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Coulomb gas ensembles and Laplacian growth
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-4971-7147
2013 (English)In: Proceedings of the London Mathematical Society, ISSN 0024-6115, E-ISSN 1460-244X, Vol. 106, no 4, 859-907 p.Article in journal (Refereed) Published
Abstract [en]

We consider weight functions Q : C -> R that are locally in a suitable Sobolev space and impose a logarithmic growth condition from below. We use Q as a confining potential in the model of one-component plasma (2-dimensional Coulomb gas) and study the configuration of the electron cloud as the number n of electrons tends to infinity, while the confining potential is rescaled: we use mQ in place of Q and let m tend to infinity as well. We show that if m and n tend to infinity in a proportional fashion, with n/m -> t, where 0 < t <+infinity is fixed, then the electrons accumulate on a compact set S-t, which we call the droplet. The set S-t can be obtained as the coincidence set of an obstacle problem, if we remove a small set (the shallow points). Moreover, on the droplet S-t, the density of electrons is asymptotically delta Q. The growth of the droplets S-t as t increases is known as the Laplacian growth. It is well known that Laplacian growth is unstable. To analyse this feature, we introduce the notion of a local droplet, which involves removing part of the obstacle away from the set S-t. The local droplets are no longer uniquely determined by the time parameter t, but at least they may be partially ordered. We show that the growth of the local droplets may be terminated in a maximal local droplet or by the droplets' growing to infinity in some direction ('fingering').

Place, publisher, year, edition, pages
2013. Vol. 106, no 4, 859-907 p.
Keyword [en]
Random Normal Matrices, Hele-Shaw, Variational-Inequalities, Hyperbolic Surfaces, Eigenvalues, Fluctuations
National Category
URN: urn:nbn:se:kth:diva-123635DOI: 10.1112/plms/pds032ISI: 000318573700004ScopusID: 2-s2.0-84877295156OAI: diva2:628450
Swedish Research CouncilGöran Gustafsson Foundation for Research in Natural Sciences and Medicine

QC 20150624

Available from: 2013-06-14 Created: 2013-06-13 Last updated: 2015-06-24Bibliographically approved

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Hedenmalm, Håkan
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