows to consider constant time basic operations, • preserves the "dynamical complexity", within P and with P-hard problems. Unbounded values bring considerations of another kind (namely, of computing fixed points from a single column of sand grains, as in [45,46,47,58]) not necessary to capture the intrinsic complexity of the problem. Cell positions are also admitted to be given using O(log(n d )) bits which is o(n d ) (see Lemma 3 for a polynomial bound on the most distant cell that can receive a grain).…”

Section: Stable Prediction Problem (S-pred)mentioning

confidence: 99%

How Hard is it to Predict Sandpiles on Lattices? A Survey

Formenti

Perrot

2019

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Since their introduction in the 80s, sandpile models have raised interest for their simple definition and their surprising dynamical properties. In this survey we focus on the computational complexity of the prediction problem, namely, the complexity of knowing, given a finite configuration c and a cell x in c, if cell x will eventually become unstable. This is an attempt to formalize the intuitive notion of "behavioral complexity" that one easily observes in simulations. However, despite many efforts and nice results, the original question remains open: how hard is it to predict the two-dimensional sandpile model of Bak, Tang and Wiesenfeld? arXiv:1909.12150v1 [cs.DM] 26 Sep 2019 * → c . However, one can consider also other types updating policies. The sequential policy consists in choosing non-deterministically a cell from the unstable ones and in updating only this chosen cell. Then, repeat the same update policy on the newly obtained configuration and so on. It is clear that the new dynamics might *

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“…Understanding the scaling limit of the sandpile identity elements (Figure 8) is another appealing problem, solved in a special case by Caracciolo, Paoletti, and Sportiello [13]. (4) As proved in [43] (and illustrated in Figure 7), the maximal elements of Γ(Z 2 ) correspond to the circles in the Apollonian band packing of R 2 . Because the radius and the coordinates of the center of each such circle are rational numbers, each maximal element of Γ(Z 2 ) is a matrix with rational entries.…”

Section: Loop Erasures Tutte Polynomial Unicyclesmentioning

confidence: 99%

Untitled

2017

Bull. Amer. Math. Soc.

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Abstract. Laplacian growth is the study of interfaces that move in proportion to harmonic measure. Physically, it arises in fluid flow and electrical problems involving a moving boundary. We survey progress over the last decade on discrete models of (internal) Laplacian growth, including the abelian sandpile, internal DLA, rotor aggregation, and the scaling limits of these models on the lattice Z d as the mesh size goes to zero. These models provide a window into the tools of discrete potential theory, including harmonic functions, martingales, obstacle problems, quadrature domains, Green functions, smoothing. We also present one new result: rotor aggregation in Z d has O(log r) fluctuations around a Euclidean ball, improving a previous power-law bound. We highlight several open questions, including whether these fluctuations are O(1).

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The Apollonian structure of integer superharmonic matrices

Cited by 31 publications

References 33 publications

Laplacian growth, sandpiles, and scaling limits

Laplacian growth, sandpiles, and scaling limits

How Hard is it to Predict Sandpiles on Lattices? A Survey

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