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Perturbed Divisible Sandpiles and Quadrature Surfaces

  • Hayk Aleksanyan
  • Henrik Shahgholian
Open Access
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Abstract

The main purpose of the present paper is to establish a link between quadrature surfaces (potential theoretic concept) and sandpile dynamics (Laplacian growth models). For this aim, we introduce a new model of Laplacian growth on the lattice \(\mathbb {Z}^{d}\) (d ≥ 2) which continuously deforms occupied regions of the divisible sandpile model of Levine and Peres (J. Anal. Math. 111(1), 151–219 2010), by redistributing the total mass of the system onto \(\frac 1m\)-sub-level sets of the odometer which is a function counting total emissions of mass from lattice vertices. In free boundary terminology this goes in parallel with singular perturbation, which is known to converge to a Bernoulli type free boundary. We prove that models, generated from a single source, have a scaling limit, if the threshold m is fixed. Moreover, this limit is a ball, and the entire mass of the system is being redistributed onto an annular ring of thickness \(\frac 1m\). By compactness argument we show that when m tends to infinity sufficiently slowly with respect to the scale of the model, then in this case also there is scaling limit which is a ball, with the mass of the system being uniformly distributed onto the boundary of that ball, and hence we recover a quadrature surface in this case. Depending on the speed of decay of 1/m, the visited set of the sandpile interpolates between spherical and polygonal shapes. Finding a precise characterisation of this shape-transition phenomenon seems to be a considerable challenge, which we cannot address at this moment.

Keywords

Singular perturbation Lattice growth model Quadrature surface Bernoulli free boundary Boundary sandpile Balayage Divisible sandpile Scaling limit 

Mathematics Subject Classification (2010)

31C20 35B25 35R35 (31C05 82C41) 

Notes

Acknowledgments

H. A. was supported by postdoctoral fellowship from Knut and Alice Wallenberg Foundation. H. Sh. was partially supported by Swedish Research Council.

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Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsKTH Royal Institute of TechnologyStockholmSweden

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