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2021 A localization theorem for the planar Coulomb gas in an external field
Yacin Ameur
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Electron. J. Probab. 26: 1-21 (2021). DOI: 10.1214/21-EJP613

Abstract

We examine a two-dimensional Coulomb gas consisting of n identical repelling point charges at an arbitrary inverse temperature β, subjected to a suitable external field.

We prove that the gas is effectively localized to a small neighbourhood of the droplet – the support of the equilibrium measure determined by the external field. More precisely, we prove that the distance between the droplet and the vacuum is with very high probability at most proportional to

lognβn.

This order of magnitude is known to be “tight” when β=1 and the external field is radially symmetric.

In addition, we prove estimates for the one-point function in a neighbourhood of the droplet, proving in particular a fast uniform decay as one moves beyond a distance roughly of the order lognβn from the droplet.

Acknowledgments

I am grateful to Seong-Mi Seo and Djalil Chafaï for comments and appreciated help.

Citation

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Yacin Ameur. "A localization theorem for the planar Coulomb gas in an external field." Electron. J. Probab. 26 1 - 21, 2021. https://doi.org/10.1214/21-EJP613

Information

Received: 8 September 2019; Accepted: 29 March 2021; Published: 2021
First available in Project Euclid: 13 April 2021

Digital Object Identifier: 10.1214/21-EJP613

Subjects:
Primary: 60K35

JOURNAL ARTICLE
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