and
ψ(z)˜ ≡
SD,R◦φ(z) :z∈E R¯◦φ(z) :z∈E.˜
We saw in prop 5.2.8 thatψis meromorphic onN. An analogous argument shows that ˜ψis meromorphic.
Recall that the transcendence degree of the field of meromorphic functions on a compact Riemann Surface is one. It follows that there exists a polynomialPsuch thatP(ψ,ψ)˜ =0. In particular, forz∈∂D=φ(∂E), we haveP(R(z),R(z))¯ =0. It then follows thatR(∂D)⊂ {P(z,z)¯ =0}and thusR(∂D) is a subset of an algebraic curve inC.
nothing) to assumeΩ = int(Ω) (so that there exist nontrivial lower bounds ons). We will do so from now on (we will eliminate the assumption later when we prove 5.30). Letbdenote the number of cusps on∂Ω,c denote the connectivity ofΩ, andddenote the number of double points of∂Ω. Gustafsson [13] proved
s≤(n−1)2+1−c−b−2d (5.31)
and Sakai [27] proved
s≥n−2+c−b, (5.32)
see also McCarthy and Yang [25]. We prove
s≤5n−5−c. (5.33)
Our proof makes use of Lemma 4.3 from [24]. The new ingredient is a result of D. Khavinson and G.
Neumann [21]. They resolved an open problem in gravitational lensing by proving that for a rational function Rof degree≥3
|{z∈C|R(z)=z}| ≤¯ 5n−5, (5.34)
where| · |denotes the cardinality. Before stating Lemma 4.3, we first need to define an important set ˆA⊂Ω used in the lemma. LetS denote the Schwarz function ofΩ. Further assume that ¯S has no critical value on
∂Ω(we will remove this assumption later). LetK≡Cˆ\Ω. Let
K=
c
G
i=1
Ki (5.35)
be the decomposition ofKinto connectivity components. By the hypothesis on ¯S, eachKiis a closed Jordan domain (recall that we are assuming for now thatΩ =int(Ω)). Let
S¯−1(K)≡ {z∈Ω : ¯S(z)∈K}. (5.36)
Then
S¯−1K=
c
G
i=1
S¯−1Ki. (5.37)
We denote byAithe component of ¯S−1Kithat contains∂Ki. This component contains a set homeomorphic to an annulus such that∂Kiis one of the boundary components of the annulus (see [24]). The set ˆAiis defined by filling in the holes of the annulus, that is
Aˆi≡Ai∪Ki. (5.38)
Aˆis defined to be the set
Aˆ≡
c
[
i=1
Aˆi. (5.39)
In the case where ¯S has critical values on∂Ω, ˆAis defined similarly but with a small modification (see [24]
section 4.3 for details). The following Lemma is from [24]:
Lemma 5.3.1. (Lee-Makarov) There exists a branched covering map G: ˆC→Cˆ of degree n such that i G=S on¯ Cˆ\A.ˆ
ii G is quasi-conformally equivalent to a rational map, i.e.
G= Φ−1◦R¯◦Φ, (5.40)
for some rational map R and some orientation-preserving quasi-conformal homeomorphism Φ: ˆC→Cˆ.
iii Each component, Kiof K contains a fixed point of G which attracts the orbits of all points of Ki. Theorem 5.3.2. LetΩbe a quadrature domain of degree n ≥ 3 satisfyingΩ = int(Ω). Let s denote the number of special points and c denote the connectivity ofΩ. Then
s≤5n−5−c.
Proof. By Lemma 4.3 in [24], ¯S|C\ˆ Aˆ = Φ−1 ◦R¯ ◦Φ|C\ˆ Aˆ for some rational map R and some orientation- preserving homeomorphismΦ: ˆC→Cˆ. We claim that ¯S has no fixed points on ˆA∩Ω. Indeed it suffices to show that ¯S has no fixed points on ˆAi∩Ω. Note that ˆAi∩Ω =Ai\∂Ki, and by definition ¯S(Ai)=Kiand thus
S¯(Ai\∂Ki)⊂Ki. Now sinceAi\∂Ki ⊂ΩandKi⊂Ωc, we have shown that ¯S has no fixed points on ˆAi∩Ω and thus ¯S has no fixed points on ˆA∩Ω.
By property (iii) from Lemma 5.3.1,Φ−1◦R¯◦Φhas at leastcfixed points on ˆA. We then have
s=|{z∈Ω|S(z)=¯z}| (5.41)
=|{z∈Ω|S¯(z)=z}| (5.42)
=|{z∈Cˆ\A|ˆ S¯(z)=z}| (5.43)
=|{z∈Cˆ\A|ˆ Φ−1◦R¯◦Φ(z)=z}| (5.44)
≤ |{z∈Cˆ|Φ−1◦R¯◦Φ(z)=z}| −c (5.45)
=|{z∈Cˆ|R¯=z}| −c (5.46)
≤5n−5−c, (5.47)
where 5.43 follows since ¯S has no fixed points on ˆA∩Ωand ˆC\Aˆ = Ω\A. 5.45 follows sinceˆ Φ−1◦R¯◦Φ has at leastcfixed points on ˆA. 5.46 follows sinceΦis a bijection. 5.47 follows from the inequality 5.34
(Theorem 3 in [21]).
We are now ready to prove 5.30.
Theorem 5.3.3. LetΩbe a quadrature domain with degree n≥3. Let c denote the connectivity ofΩ. Then
c≤5n−5. (5.48)
Proof. LetΛ≡int(Ω), let ˜cdenote the connectivity ofΛ, and let ˜sdenote the number of special points inΛ. Thenint(Λ)= Λand so by Theorem 5.3.2,
˜
s≤5n−5−c.˜ (5.49)
Recall that∂Ωis subset of an algebraic curve,Γ, of degree 2n. MoreoverΓ\∂Ωis the set of special points of Ω. LetSΩ,SΛdenote the sets of special points inΩandΛrespectively. Note thatSΩ⊂SΛand that
|SΛ\SΩ|=|∂Ω\∂Λ|. (5.50)
By hypothesis onΛ,∂Λdoes not contain any special points. It then follows that
|∂Ω\∂Λ| ≤ |Γ\∂Λ|=s.˜ (5.51)
Note thatΩis obtained fromΛby removing fromΛthe set of special points inΛ\Ω. That is
Ω = Λ\{SΛ\SΩ}. (5.52)
Since removing a point fromΩincreases the connectivity by one, it follows thatc−c˜=|{SΛ\SΩ}|. Putting everything together we have:
c=|{SΛ\SΩ}|+c˜ (5.53)
=|∂Ω\∂Λ|+c˜ (5.54)
≤s˜+c˜ (5.55)
≤5n−5, (5.56)
where 5.54 follows from 5.50, 5.55 follows from 5.51, and 5.56 follows from Theorem 5.3.2.
Chapter 6
Coulomb Gas Ensembles and CFT
6.1 Introduction
LetQ:C→R∪ {∞}andβ >0. The following sequence of probability measures
Πn ≡ 1 Zn
Z
Cn
e−βHndA⊗n, (6.1)
where
Hn(z1, ...,zn)≡ X
1≤i<j≤n
log 1
|zi−zj|+(n−1)
n
X
i=1
Q(zi),
andZnis a normalizing constant is called a Coulomb gas ensemble orβ-ensemble. Coulomb gas ensembles occur frequently in random matrix theory. For example, whenβ=2 they are the eigenvalue distributions of certain classes of random normal matrices. There is also a connection to conformal field theory. The internal energy component of the integrand of 6.1 is related to the vacuum expectation of vertex operators. Coulomb gas ensembles generalize nicely to Riemann surfaces. Let
Hnν,Q(z1, ...,zn)≡ X
1≤i<j≤n
[Gν(zi,zj)+Q(zi)+Q(zj)].
Physically,Hcan be interpreted as the energy of configuration ofnunit point charges placed at{z1, ...,zn}, in the presence of an external field (n−1)Q. We define the partition function
ZnQ,ν,β≡ Z
Mn
e−βHndvol⊗ng .
In analogy with statistical mechanics, the Boltzmann Gibbs measure corresponding to the energy functionH
is defined to be
Πn ≡ 1 Zn
e−βHndvol⊗ng . (6.2)
For certainν,Gν(z,w) =E[Φ(z)Φ(w)], whereΦis a Free Bosonic Field onM. Letk≤n, and letA⊂ Mk. The k-th marginal measure ofΠnis defined by
Πn,k(A)≡Πn(A×Mn−k).
The main result of this section is:
Theorem 6.3.3. Let Q be an admissible potential. If Qν,gis continuous then
Πn,k* µ⊗keq
as n→ ∞.
Johansson [18] proved this result onRand Hedenmalm and Makarov [15] later proved the analogue in the complex plane. Our proof follows the general structure of Johansson’s although some portions are more technical in this setting (e.g. Lemma 6.3.1). The proof follows the structure of Johansson’s [18] proof of the convergence of the marginal measures forβ-ensembles onR. Hedenmalm and Makarov [15] later proved the analogue in the plane.
We then show that the Bosonic free field on the cylinder can be realized as a limit of Fluctuations of a Coulomb gas ensemble on the cylinder.