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

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.