**KERNEL ESTIMATES**

YACIN AMEUR AND JOAQUIM ORTEGA-CERD `A

Abstract. LetQbe a suitable real valued function onCwhich increases sufficiently rapidly asz→ ∞. Ann-Fekete set corresponding toQis a subset{zn1, . . . ,znn}ofCwhich maximizes the weighted Vandermonde determinantQn

i<j

zni−zn j

2e^{−}^{n(Q(z}^{n1}^{)+}^{···}^{+Q(z}^{nn}^{))}. It is well known that there exists a
compact setSknown as the “droplet” such that the sequence of measuresµn=n^{−}^{1}(δzn1+· · ·+δznn)
converges to the equilibrium measure∆Q·**1**SdAasn→ ∞. In this note we consider a related topic,
proving 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 Ginibre potentialQ=|z|^{2}we obtain results which
hold globally; 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 method combines Landau’s idea to relate the density of a family of discrete sets to properties of the spectrum of the concentration operator, with estimates for the correlation kernel of the corresponding random normal matrix ensemble.

1. Fekete sets

1.1. **Potentials and droplets.** We recapture some notions and results from weighted potential
theory. Proofs and further results can be found in [23]. Cf. also [2] and [17] where the setting is
more tuned to fit the present discussion.

LetQ:C→R∪ {+∞}be a suitable function (the“potential” or “external field”) satisfying lim inf

z→∞

Q(z)

log|z|^{2} = +∞.

(In detail: we require in addition that the functionw:=e^{−}^{Q}^{/}^{2}satisfy the mild condition of being
an “admissible weight” in the sense of [23], p. 26. This means thatwis upper semi-continuous
and the set{w>0}has positive logarithmic capacity.)

We associate toQthe “equilibrium potential”Qbin the following way: Let SHQbe the set of all
subharmonic functionsu:C→Rsuch thatu(z)≤log_{+}|z|^{2}+const. andu≤QonC. One defines

2010Mathematics Subject Classification. 31C20; 82B20; 30E05; 94A20.

Key words and phrases. Weighted Fekete set; droplet; equidistribution; concentration operator; correlation kernel.

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. The first author was supported by grants from Magnussons fond, Vetenskapsrådet, SveFum, and the European Science Foundation. The second author was supported by grants MTM2008-05561-C02-01 and 2009 SGR 1303.

1

the equilibrium potential asQ(z)b =sup{u(z) ; u∈SHQ}.The droplet associated toQis the set S={z∈C;Q(z)=Q(z)b }.

This is a compact set; one has that∆Q≥0 onSand that theequilibrium measure

(1.1) dσ(z)=**1**S(z)∆Q(z)dA(z)

is a probability measure onC. Here we agree that dAis normalized area measure dA= _{π}^{1}dxdy,
while∆ = ∂∂ = ^{1}_{4}(∂^{2}/∂x^{2}+∂^{2}/∂y^{2}) is the normalized Laplacian; ∂ = ^{1}_{2}(∂/∂x−i∂/∂y) and∂ =

1

2(∂/∂x+i∂/∂y) are the complex derivatives.

We will make the standing assumption thatQbeC^{3}-smooth and strictly subharmonic in some
neighbourhoodΛofS. In other words, we assume that theconformal metricds^{2}(z)= ∆Q(z)|dz|^{2}is
comparable to the Euclidean metric onΛ.

1.2. **Fekete sets.** Consider the weighted Vandermonde determinant
Vn(z1, . . . ,zn)=Y

i<j

zi−zj

2e^{−}^{n(Q(z}^{1}^{)}^{+}^{···}^{+}^{Q(z}^{n}^{))}, z1, . . . ,zn∈C.

A setF_{n}={zn1, . . . ,znn}which maximizesVnis called ann-Fekete setcorresponding toQ. Notice
that Fekete sets are not unique.

Equivalently, the setF_{n}minimizes the weighted energy
(1.2) Hn(z1, . . . ,zn)=X

i,j

log zi−zj

−1

+n

n

X

j=1

Q(zj)
over all configurations{zj}^{n}

j=1 ⊂C. If we think of the pointszjas giving locations fornidentical repelling point charges with total charge 1 confined toC under the influence of the external magnetic fieldnQ, thenHncan be regarded as the the energy of the system.

The following classical result displays some fundamental and well-known properties of Fekete sets.

**Theorem 1.1.** For any Fekete setF_{n}={zn1, . . . ,znn}holds:

(1) F_{n}⊂S

(2) Letσbe the equilibrium measure(1.1). We then have convergence in the sense of measures 1

n

n

X

j=1

δzn j→σ, as n→ ∞.

A proof can be found in [23], theorems III.1.2 and III.1.3. (Notice that our assumptions onQ
imply thatS=S^{∗}in the notation of [23].) The theorem 1.1 was generalized to line bundles over
complex manifolds in [9], [10].

We remark that the property (1) is essential to the analysis in this paper, and that the standard proof of (1) (e.g. in [23]) depends on the “maximum principle for weighted polynomials”, which is reproduced in Lemma 2.8 below.

We will consider related questions concerning the distribution of Fekete points. In a sense, we will prove that these points are maximally spread out with respect to the conformal metric. To quantify this assertion, we introduce some definitions.

**Definition 1.2.** LetF ={F_{n}}^{∞}

n=1be a family ofn-Fekete sets. Also letζ=(zn)^{∞}_{1} be a sequence of
points inS. We define thelower Beurling–Landau’s densityofF with respect toζby

D^{−}(F;ζ)=lim inf

R→∞ lim inf

n→∞

#

F_{n}∩D
zn;R/√

n
R^{2}∆Q(zn) ,
and we define the corresponding upper density by

D^{+}(F;ζ)=lim sup

R→∞

lim sup

n→∞

#

F_{n}∩D
zn;R/√

n
R^{2}∆Q(zn) .
We also put

dn(ζ)=dist(zn,C\S).

Here “dist” denotes the Euclidean distance in the plane, andD(ζ;r) is the open disk with center ζand radiusr.

We have the following theorem.

**Theorem 1.3.** Putδn=log^{2}n/√

n,and suppose that dn(ζ)≥3δnfor all n. Then

(1.3) D^{−}(F;ζ)=D^{+}(F;ζ)=1.

A proof is given in§2.3.

Remark1.4. The function%n(z)^{−}^{2}defined bynσ(D(z;%n(z)))=1 can be considered as a regularized
version of the Laplacian ∆Q(z). Replacing ∆Q(zn) by %n(zn)^{−}^{2} in our definition of Beurling–

Landau’s densities, it becomes possible to extend our results to cover some situations in which

∆Q=0 at isolated points of the droplet.

1.3. **The Ginibre case.** The potentialQ(z)=|z|^{2}is known as theGinibre potential. It is easy to see
that for this potential, the droplet isS=D, i.e. the closed unit disk with center 0.

**Theorem 1.5.** Suppose that Q(z) = |z|^{2}. Letζ = (zn) be a sequence inDand assume that the limit
L=limn→∞

√

n(1− |zn|)exists. Then (1) If L= +∞, then(1.3)holds (2) If L<+∞, then

(1.4) D^{−}(F;ζ)=D^{+}(F;ζ)= 1

2. A proof is given in§2.4.

Remark 1.6. The condition that the limitLexists is really superfluous and is made merely for technical convenience. Indeed, we can assert that lim infn→∞

√

n(1− |zn|)= +∞then (1.3) holds
while if lim sup_{n}→∞

√

n(1− |zn|)<+∞then (1.4) holds. These somewhat more general statements can be proved without difficulty by using the arguments below.

1.4. **A conjecture.** The boundary of a droplet corresponding to a smooth potential is in general
a quite complicated set. However, owing to Sakai’s theory [24], it is known that the situation
is more manageable for potentialsQwhich arereal-analyticin a neighbourhood of the droplet.

Namely, for a real analytic potentialQ, the boundary ofSis a finite union of real analytic arcs and possibly a finite number of isolated points. The boundary ofSmay also have finitely many singularities which can be either cusps or double-points. This result can easily be proved using arguments from [18], Section 4.

Suppose thatQis real-analytic and strictly subharmonic in a neighbourhood ofS, and assume
that∂Shas no singularities. LetS^{∗}denote the setSwith eventual singularities and isolated points
removed. Also letζ=(zn)^{∞}_{1} be a sequence of points inS^{∗}and assume for simplicity that the limit
L = limn→∞

√

ndn(ζ) exists, wheredn(ζ) is the distance ofzn to∂S. We conjecture that for any
sequenceF ={F_{n}}of weighted Fekete sets, we have (i) ifL= +∞, thenD^{−}(F, ζ)=D^{+}(F, ζ)=1
and (ii) ifL<+∞, thenD^{−}(F, ζ)=D^{+}(F, ζ)=1/2.

The conjecture is supported by the results of the forthcoming paper [5].

1.5. **Earlier work and related topics.** The topics considered in this note, as well as our basic
strategy, were inspired by the paper [19] by Landau, which concerns questions about interpo-
lation and sampling for functions in Paley–Wiener spaces. In particular, our “Beurling–Landau
densities” can be seen as straightforward adaptations of the densities defined in [19], and our re-
sults below are parallel to those of Landau. The historically interested reader should also consult
Beurling’s lecture notes (see the references in [19]), where some of the basic concepts appeared
earlier; in fact Landau’s exposition depends in an essential way on Beurling’s earlier work.

In the one-component plasma (or “OCP”) setting, one introduces a temperature 1/β, where
β > 0. The probability measure dP^{β}_{n}(z) =(Z^{β}_{n})^{−}^{1}e^{−}^{β}^{H}^{n}^{(z)}dVn(z) onC^{n}is known as the density of
states at the temperature 1/β. Here dVnis Lebesgue measure onC^{n},Hnis the Hamiltonian (1.2),
andZ^{β}_{n}is a normalizing constant. One then considers configurationsΨ^{β}n={zi}^{n}

1picked randomly
with respect to**P**^{β}n.

Intuitively, Fekete sets should correspond to particle configurations at temperature zero, or
rather, the “limiting configurations” as 1/β → 0, although the latter “limit” so far has been
understood mostly on a physical level. In this interpretation, the methods of the present note
prove that the Beurling–Landau density of temperature zero configurations is in fact completely
determined by properties atβ=1. (More precisely: it is determined by the one- and two-point
functions of**P**^{1}_{n}.)

A more subtle problem is to characterise Fekete sets amongst all configurations of Beurling–

Landau density one. It is believed that a certain crystalline structure will manifest itself (known as the “Abrikosov lattice”). We refer to [14], [25] and the references therein for further details on this topic. A survey of related questions for minimum energy points on manifolds is found in [16].

2. Weighted polynomials and triangular lattices

Our approach combines the method for characterizing Fekete sets and triangular lattices from the papers [21] and [22] with correlation kernel estimates of the type found in [6], [1], [2], [4].

In the Ginibre case, we use the explicit representation of the correlation kernel available for that potential, as well as estimates from the papers [27], [15], [12], and [11].

2.1. **Weighted polynomials.** LetHnbe the space of polynomialspof degree at mostn−1, normed
by

p

2 nQ := R

C

p(z)

2e^{−}^{nQ(z)}dA(z). The reproducing kernel for Hn is Kn(z,w) = Pn−1

j=0ej(z)ej(w),
where{ej}^{n}^{−}^{1}

0 is an orthonormal basis forHn.

For our purposes, it is advantageous to work with spaces ˜Hnof weighted polynomials f =
p·e^{−}^{nQ}^{/}^{2}, wherepis a polynomial of degree≤n−1, and one defines the norm in ˜Hnas the usual
L^{2}(dA)-norm. The reproducing kernel for ˜Hnis given by

**K**n(z,w)=Kn(z,w)e^{−}^{nQ(z)}^{/}^{2}^{−}^{nQ(w)}^{/}^{2}.

The function**K**nis known as thecorrelation kernelcorresponding to the potentialQ; the reproduc-
ing property means that

f(z)=hf,**K**n,zi, f ∈H˜n, z∈C,

where**K**n,z(ζ)=**K**n(ζ,z), and the inner product is the usual one inL^{2}=L^{2}(C,dA).

When ρn is not an integer, we interpret ˜H_{ρn} as the space Hk wherek is the largest integer
satisfying k < ρn. All statements below shall be understood in terms of this convention; in
particular,**K**_{ρn}(z,w) :=Kk(z,w)e^{−}k(Q(z)+Q(w))/2.

2.2. **Triangular lattices.** LetZ=n
Z_{j}o^{∞}

j=1be a triangular lattice of points inC. We write
Z_{n}={zn1,zn2, . . . ,znmn}.

It will be convenient to introduce some classes of lattices.

Letρ >0. A familyZis said to beρ-interpolatingif there is some constantCsuch that, for all
families of valuesc={cn}^{∞}

1 ,cn={cn j}^{m}^{n}

j=1, such that sup

n

1 nρ

mn

X

j=1

cn j

2<∞,

there exists a sequence fn∈H˜_{ρn}such that fn(zn j)=cn j, 1≤ j≤mn, and

fn

2≤C 1 nρ

mn

X

j=1

cn j

2.

We say that a familyZisuniformly separatedif there is a numbers>0 such that for any two
distinct pointsz,w∈ Z_{n}we have|z−w|>s/√

n. The following simple lemma holds.

**Lemma 2.1.** Any interpolating family which is contained in S is uniformly separated.

A proof is given in§3.2.

Intuitively, an interpolating family should be “sparse”. We will also need a notion which implies the “density” of a family contained inS. For this purpose, the following classes have turned out to be convenient.

**Definition 2.2.** WriteS^{+} =S+D(0;s/√

n),wheresis some fixed positive number. LetZ ⊂Sbe a triangular family. We hay thatZis of classMS,ρifZis uniformly 2s-separated and

Z

S^{+}

f

2 ≤C 1 nρ

X

zn j∈Z_{n}

f(zn j)

2, f ∈H˜nρ

for all largen.

**Definition 2.3.** Letδn=log^{2}n/√

nand putSn={z∈S; dist(z, ∂S)≥2δn}.We say that a triangular familyZ ⊂Sis of classMSn,ρifZis uniformly separated and

Z

Sn

f

2≤C 1 nρ

X

zn j∈Z_{n}

f(zn j)

2, f ∈H˜nρ

for all largen.

2.3. **Results in the interior of the droplet.** We have the following lemma.

**Lemma 2.4.** Letζ=(zn)be a convergent sequence in S withdist(zn, ∂S)≥3δnfor all n. Then
(i) IfZis of class MSn,ρ, then D^{−}(Z;ζ)≥ρ,

(ii) IfZisρ-interpolating, then D^{+}(Z;ζ)≤ρ.

A proof is given in Section 5.

WhenF_{n}is a Fekete set, we writeF^{0}

n=F_{n}∩SnandF^{0}={F^{0}

n}.
**Lemma 2.5.** One has that

(1) F is uniformly separated,

(2) F^{0}isρ-interpolating for anyρ >1,
(3) F is of class MSn,ρwheneverρ <1.

A proof is given in Section 6.

Using lemmas 2.4 and 2.5, we infer that forζ=(zn) with dist(zn, ∂S) ≥3δn, we have for any
ε >0 thatD^{−}(F;ζ) ≥1−εandD^{+}(F;ζ) ≤1+ε. This finishes the proof of Theorem 1.3, since
evidentlyD^{−}≤D^{+}. q.e.d.

2.4. **The Ginibre case.** Now letQ=|z|^{2}so thatS=D, and fix a convergent sequenceζ=(zj) in
Dsuch that the limitL=limn→∞

√

n(1− |zn|) exists.

**Lemma 2.6.** Suppose that Q=|z|^{2}, and letZbe a triangular family contained inD.

(1) IfZis of class M_{D,ρ}, then

D^{−}(Z;ζ)≥

ρ if L= +∞, ρ/2 if L<+∞. (2) IfZisρ-interpolating, then

D^{+}(Z;ζ)≤

ρ if L= +∞, ρ/2 if L<+∞. A proof is given in§8.3.

**Lemma 2.7.** LetF ={F_{n}}be a family of Fekete sets with respect to the potential Q=|z|^{2}. ThenF is of
class M_{D,ρ}for anyρ <1andρ-interpolating for anyρ >1.

A proof is given in§8.5.

To finish the proof of Theorem 1.5 it suffices to combine Lemma 2.6 and Lemma 2.7. q.e.d.

2.5. **Auxiliary lemmas.** We state a couple of known facts which are used frequently in the
following. The following uniform estimate is well-known (see e.g. [23]).

**Lemma 2.8.** Let f ∈H˜nand z∈C\S. Assume that
f

≤1on S. Then f(z)

≤e^{−}^{n}

Q(z)−bQ(z)

/2, z∈C.

(Proof: Let f =p·e^{−}^{nQ/2}. The assumption gives that ^{1}_{n}log
p

2is a subharmonic minorant ofQ
which grows no faster than log|z|^{2}+const. asz→ ∞. Thus ^{1}_{n}log

p

2≤Q.)b
We will also use the following well-known pointwise-L^{2}estimate.

**Lemma 2.9.** Let f = ue^{−}^{nQ}^{/}^{2} where u is holomorphic and bounded in D(z0;c/√

n) for some c > 0.

Suppose that∆Q(z)≤K for all z∈D(z0;c/√

n). Then

(2.1)

f(z0)

2≤n·e^{Kc}^{2}c^{−}^{2}
Z

D(z0;c/√ n)

f

2dA.

In particular, ifZis2s-separated, then for all f ∈H˜n

(2.2) 1

n X

zn j∈Ω

f(zn j)

2 ≤Cs^{−}^{2}
Z

Ω^{+}

f(ζ)

2dA(ζ),

where C depends only on the upper bound of∆Q on S^{+}andΩ^{+}=n

ζ∈C; dist(ζ,Ω)≤s/√ no

. A proof of (2.1) can be found e.g. in [2], Section 3. The estimate (2.2) is immediate from this.

We will also need the following lemma on uniform estimates and “off-diagonal damping” for correlation kernels.

**Lemma 2.10.** (i) There is a constant C such that for all z,w∈C,

|**K**n(z,w)| ≤Cne^{−}^{n(Q(z)}^{−b}^{Q(z))/2}e^{−}^{n(Q(w)}^{−b}^{Q(w))/2}.

(ii) Suppose that z∈S and letδ=dist(z, ∂S). There are then positive constants C and c such that

|**K**n(z,w)| ≤Cnexp

−c

√

nmin{|z−w|, δ}

·e^{−}^{n(Q(w)}^{−b}^{Q(w))/2}, w∈C.

Part (i) is standard, see e.g. [2], Sect. 3. For a proof of (ii) we refer to [2], Corollary 8.2 (which also shows that the constantccan be taken proportional to inf{p

∆Q(z); z∈S}).

2.6. **Notation.** We use the same letter**K**to denote a kernel**K(z,**w) and its corresponding integral
operator **K(**f)(z) = R

C f(w)K(z,w)dA(w). We will denote by the same symbol “C” a constant independent ofn, which can change meaning as we go along. The notation “An -Bn” means thatAn≤CBn. We shall write

(2.3) An(z)=D(z;R/√

n) , A^{+}_{n}(z)=D(z; (R+s)/√

n) , A^{−}_{n}(z)=D(z; (R−s)/√
n).

3. Preliminary estimates

In this section, we discuss gradient estimates for weighted polynomials; these will be useful in the following. In particular they imply that interpolating families are uniformly separated.

3.1. **Inequalities of Bernstein type.** The following lemma is analogous to Lemma 18 in [20].

**Lemma 3.1.** Let p be a polynomial of degree at most n. Fix a point z such that p(z),0and|∆Q(z)|<K.

Then (3.1)

∇ p

e^{−}^{nQ}^{/}^{2}

(z)

≤C

√ n

pe^{−}^{nQ}^{/}^{2}
_{L}^{∞},
and

(3.2)

∇ p

e^{−}^{nQ}^{/}^{2}

(z)

≤Cn

pe^{−}^{nQ}^{/}^{2}
_{L}2,
where the constant C depends only on K.

Proof. LetHz(ζ)=Q(z)+2∂Q(z)·(ζ−z)+∂^{2}Q(z)(ζ−z)^{2}andhz(ζ)=ReHz(ζ),so that
Q(ζ)=hz(ζ)+ ∆Q(z)|ζ−z|^{2}+O(|z−ζ|^{3}).

In particular, there is a constantCsuch that

(3.3) n|Q(ζ)−hz(ζ)| ≤C when |ζ−z| ≤1/√ n, whereCdepends only onK.

Now observe that, (3.4)

∇ p

e^{−}^{nQ}^{/}^{2}

(ζ) =

p^{0}(ζ)−n·∂Q(ζ)·p(ζ)

e^{−}^{nQ(}^{ζ}^{)}^{/}^{2},
and

∇ p

e^{−}^{nh}^{z}^{/}^{2}

(ζ) =

p^{0}(ζ)−n·∂hz(ζ)·p(ζ)

e^{−}^{nh}^{z}^{(}^{ζ}^{)}^{/}^{2}=

=

d

dζ(pe^{−}^{nH}^{z}^{/2})(ζ)
.
(3.5)

The expressions (3.4) and (3.5) are identical whenζ=z.

By Cauchy’s estimate applied to the circleC_{1/}^{√}_{n}(z) with centerzand radius 1/√
n,
(3.6)

d

dζ(pe^{−}^{nH}^{z}^{/}^{2})(z)
= 1

2π Z

C_{1/}^{√}_{n}(z)

p(ζ)e^{−}^{nH}^{z}^{(ζ)/2}
(z−ζ)^{2} dζ

≤ n 2π

Z

C_{1/}^{√}_{n}(z)

p(ζ)

e^{−}^{nh}^{z}^{(}^{ζ}^{)}^{/}^{2}|dζ|.
In view of (3.3), the right side can be estimated by a constant depending only onK, times

(3.7) n

Z

C_{1/}^{√}_{n}(z)

p(ζ)

e^{−nQ(ζ)/2}|dζ|.

To prove (3.1), it suffices to notice that (3.7) can be estimated by 2π√ n

pe^{−}^{nQ}^{/}^{2}
_{L}^{∞}.
Next notice that, by Lemma 2.9,

p(ζ)

2e^{−}^{nQ(ζ)} ≤C^{0}n
Z

D(ζ;1/√ n)

p(ξ)

2e^{−}^{nQ(ξ)}dA(ξ)≤C^{0}n

pe^{−}^{nQ/2}

2
L^{2}

with another constantC^{0}depending only onK. We conclude that

∇ p

e^{−}^{nQ/2}

(ζ) -n

√ n

pe^{−}^{nQ/2}
_{L}2

Z

C_{1/}^{√}_{n}(z)

|dζ|-n

pe^{−}^{nQ/2}
_{L}2,

with a constant depending only onK. This proves (3.2).

3.2. **Proof of Lemma 2.1.** LetZbe an interpolating family contained inS. (W.l.o.g. putρ=1.)
Fix an index j, 1≤ j≤mn. SinceZis interpolating, we can find a function f = fn ∈H˜nsuch
that f(zn j^{0}) = δj j^{0} andkfk^{2} ≤ C/n. Letδ > 0 be small enough thatD(zn j;δ) ⊂ Λ. Also assume
w.l.o.g. that a pointzn j^{0}satisfies

zn j−zn j^{0}

< δ; if there is no such j^{0}there is nothing to prove.

Evidently,

1= f(zn j)

−

f(zn j^{0})

≤

∇

f

_{L}^{∞}_{(}_{Λ}_{)}

zn j−zn j^{0}

. Thus Lemma 3.1 gives

1≤C1n f

zn j−zn j^{0}

≤CC1

√ n

zn j−zn j^{0}

.

This proves thatZiss-separated withs=1/(CC1).

4. The spectrum of the concentration operator

LetΩbe a measurable subset of the plane. Theconcentration operator**K**^{Ω}nρis defined by
**K**^{Ω}_{n}_{ρ}(f)(z)=

Z

Ωf(w)K_{nρ}(z,w)dA(w)=**K**_{nρ}(1_{Ω}·f)(z).

This is a positive contraction on ˜H_{nρ}.

In this section, we apply a technique which relates the spectrum of the concentration operator
to the number of points inΩ∩ Z_{n}whenZis either an interpolating family or an M-family; the
technique essentially goes back to Landau’s paper [19]. We here follow the strategy in [22], in a
suitably modified form.

We first turn toM-families. We will consider the cases ofMSand ofMSnfamilies separately.

4.1. MS,ρ**-families.** Fix a pointz∈Sand letλ^{nρ}_{j} =λ^{nρ}_{j} (z) denote the eigenvalues of**K**^{A}nρ^{n}^{(z)}: ˜Hnρ→
H˜_{nρ}, taken in decreasing order. Letφ^{n}_{j}^{ρ}be corresponding normalized eigenvectors. We write

N^{+}_{nρ}=N^{+}_{nρ}(z)=# Z_{n}∩A^{+}_{n}(z).
(See (2.3) for the definitions of the setsAnandA^{+}_{n}.)

**Lemma 4.1.** Suppose thatZ ⊂S is of class MS,ρ. There is then a constantγ <1and a number n0such
that for all z∈S and n≥n0, we haveλ^{nρ}_{N}+

nρ(z)+1 < γ.

Proof. W.l.o.g. put ρ = 1. Fix z ∈ S and suppose that f ∈ H˜n is such that f(zn j) = 0 when
zn j∈A^{+}_{n}(z). SinceZis 2s-separated,

(4.1)

Z

S^{+}

f

2≤C1 n

X

zn j∈S\A^{+}n(z)

f(zn j)

2≤Cs^{−}^{2}
Z

S^{+}\An(z)

f

2.

Now define f =PN_{n}^{+}+1

j=1 c^{n}_{j}φ^{n}_{j},where the numbersc^{n}_{j} (not all zero) are chosen so that f(zn j)=0
for allzn j∈A^{+}_{n}(z). This is possible for all largen, because the separation ofZensures thatN^{+}_{n} ≤C
for some constantC=C(R,s).

Since the operator**K**nis the orthogonal projection ofL^{2}onto ˜Hn, we have

N_{n}^{+}+1

X

j=1

λ^{n}_{j}
c^{n}_{j}

2=D

**K**^{A}_{n}^{n}^{(z)}f,fE

=D

**1**An(z)·**K**n(f),**K**n(f)E

= Z

An(z)

f

2dA.

(4.2)

We infer by means of (4.1) and (4.2) that
λ^{n}_{N}n++1

N^{+}_{n}+1

X

j=1

c^{n}_{j}

2

≤

N^{+}_{n}+1

X

j=1

λ^{n}_{j}
c^{n}_{j}

2= Z

S^{+}

− Z

S^{+}\An(z)

! f

2dA

≤ 1−s^{2}
C

! Z

S^{+}

f

2≤ 1−s^{2}
C

! f

2≤ 1−s^{2}
C

!^{N}n^{+}+1

X

j=1

c^{n}_{j}

2,

which proves thatλN^{+}n+1≤1−s^{2}/C.

Notice that the separability ofZimplies that #(Z_{n}∩(D(z; (R+s)/√

n)\D(z;R/√

n))) ≤ CR.

Therefore Lemma 4.1 implies the estimate

(4.3) #(Z_{n}∩D(z;R/√

n))≥#{j; λ^{nρ}_{j} ≥γ}+O(R),
where theO-constant is independent ofn.

4.2. MSn,ρ**-families.** We now modify the construction in the previous subsection.

Recall that Sn = {z ∈ S; dist(z, ∂S) ≥ 2δn} where δn = log^{2}n/√

n. Fix a sequence zn ∈ Sn

satisfying

dist(zn, ∂S)≥3δn.

As before, we consider the eigenvaluesλ^{nρ}_{j} (decreasing order) and corresponding eigenfunctions
φ^{nρ}_{j} of the concentration operator**K**^{A}nρ^{n}^{(z}^{n}^{)}. We will use the following lemma.

**Lemma 4.2.** For any positive integer K there is a constant CKand a number n0 =n0(R)such that for all j
λ^{n}_{j}^{ρ}

Z

C\Sn

φ^{n}_{j}^{ρ}

2

≤CKn^{−}^{K}, n≥n0.

Proof. W.l.o.g. letρ=1. Choosen0such that dist(An(zn),C\Sn)≥δn/2 whenn≥n0. By Lemma 2.10, we then have an estimate

|**K**n(ζ,w)| ≤Cne^{−}^{c}^{log}^{2}^{n}e^{−}^{n(Q(w)}^{−b}^{Q(w))/2}, w∈C\Sn, ζ∈An(zn),
wherecandCare positive constants. This gives

Z

C\Sn

Z

An(zn)

φj(w)Kn(ζ,w)φj(ζ)dA(ζ)dA(w)

≤Cne^{−}^{c}^{log}^{2}^{n}
Z

φj(z)

e^{−}^{n(Q(z)}^{−b}^{Q(z))/2}

!2

≤

≤CKn^{−}^{K}kφjk^{2}=CKn^{−}^{K},
where we have used the Cauchy-Schwarz inequality and thatR

e^{−}^{n(Q}^{−}^{Q)}^{b} =1+o(1).

**Lemma 4.3.** Suppose thatZ ⊂ S is of class MSn,ρ. There is then a constant γ < 1 and a number
n0 =n0(R)such that for all z∈S satisfyingdist(z, ∂S)≥3δnand all n≥n0, we haveλ^{nρ}_{N}n+ρ(z)+1< γ.

Proof. W.l.o.g. putρ=1. Assume that f ∈H˜nis such thatf(zn j)=0 for allzn j∈ Z_{n}∩A^{+}_{n}(z). Then
sinceZis 2s-separated for a sufficiently smalls,

Z

Sn

f

2≤C1 n

X

zn j∈S\A^{+}_{n}(z)

f(zn j)

2 ≤Cs^{−}^{2}
Z

S^{+}\An(z)

f

2dA.

(4.4)

We again define f =PN^{+}_{n}+1

j=1 c^{n}_{j}φ^{n}_{j},where the numbersc^{n}_{j} (not all zero) are chosen so thatf(zn j)=0
for allzn j∈ Z_{n}∩A^{+}_{n}(z).

This time, observe that Lemma 4.2 and the Cauchy-Schwarz inequality implies
λ^{n}_{N}+

n+1

Z

C\Sn

f

2≤

N^{+}_{n}+1

X

j,k=1

qλ^{n}_{j}λ^{n}_{k}
Z

C\Sn

c^{n}_{j}φ^{n}_{j}·c^{n}_{k}φ^{n}_{k}

≤

≤CKn^{−}^{K}

N^{+}_{n}+1

X

j,k=1

c^{n}_{j}c^{n}_{k}

≤CKn^{−}^{K}(N_{n}^{+}+1)

N_{n}^{+}+1

X

j=1

c^{n}_{j}

2

≤C^{0}n^{−}^{k}

N^{+}_{n}+1

X

j=1

c^{n}_{j}

2.

In view of (4.4), we now conclude that
λ^{n}_{N}+

n+1
N_{n}^{+}+1

X

j=1

c^{n}_{j}

2

≤

N_{n}^{+}+1

X

j=1

λ^{n}_{j}
c^{n}_{j}

2 = Z

S^{+}

− Z

S^{+}\An(z)

! f

2dA≤

≤ 1−s^{2}
C

! Z

Sn

f

2+ Z

S^{+}\Sn

f

2≤ 1−s^{2}
C

! f

2+ Z

S^{+}\Sn

f

2≤

≤

1−s^{2}

C + CKn^{−}^{K}
λ^{n}_{N}+

n+1

N_{n}^{+}+1

X

j=1

c^{n}_{j}

2,

whereCKdepends only onK,R, ands.

Withα=1−s^{2}/C, this impliesλ^{n}_{N}+
n+1 < p

α^{2}+4CKn^{−}^{K}.Thus if we defineγas any number in
the interval (α,1), we obtainλ^{n}_{N}+

n+1≤γfor allnlarge enough.

As a corollary, we obtain the following estimate: Let Zbe as in Lemma 4.3. Then for all n≥n0(R)

(4.5) #(Z_{n}∩D(z;R/√

n))≥#{j; λ^{n}_{j}^{ρ}≥γ}+O(R),
where theO-constant is independent ofn.

4.3. **Interpolating families.** Assume thatZbe aρ-interpolating sequence contained inS, and
let 2sbe a separation constant forZ. We can w.l.o.g. assume thatρ=1.

Fixz∈ S. We defineI_{n}as the set of indicesjsuch thatzn j ∈A^{−}_{n}(z) and letN_{n}^{−}=N^{−}_{n}(z) be the
cardinality ofI_{n}. By the separation we have a uniform boundN_{n}^{−}≤C=C(R,s).

Now let{cj}^{m}^{n}

1 be a sequence withcj=0 whenj<I_{n}. SinceZis interpolating we can choose
fn j ∈ H˜n such that fn j(zn j^{0})= δj j^{0} and

fn j

2 ≤C/nfor allnand j. The functions fn j, j∈ I_{n}are
linearly independent and span anN^{−}_{n}-dimensional subspace of ˜Hn. We denote this subspace by

F=spann

fn j; j∈ I_{n}o
.
Note that an arbitrary f =P

j∈Incjfn j∈Fsatisfies

f

2≤CN_{n}^{−}1
n

X

j∈I_{n}

cj

2 ≤C^{0}1
n

X

j∈I_{n}

f(zn j)

2.

Applying (2.2) now gives f

2≤C Z

An(z)

f

2=CD

**K**^{A}_{n}^{n}^{(z)}f,fE
.

Withδ=1/C, we have shown that

h**K**^{A}n^{n}^{(z)}f,fi

hf,fi ≥δ, f ∈F, f .0.

Letλ^{n}_{j} be the eigenvalues of the operator**K**^{A}_{n}^{n}^{(z)} on ˜Hnarranged in decreasing order. By the
Weyl–Courant lemma (see [13], p. 908) we have

λ^{n}_{j}−1≥ inf

g∈Ej

h**K**^{A}_{n}^{n}^{(z)}g,gi
hg,gi ,

whereEjranges over all j-dimensional subspaces of ˜Hn. Since dimF=N^{−}_{n}, we obtainλ^{n}_{N}^{−}

n−1 ≥δ.

The construction can obviously be carried out forρ ,1 as well. We have proved the following lemma.

**Lemma 4.4.** Suppose thatZisρ-interpolating, and letλ^{n}_{j}^{ρ}be the eigenvalues of the operator**K**^{A}_{nρ}^{n}^{(z)}on
H˜nρ, where z ∈ S. Also let N^{−}_{nρ}be the number of points inZ_{n}∩A^{−}_{n}(z). Then there is a numberδ >0
independent of n and z such that

#{j; λ^{nρ}_{j} ≥δ} ≥N_{nρ}^{−} −1.

Next notice that sinceZis 2s-separated (Lemma 2.1), there is a constantCsuch that

#(Z_{n}∩D(z;R/√

n))−N_{nρ}^{−} ≤C(R^{2}−(R−s)^{2})/s^{2}.
Using Lemma 4.4, we conclude that

(4.6) #(Z_{n}∩D(z;R/√

n))≤O(R)+#{j; λ^{nρ}_{j} ≥δ}, as R→ ∞,
where theO-constant is independent ofn.

5. Beurling–Landau densities ofMfamilies and of interpolating families

In this section, we prove Lemma 2.4. Our proof depends partly on trace estimates for the concentration operator, which are proved in Section 7.

5.1. **Proof of Lemma 2.4(i).** Let Z ⊂ Sbe of class MSn,ρ, and letζ = (zn) be a sequence with
dist(zn,C\S)≥3δn.

Consider the eigenvalues λ^{n}_{j}^{ρ} = λ^{n}_{j}^{ρ}(zn) of the concentration operator **K**^{A}nρ^{n}^{(z}^{n}^{)}, and putµn =
Pmn

j=1δ_{λ}^{n}^{ρ}

j whereδzis the Dirac measure atz. We then have trace

**K**^{A}nρ^{n}^{(z}^{n}^{)}

= Z 1

0

xdµn(x) , trace

**K**^{A}n^{n}^{(z}^{n}^{)}◦**K**^{A}n^{n}^{(z}^{n}^{)}

= Z 1

0

x^{2}dµn(x).

Letγandn0be given by Lemma 4.3. We then have, for alln≥n0(R),

#{j;λ^{n}_{j}^{ρ}> γ}=
Z 1

γ dµn(x)≥ Z 1

0

xdµn(x)− 1 1−γ

Z 1

0

x(1−x)dµn(x)=

=trace
**K**^{A}nρ^{n}^{(z}^{n}^{)}

− 1 1−γ

htrace
**K**^{A}nρ^{n}^{(z}^{n}^{)}

−trace

**K**^{A}nρ^{n}^{(z}^{n}^{)}◦**K**^{A}nρ^{n}^{(z}^{n}^{)}

i.

By the estimate (4.5) followed by the trace estimates in lemmas 7.1 and 7.3, we now get lim inf

n→∞

#(Z_{n}∩D(zn;R/√
n))

R^{2}∆Q(zn) ≥lim inf

n→∞

#{j;λ^{nρ}_{j} > γ}+O(R)
R^{2}∆Q(zn) ≥

≥lim inf

n→∞

trace

**K**^{A}nρ^{n}^{(z}^{n}^{)}

R^{2}∆Q(zn) − 1
1−γ

trace
**K**^{A}nρ^{n}^{(z}^{n}^{)}

−trace

**K**^{A}nρ^{n}^{(z}^{n}^{)}◦**K**^{A}nρ^{n}^{(z}^{n}^{)}

R^{2}∆Q(zn)

+O(1/R)=

=ρ(R+s)^{2}/R^{2}+O(1/R).

SendingR→ ∞, we obtainD^{−}(Z;ζ)≥ρ, and the proof of Lemma 2.4(i) is finished.

5.2. **Proof of Lemma 2.4(ii).** LetZbe aρ-interpolating family and letζ=(zn) be a sequence with
zn∈Snfor alln. Again letλ^{n}_{j}^{ρ}be the eigenvalues of the concentration operator**K**^{A}_{nρ}^{n}^{(z}^{n}^{)}.

Letµnbe the measureµn=Pmn

j=1δ_{λ}^{nρ}

j . Then for anyδ∈(0,1)
{j;λ^{nρ}_{j} ≥δ}=

Z 1

δ dµn(x)≤ Z 1

0

xdµn(x)+1 δ

Z 1

0

x(1−x)dµn(x).

In view of the estimate (4.6), we can pickδ=δ(R,s)>0 so that

#(Z_{n}∩D(z;R/√

n))≤O(R)+trace
**K**^{A}_{nρ}^{n}^{(z}^{n}^{)}

+1 δ

htrace
**K**^{A}_{nρ}^{n}^{(z}^{n}^{)}

−trace

**K**^{A}_{nρ}^{n}^{(z}^{n}^{)}◦**K**^{A}_{nρ}^{n}^{(z}^{n}^{)}i
.
Forzn∈Sn, the trace estimates in lemmas 7.3 and 7.1 now imply

lim sup

n→∞

#(Z_{n}∩D(zn;R/√
n))
R^{2}∆Q(zn) ≤

≤lim sup

n→∞

trace(K^{A}nρ^{n}^{(z}^{n}^{)})
R^{2}∆Q(zn) +1

δlim sup

n→∞

trace
**K**^{A}nρ^{n}^{(z}^{n}^{)}

−trace

**K**^{A}nρ^{n}^{(z}^{n}^{)}◦**K**^{A}nρ^{n}^{(z}^{n}^{)}

R^{2}∆Q(zn) +O(1/R)=

=ρ+O(1/R).

LettingR→ ∞now shows thatD^{+}(Z;ζ)≤ρ, which finishes the proof of Lemma 2.4(ii).

6. Equidistribution of the bulk part of aFekete set

In this section we prove Lemma 2.5. The proof is given modulo some estimates for the correlation kernel, which are postponed to the next section.

6.1. **Proof of Lemma 2.5(1).** LetF_{n} = {zn1, . . . ,znn} be a Fekete set and consider the Lagrange
interpolation polynomials

ln j(z)=Y

i,j

(z−zni)/Y

i,j

(zn j−zni).

To avoid bulky notation, from now on writezj:=zn jetc.

Now consider theLeja–Siciak functioncorresponding toF_{n},
Φn(z)=max

lj(z)

2e^{nQ(z}^{j}^{)}; j=1, . . . ,n
.
It is known that for allz∈C

(6.1) Φn(z)^{1}^{/}^{n}≤e^{Q(z)}^{b} and Φn(z)^{1}^{/}^{n}→e^{Q(z)}^{b} , as n→ ∞.

We refer to [23],§III.5, notably eq. (5.3) and Corollary 5.3, for proofs of these statements.

Let us write

(6.2) `j(z)=lj(z)e^{−}^{n(Q(z)}^{−}^{Q(z}^{j}^{))/2},
and notice that (6.1) implies that

(6.3)

`j(z)

≤e^{−}^{n(Q(z)}^{−}^{Q(z))/2}^{b} .

The following lemma concludes our proof for part (1) of Lemma 2.5.

**Lemma 6.1.** LetF ={F_{n}}be a family of Fekete sets. ThenF is uniformly separated.

Proof. By (6.3) we havek`jk_{L}^{∞}≤1 for allj. Hence Lemma 3.1 implies that there is a neighbourhood
ΛofSsuch that

∇ `j

_{L}^{∞}_{(}_{Λ}_{)}

≤C

√ n for some constantCindependent ofnand j.

Fixzn j∈ F_{n}and assume that a pointznk ∈ F_{n}is sufficiently close tozn j. Then
1=

`j(zn j)

−

`j(znk) ≤

∇

`j

_{L}^{∞}_{(Λ)}

zn j−znk

≤C

√ n

zn j−znk

.

We have shown thatF is uniformly separated with best separation constant≥1/C.

6.2. **Proof of Lemma 2.5(2).** We now modify the weighted Lagrangian polynomials`j(6.2), by
multiplying by certain “peak polynomials”, to localize to a small neighbourhood ofzj.

Takeε >0 small; consider the corresponding kernel**K**_{εn}(z,w), and put

(6.4) Lj(z)= **K**_{ε}n(z,zj)

**K**_{ε}n(zj,zj)

!2

·`j(z).

These are weighted polynomials of degree (1+2ε)n; evidentlyLj(zk)=δjk.We have the following lemma.

**Lemma 6.2.** There is a constant C depending onεbut not on n such that for all zj∈ F_{n}∩Sn, we have

Lj

_{L}1≤ C

n.
Proof. Fixzj∈ F_{n}∩Sn. By (6.1) we have

Lj(z)

≤

K_{ε}n(z,zj)

2

**K**_{εn}(zj,zj)^{2}e^{−}^{n(Q(z)}^{−b}^{Q(z))/2}.

Using the asymptotics in Lemma 7.4 and the fact thatQb≤Qeverywhere, we conclude that

Lj

_{L}1≤ C
(εn)^{2}

Z

C

Knε(z,zj)

2dA(z)= C

(εn)^{2}**K**nε(zj,zj)≤ C^{0}

εn.

**Lemma 6.3.** Let

Fn(z)= X

zj∈Sn

Lj(z)

.

There are then constants C=C(ε,s)and n0=n0(ε)such thatkFnk_{L}∞≤C when n≥n0.

Proof. Forzj∈Snwe have**K**n(zj,zj)≥cnwherec>0, by Lemma 7.4. Using Lemma 2.10 we find
that

Lj(z)

≤CVj(z), zj∈Sn, z∈C, where

(6.5) Vj(z)=exp

−c

√nεminn z−zj

, δn

o, wherecis a positive constant.

Observe thatVj(z)≤e^{−}^{c}

√εlog^{2}n≤1/nwhen
z−zj

≥δnandnis large enough. This gives that there isn0=n0(ε) such that

Fn(z)≤C X

zj∈D(z;δn)

Vj(z)+1, n≥n0.

Now whenzj ∈ D(z;δn) we haveVj(z) = e^{−}^{c}

√nε|z−zj|. Hence when w−zj

≤ s/√

n we have
Vj(z)≤Ce^{−}^{c}

√ nε|z−w|

, whereC=e^{cs}

√ε. This gives that Vj(z)≤Cn

s^{2}
Z

D(zj;s/√ n)

e^{−}^{c}

√ nε|z−w|

dA(w).

By the separation, we then obtain that, whenn≥n0, Fn(z)≤1+Cn

s^{2}
Z

C

e^{−}^{c}

√nε|z−w|

dA(w)=1+C 1
s^{2}ε

Z

C

e^{−}^{c}^{|}^{ζ}^{|}dA(ζ)<∞.

The proof of the lemma is finished.

The following lemma concludes our proof for part (1) of Lemma 2.5.

**Lemma 6.4.** Let F = {F_{n}}^{∞}

n=1 be a sequence of Fekete sets. Then the triangular familyF^{0} given by
F^{0}

n=F_{n}∩Snis(1+2ε)-interpolating for anyε >0.

Proof. Write F_{n}∩Sn = {zn1, . . . ,znmn} and take a sequence c = (cj)^{m}_{j}_{=}^{n}_{1}. Consider the operator
T:C^{m}^{n} →L^{1}+L^{∞}defined byT(c)=P

jcjLj, whereLjare given by (6.4). In view of Lemma 6.2 kTk

`^{1}mn→L^{1}≤supkLjk_{L}_{1} ≤C/n,
and by Lemma 6.3,

kTk_{`}^{∞}

mn→L^{∞}≤ kFnk_{L}∞≤C.

By the Riesz–Thorin theorem, we conclude that
kTk_{`}2

mn→L^{2}≤C/√
n.

We have shown that, if f =T(c), then f ∈H˜n(1+2ε), f(zn j)=cjfor allj≤mnand Z

f

2≤ C n

mn

X

j=1

f(zn j)

2.

I.e.,F^{0}is (1+2ε)-interpolating.