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Beurling-Landau densities of weighted Fekete sets and correlation kernel estimates

Ameur, Yacin LU and Ortega-Cerda, Joaquim (2012) In Journal of Functional Analysis 263(7). p.1825-1861
Abstract
Let Q be a suitable real function on C. An n-Fekete set corresponding to Q is a subset {z(n vertical bar) , . . . , z(nn)} of C which maximizes the expression Pi(n)(i<j) vertical bar z(ni) - z(nj)vertical bar(2)e(-n(Q(zn1)) + . . . +Q(z(nn))). It is well known that, under reasonable conditions on Q. there is a compact set S known as the "droplet" such that the measures mu(n) = n(-1) (delta(zn vertical bar) + . . . + delta(znn)) converges to the equilibrium measure Delta Q . 1(s) dA as n -> infinity. In this note we prove that Fekete sets are, in a sense, maximally spread out with respect to the equilibrium measure. In general, our results apply only to a part of the Fekete set, which is at a certain distance away from the boundary of... (More)
Let Q be a suitable real function on C. An n-Fekete set corresponding to Q is a subset {z(n vertical bar) , . . . , z(nn)} of C which maximizes the expression Pi(n)(i<j) vertical bar z(ni) - z(nj)vertical bar(2)e(-n(Q(zn1)) + . . . +Q(z(nn))). It is well known that, under reasonable conditions on Q. there is a compact set S known as the "droplet" such that the measures mu(n) = n(-1) (delta(zn vertical bar) + . . . + delta(znn)) converges to the equilibrium measure Delta Q . 1(s) dA as n -> infinity. In this note we prove that Fekete sets are, in a sense, maximally spread out with respect to the equilibrium measure. In general, our results apply only to a part of the Fekete set, which is at a certain distance away from the boundary of the droplet. However, for the potential Q = vertical bar z vertical bar(2) we obtain results which hold globally, and we conjecture that such global results are true for a wide range of potentials. (C) 2012 Elsevier Inc. All rights reserved. (Less)
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author
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type
Contribution to journal
publication status
published
subject
keywords
Weighted Fekete set, Droplet, Equidistribution, Concentration operator, Correlation kernel
in
Journal of Functional Analysis
volume
263
issue
7
pages
1825 - 1861
publisher
Elsevier
external identifiers
  • wos:000307906800003
  • scopus:84864839327
ISSN
0022-1236
DOI
10.1016/j.jfa.2012.06.01
language
English
LU publication?
yes
id
ab4efc7c-b8ba-4bfe-a4f9-973556b5489f (old id 3146789)
date added to LUP
2016-04-01 14:17:51
date last changed
2022-01-27 23:51:02
@article{ab4efc7c-b8ba-4bfe-a4f9-973556b5489f,
  abstract     = {{Let Q be a suitable real function on C. An n-Fekete set corresponding to Q is a subset {z(n vertical bar) , . . . , z(nn)} of C which maximizes the expression Pi(n)(i&lt;j) vertical bar z(ni) - z(nj)vertical bar(2)e(-n(Q(zn1)) + . . . +Q(z(nn))). It is well known that, under reasonable conditions on Q. there is a compact set S known as the "droplet" such that the measures mu(n) = n(-1) (delta(zn vertical bar) + . . . + delta(znn)) converges to the equilibrium measure Delta Q . 1(s) dA as n -&gt; infinity. In this note we prove that Fekete sets are, in a sense, maximally spread out with respect to the equilibrium measure. In general, our results apply only to a part of the Fekete set, which is at a certain distance away from the boundary of the droplet. However, for the potential Q = vertical bar z vertical bar(2) we obtain results which hold globally, and we conjecture that such global results are true for a wide range of potentials. (C) 2012 Elsevier Inc. All rights reserved.}},
  author       = {{Ameur, Yacin and Ortega-Cerda, Joaquim}},
  issn         = {{0022-1236}},
  keywords     = {{Weighted Fekete set; Droplet; Equidistribution; Concentration operator; Correlation kernel}},
  language     = {{eng}},
  number       = {{7}},
  pages        = {{1825--1861}},
  publisher    = {{Elsevier}},
  series       = {{Journal of Functional Analysis}},
  title        = {{Beurling-Landau densities of weighted Fekete sets and correlation kernel estimates}},
  url          = {{http://dx.doi.org/10.1016/j.jfa.2012.06.01}},
  doi          = {{10.1016/j.jfa.2012.06.01}},
  volume       = {{263}},
  year         = {{2012}},
}