An asymptotic expansion for the discrete harmonic potential

Kozma, Gady; Schreiber, E.

doi:10.1214/ejp.v9-170

Cited by 26 publications

(37 citation statements)

References 7 publications

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“…Moreover, g(z) > 0 for all z = 0. Exact values of g near the origin can be computed using the McCrea-Whipple algorithm, described in [KS04]. In particular,…”

Section: The Discrete Harmonic Function H ζ (Z)mentioning

confidence: 99%

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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“…Moreover, g(z) > 0 for all z = 0. Exact values of g near the origin can be computed using the McCrea-Whipple algorithm, described in [KS04]. In particular,…”

Section: The Discrete Harmonic Function H ζ (Z)mentioning

confidence: 99%

Logarithmic fluctuations for internal DLA

Jerison

Levine

Sheffield⋆

2011

J. Amer. Math. Soc.

135

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“…In particular, we would like to know the asymptotic order of the Green function g(x) when x is far from the origin. It turns out [22,36,67] that…”

Section: Laplacian Growth Sandpiles and Scaling Limits 361mentioning

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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“…Proof of (10): The proof is based on an asymptotic expansion of the potential function of the random walk (cf. [3], [12], [7]) from which an application of the reflection principle immediately yields…”

Section: Thus (V) Is Provedmentioning

confidence: 99%