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Perturbed Divisible Sandpiles and Quadrature Surfaces
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0003-4834-6476
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-1316-7913
2019 (English)In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 51, no 4, p. 511-540Article in journal (Refereed) Published
Abstract [en]

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 DOUBLE-STRUCK CAPITAL Zd (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 1/m-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 1/m. 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.

Place, publisher, year, edition, pages
SPRINGER , 2019. Vol. 51, no 4, p. 511-540
Keywords [en]
Singular perturbation, Lattice growth model, Quadrature surface, Bernoulli free boundary, Boundary sandpile, Balayage, Divisible sandpile, Scaling limit
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-264863DOI: 10.1007/s11118-018-9722-6ISI: 000496293600003Scopus ID: 2-s2.0-85051720742OAI: oai:DiVA.org:kth-264863DiVA, id: diva2:1379594
Note

QC 20191217

Available from: 2019-12-17 Created: 2019-12-17 Last updated: 2020-01-03Bibliographically approved

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Aleksanyan, HaykShahgholian, Henrik

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