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


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


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.


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


Download Citation

Yacin Ameur. "A localization theorem for the planar Coulomb gas in an external field." Electron. J. Probab. 26 1 - 21, 2021.


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

Primary: 60K35

Keywords: Coulomb gas , droplet , external potential , Localization


Vol.26 • 2021
Back to Top