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Partial Balayage and a Generalization of the Divisible Sandpile Model
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-6942-8959
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In recent work by L. Levine and Y. Peres, it was observed that three models for particle aggregation on the lattice—the divisible sandpile, rotor-router aggregation, and internal diffusion limited aggregation—share a common scaling limit as the lattice spacing tends to zero, if they are started with the same initial mass configuration. It is straightforward to observe that this scaling limit is precisely the same as the potential-theoretic operation of taking the partial balayage of this initial mass configuration to the Lebesgue measure. However, from the theory of the partial balayage operation it is clear that one may take the partial balayage of a mass configuration to a more general measure than the Lebesgue measure, which one cannot do for the three aggregation models described by Levine and Peres. In this paper we therefore generalize one of these models, the divisible sandpile model, in mainly a bounded setting, and show that a natural scaling limit of this generalization is given by a general partial balayage operation.

Keyword [en]
Divisible sandpile, partial balayage, obstacle problem
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-186345OAI: oai:DiVA.org:kth-186345DiVA: diva2:927042
Note

QC 20160525

Available from: 2016-05-10 Created: 2016-05-10 Last updated: 2016-05-25Bibliographically approved
In thesis
1. Partial Balayage and Related Concepts in Potential Theory
Open this publication in new window or tab >>Partial Balayage and Related Concepts in Potential Theory
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of three papers, all treating various aspects of the operation partial balayage from potential theory.

The first paper concerns the equilibrium measure in the setting of two dimensional weighted potential theory, an important measure arising in various mathematical areas, e.g. random matrix theory and the theory of orthogonal polynomials. In this paper we show that the equilibrium measure satisfies a complementary relation with a partial balayage measure if the weight function is of a certain type.

The second paper treats the connection between partial balayage measures and measures arising from scaling limits of a generalisation of the so-called divisible sandpile model on lattices. The standard divisible sandpile can, in a natural way, be considered a discrete version of the partial balayage operation with respect to the Lebesgue measure. The generalisation that is developed in this paper is essentially a discrete version of the partial balayage operation with respect to more general measures than the Lebesgue measure.

In the third paper we develop a version of partial balayage on Riemannian manifolds, using the theory of currents. Several known properties of partial balayage measures are shown to have corresponding results in the Riemannian manifold setting, one of which being the main result of the first paper. Moreover, we utilize the developed framework to show that for manifolds of dimension two, harmonic and geodesic balls are locally equivalent if and only if the manifold locally has constant curvature.

Abstract [sv]

Denna avhandling består av tre artiklar som alla behandlar olika aspekter av den potentialteoretiska operationen partiell balayage.

Den första artikeln betraktar jämviktsmåttet i tvådimensionell viktad potentialteori, ett viktigt mått inom flertalet matematiska inriktningar såsom slumpmatristeori och teorin om ortogonalpolynom. I denna artikel visas att jämviktsmåttet uppfyller en komplementaritetsrelation med ett partiell balayage-mått om viktfunktionen är av en viss typ.

Den andra artikeln behandlar relationen mellan partiell balayage-mått och mått som uppstår från skalningsgränser av en generalisering av den så kallade "delbara sandhögen", en diskret modell för partikelaggregation på gitter. Den vanliga delbara sandhögen kan på ett naturligt sätt betraktas som en diskret version av partiell balayage-operatorn med avseende på Lebesguemåttet. Generaliseringen som utarbetas i denna artikel är väsentligen en diskret version av partiell balayage-operatorn med avseende på mer allmänna mått än Lebesguemåttet.

I den tredje artikeln formuleras en version av partiell balayage på riemannska mångfalder utifrån teorin om strömmar. Åtskilliga tidigare kända egenskaper om partiella balayage-mått visas ha motsvarande formuleringar i formuleringen på riemannska mångfalder, bland annat huvudresultatet från den första artikeln. Vidare så utnyttjas det utarbetade ramverket för att visa att tvådimensionella riemannska mångfalder har egenskapen att harmoniska och geodetiska bollar lokalt är ekvivalenta om och endast om mångfalden lokalt har konstant krökning.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2016. 47 p.
Series
TRITA-MAT-A, 2016:07
Keyword
Partial balayage, equilibrium measure, obstacle problem, divisible sandpile, Riemannian manifold, quadrature domain, Laplacian growth, mother body
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-186367 (URN)978-91-7729-025-4 (ISBN)
Public defence
2016-08-30, F3, Lindstedtsvägen 26, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

QC 20160524

Available from: 2016-05-24 Created: 2016-05-10 Last updated: 2016-05-24Bibliographically approved

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