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

Abstract

Let Q be a suitable real function on C. An n-Fekete set corresponding to Q is a subset {z n1 , . . . , z nn } of C which maximizes the expression n i<j |z ni − z nj | 2 e −n(Q(z n1 )+···+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 μ n = n −1 (δ z n1 + · · · + δ z nn ) converges to the equilibrium measure Q · 1 S dA as n → ∞. 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 = |z| 2 we obtain results which hold globally, and we conjecture that such global results are true for a wide range of potentials.In this paper we discuss equidistribution results for weighted Fekete sets in subsets of the plane. More precisely, we show that Fekete sets are maximally spread out relative to a rescaled version of the Beurling-Landau density, in the "droplet" corresponding to the given weight. Our ✩ This work is a contribution to the research program on "Complex Analysis and Spectral Problems" which was conducted at the CRM in Barcelona during the spring semester of 2011.

Repulsion in Low Temperature $${\beta}$$ β -Ensembles

“…Further, the distance tends to increase with β, and as β → ∞, we recover formally the separation theorem for Fekete sets from the papers [1,5].…”

“…The analysis of Fekete configurations in [1,5] depends on the inequality | j | ≤ 1 for the associated weighted Lagrange polynomials j . This bound plays a similar role when β = ∞ as the L 2β -estimate in Lemma 2 does in the present case.…”

“…Our estimate for the constant c should be compared with the asymptotic lower bound 1/ √ e for the distance between Fekete points obtained in [1,Theorem 1]. In fact, our method of proof is somewhat related to the approach in [1,5], see concluding remarks below.…”

“…[5] for an independent proof. Our estimate for the constant c should be compared with the asymptotic lower bound 1/ √ e for the distance between Fekete points obtained in [1,Theorem 1].…”

Local incompressibility estimates for the Laughlin phase

“…One expects the true minimizers and low-temperature Gibbs states to follow this profile on any microscopic length scale ≫ N −1/2 (in the scaled variables used in (2.22)). This has recently been proved in [4,53] for ground states (see also [50] for higher dimensional Coulomb and Riesz gases) and in [7,8,34,36] for Gibbs states. See also earlier partial results (not on the optimal scale) in [65,56,51] and [2,3,10,16,17,37,35,64] for related recent literature.…”