On boundary confinements for the Coulomb gas
Yacin Ameur Nam-Gyu Kang Seong-Mi Seo


A scale of boundary confinements]On boundary confinements for the Coulomb gas




Yacin Ameur

Department of Mathematics

Faculty of Science

Lund University

P.O. BOX 118

221 00 Lund

Sweden
Yacin.Ameur@maths.lth.se


Nam-Gyu Kang

School of Mathematics

Korea Institute for Advanced Study

85 Hoegiro

Dongdaemun-gu

Seoul 02455

Republic of Korea
namgyu@kias.re.kr



Seong-Mi Seo

School of Mathematics

Korea Institute for Advanced Study

85 Hoegiro

Dongdaemun-gu

Seoul 02455

Republic of Korea
seongmi@kias.re.kr



[2010]82D10, 60G55, 46E22, 42C05, 30D15

Nam-Gyu Kang was partially supported by Samsung Science and Technology Foundation (SSTF-BA1401-51) and by a KIAS Individual Grant(MG058103) at Korea Institute for Advanced Study.
Seong-Mi Seo was partially supported by a KIAS Individual Grant (MG063103) at Korea Institute for Advanced Study and by a National Research Foundation of Korea Grant funded by the Korea government (No. 2019R1F1A1058006). 


 We introduce a family of boundary confinements for Coulomb gas ensembles, and study them in the two-dimensional determinantal case of random normal matrices.
The family interpolates between the free boundary and hard edge cases, which
have been well studied in various random matrix theories. The confinement can also be relaxed beyond the free boundary to produce ensembles with fuzzier boundaries, i.e.,
where the particles are more and more likely to be found outside of the boundary. The resulting ensembles are investigated with respect to
scaling limits and distribution of the maximum modulus. In particular, we prove existence of a new point field - a limit of scaling limits to the ultraweak
point when the droplet ceases
to be well defined.


[
    Seong-Mi Seo
    December 30, 2023
=====================





§ INTRODUCTION AND MAIN RESULTS


In the theory of Coulomb gas ensembles, it is natural to consider different kinds of boundary confinements. The most well-known examples are the “free boundary”, where particles are admitted to range freely outside of the droplet,
and the “hard edge”, where they are completely confined to it. On the other hand, notions of weakly confining potentials have attracted attention recently, where
the boundary is softer than a free boundary, i.e., particles are more likely to be found outside of the boundary. In this note,
we introduce a one-parameter family of edge confinements, ranging all the way between an idealized “ultraweak” edge and a hard edge.






Our construction can be applied to general
Coulomb gas ensembles in any dimension and for any inverse temperature β.
However, we shall here be content to develop the theory only
in the determinantal, two-dimensional case,
i.e., we will consider ensembles of
eigenvalues of random normal matrices.


The study of universality in free boundary ensembles has been the focus of several recent works <cit.>. Notably, in the paper <cit.>, it is shown that with free boundary confinement, the point field with intensity function R(z)=(z+z̅) appears universally (i.e., for a “sufficiently large” class of ensembles) when rescaling about a regular boundary point, where , the “free boundary function”, is given by

    (z)=_1(z):=1/√(2π)∫_-∞^0e^-(z-t)^2/2  dt=1/2z/√(2).







For the hard edge Ginibre ensemble,
a direct computation with the orthogonal polynomials in <cit.> shows that,
under a natural scaling about a boundary point, the point process
of eigenvalues converges to the determinantal point field determined by the 1-point function R(z)=_∞(z+z̅)·_Ł(z)
where _Ł is the indicator function of the left half plane Ł={ z<0} and where _∞, the “hard edge plasma function”, is defined by

    _∞(z)=1/√(2π)∫_-∞^0e^-(z-t)^2/2/(t)  dt.

As far as we know, this function appeared first in the physical paper <cit.> from 1982, cf. <cit.>;
see Figure 1.



We shall introduce a scale of point-processes depending on a confinement-parameter  such that the values =1 and =∞ correspond to the free boundary and the hard edge, respectively.


We note at this point that the functions _1 and _∞ both have the convolution structure

    _(z) = 1/√(2π)∫_-∞^0e^-(z-t)^2/2/Φ_(t)  dt,

where Φ_1=1 and Φ_∞=. We shall show that if >0, then the choice

    Φ_(t):=(t)+1/√()(1-(t/√()))exp{(1-)t^2/2}

leads to a scale of new determinantal point fields with 1-point intensities

    R(z)=R^()(z)=_(z+z̅)exp{2(1-)( z)_+^2},

where we write x_+=max{x,0}  and (to avoid bulky notation) x_+^2 instead of (x_+)^2.

We shall find that these point fields emerge naturally as scaling limits about regular boundary points of the droplet, if we set up appropriate boundary confinements.
In the limit as → 0+ we will establish existence of a new point field in the critical case where the droplet ceases to be well defined. This
point field might be said
to model an idealized ultraweak edge; its density profile is depicted in Figure <ref> and also in Figure 3.








Hard edge ensembles are well known in the Hermitian theory, where they are usually associated with the Bessel kernel <cit.>. Another possibility, a “soft/hard edge”, appears when a soft edge is replaced by a hard edge cut. This situation was studied by Claeys and Kuijlaars in the paper <cit.>. In this case a Painlevé II kernel arises instead of a Bessel kernel. The hard edges in the present note (in the case =∞) are actually of the soft/hard type, but
to keep our terminology simple, we prefer to use the adjective “hard”. (We follow the papers <cit.> in this connection.)




 §.§ Basic setup

Fix a function (“external potential”) Q:→∪{+∞} and write Σ={Q<+∞}. We assume that Σ be dense in Σ and that Q be lower semicontinuous on  and real-analytic on Σ and “large” near ∞:

    lim inf_ζ→∞Q(ζ)/log|ζ|^2>1.


We next form the equilibrium measure σ in external potential Q, namely the measure μ that minimizes the weighted energy

    I_Q[μ]=∬_^2log1/|ζ-η|  dμ(ζ)dμ(η)+∫_ Q  dμ

amongst all compactly supported Borel probability measures μ on .
It is well known <cit.> that σ is unique, is absolutely continuous, and takes the form

    dσ(ζ)= Q(ζ)·_S(ζ)  dA(ζ),

where S is a compact set which we call the droplet in potential Q. Here and henceforth we use the convention that = denotes 1/4 times the usual Laplacian, while dA=dxdy/π is Lebesgue measure normalized so that the unit disk has measure 1.

We will in the following assume that S⊂Σ and that S be connected.
Under our assumptions, S is finitely connected and the boundary Ṣ is a union of a finite number of real-analytic arcs, possibly with finitely many singular points which can be certain types of cusps and/or double points. See e.g., <cit.>.

We shall consider the outer boundary
Γ=S,
where the polynomially convex hull S is the
union of S and the bounded components of ∖ S.
Thus Γ is a Jordan curve having possibly finitely many singular points. We shall assume that Γ be everywhere regular, i.e., that there are no singular points on Γ.

 We can do with weaker assumptions: for instance concerning regularity it suffices to assume that Q be real-analytic in a neighborhood of Γ and, say, C^2-smooth on Σ.


We next recall a basic potential-theoretic construct: the obstacle function Q̌(ζ) in external potential Q. This function can be defined in several ways, e.g., as the maximal subharmonic function which is less than Q on  and grows at most like log|ζ|^2 + O(1) at infinity,
or as γ-2U^σ(ζ) where
U^σ(ζ)=∫_log1/|ζ-η|  dσ(η) is the logarithmic potential of σ, and γ is a (“modified Robin's”) constant.

The properties of Q̌ to be used below are (i) Q̌=Q on S and Q̌≤ Q everywhere, (ii) Q̌ is C^1,1-smooth on  and harmonic
on the complement S^c of S, (iii) Q̌(ζ)= log|ζ|^2+O(1) as ζ→∞. (See <cit.> for proofs.)

In general, it might happen that Q̌=Q on some points of the complement S^c, called “shallow points” in <cit.>. With a mild restriction, we will assume that no such points exist, i.e., we assume that the droplet S equals to the coincidence set
{Q=Q̌} and that Q̌<Q everywhere on S^c.

After these proviso, we introduce our main object of study.

We fix a number (the “confinement-constant”)  with 0<<∞
and consider the modified (C^1,1-smooth) potential Q^() defined by

    Q^()(ζ)=Q(ζ)+(1-)Q̌(ζ).



Observe that when =1 we have Q^()=Q and if =∞ we recover the hard edge potential Q^(∞)=Q+∞·_S which has been denoted Q^S in papers such as <cit.>.
As → 0 we recover the obstacle function Q^(0)=Q̌, which is not an admissible potential since it fails to satisfy the growth condition (<ref>).


Given a positive value of the confinement constant  it is natural to study corresponding planar Coulomb gas ensembles determined by a partition function of the form

    Z_n,^β:=∫_^n e^-β H_n,  dV_n,     H_n,(ζ_1,…,ζ_n)=∑_j k^nlog1/|ζ_j-ζ_k|
    +n∑_j=1^nQ^()(ζ_j),

where we write dV_n for the usual Lebesgue measure in ^n divided by π^n. Here β is an arbitrary positive constant
(the “inverse temperature”).

We next introduce a Boltzmann-Gibbs type probability measure on ^n by

    d_n,^β=1/Z_n,^β· e^-β H_n,  dV_n

and consider configurations {ζ_j}_1^n of points in , picked randomly with respect to this measure.

By arguing as in the free boundary case (see <cit.>) it is easy to verify that the system {ζ_j}_1^n roughly tends to follow the equilibrium distribution, in the sense that, for each bounded and continuous function f on , one has the convergence

    1/n_n,^β[f(ζ_1)+⋯ +f(ζ_n)]→σ (f),     (n→∞).




 As for any point process, the system {ζ_j}_1^n is determined by the collection of its k-point intensity functions _n,k=_n,k^(). A characterization of these functions seems to be quite a hard enterprise. In this note, we shall henceforth restrict to the important determinantal case β=1, leaving other β to a future investigation.

In the case β=1, we have the basic determinant formula

    _n,k^(η_1,…,η_k)=(_n^(η_i,η_j))_i,j=1^k,

where the correlation kernel _n^ can be taken as the reproducing kernel for the subspace _n^ of L^2=L^2(,dA) consisting of all “weighted polynomials” w=pe^-nQ^S/2 where p is an analytic polynomial of degree at most n-1. (This canonical correlation kernel is used
without exception below.)

We will write _n^=_n,1^ for the 1-point function, which is the key player in our discussion below.




 §.§ Scaling limit

Let us now fix a (regular) point on the outer boundary Γ, without loss of generality we place it at the origin, and so that the outwards normal to Γ at 0 points in the positive real direction.

We define a rescaled process {z_j}_1^n by magnifying distances about 0 by a factor √(n Q(0)),

    z_j=√(n Q(0)) ζ_j,         (j=1,…,n).


In general, we will denote by ζ,z two complex variables related by z=√(n Q(0)) ζ.
We regard the droplet S as a subset of the ζ-plane.
 Restricting to a fixed bounded subset of the z-plane, the image of the droplet then more and more resembles left half plane Ł, as n→∞; see Figure 2.



Following <cit.> we denote by plain symbols R_n^,K_n^, etc., the 1-point function, canonical correlation kernel, etc., with respect to the process {z_j}_1^n. Here the canonical kernel K_n is, by definition

    K_n^(z,w)=1/n Q(0)_n^(ζ,η),   z=√(n Q(0)) ζ,   w=√(n Q(0)) η.




Recall that a function of the form c(ζ,η)=g(ζ)g̅(η), where g is a continuous unimodular function, is called a cocycle. A function h(z,w) is Hermitian if h̅(w,z)=h(z,w), and Hermitian-analytic if furthermore h is analytic in z and in w̅. Finally, the Ginibre kernel is

    G(z,w)=e^-|z|^2/2-|w|^2/2+zw̅.

This is the correlation kernel of the infinite Ginibre ensemble, which emerges by rescaling about a regular bulk point, see e.g., <cit.>.






 (“Structure of limiting kernels, Ward's equation”).

 
  *  There exists a sequence of cocycles c_n such that each subsequence of the sequence (c_nK_n) has a subsequence converging boundedly in ^2 and
locally uniformly in (∖ i)^2 to a Hermitian limit K.


  *  Each limiting kernel K in (i) is of the form

    K(z,w)=G(z,w)Ψ(z,w)exp{(1-)(( z)_+^2+( w)_+^2)}

where Ψ is some Hermitian-entire function.


  *  Each limiting 1-point function R(z)=K(z,z) is everywhere strictly positive and satisfies the modified Ward equation

    C=R-1-log R+(1-)_ z>0,

where C(z) is the Cauchy-transform of the Berezin kernel corresponding to R (see Section <ref>).







By Theorem <ref> and standard arguments (e.g., Macchi-Soshnikov's theorem, see <cit.>) we obtain immediately the existence and uniqueness 
of a non-trivial limiting point field
 {z_j}_1^∞ corresponding to each limiting 1-point function R in Theorem <ref>.


 We have the following theorem.


If Q is radially symmetric, or more generally, if conditions (P1) and (P2) in Section <ref> are satisfied, then
the rescaled process {z_j}_1^n converges as n→∞ to a unique determinantal point field {z_j}_1^∞ determined by the 1-point function R^() in (<ref>).


Note in particular that Theorem <ref> proves existence of a determinantal point field with 1-point function R^().



Observe also that the 1-point intensities R^()(z) converge locally uniformly as →0+ to the function

    R^(0)(z)   =_0(z+z̅)exp{2( z)_+^2},
    _0(z)   =∫_-∞^0 e^-(z-t)^2/2/√(2π)(t)-t^-1e^-t^2/2  dt.


In view of this convergence, standard arguments imply that
 the point fields {z_j}_1^∞ with 1-point functions R^() converge (in the sense of point fields) to a new determinantal point
field with 1-point function R^(0). It is also easy to establish convergence on the level of Ward equations. We summarize this in the form of a theorem.

 There exists a unique determinantal point field in  with 1-point function R^(0). The 1-point function R^(0) gives rise to
a solution to the generalized Ward equation (<ref>) with parameter value =0.


It is interesting to note that R^(0) has a heavy tail in the sense that

    R^(0)(x):    = _0(2x)   e^2x^2 = 1/4x^2 + O(x^-3),    x → +∞.

By contrast, if >0 then R^()(x)≲ Ce^-2 x^2 as x→+∞, see Fig. 3.




 Likewise, by letting →+∞, we can recover the results concerning the hard edge point field with 1-point function R^(∞) from the note <cit.>. We refer to <cit.> for further details about hard edge point fields and their applications.






 §.§ Distribution of the maximum modulus
 Let us now assume that the external potential Q is radially symmetric, and let {ζ_j}_1^n be a random sample from the corresponding Gibbs' distribution.

We shall denote by the symbol “|ζ|_n” the maximum modulus of the sample, i.e., the random variable

    |ζ|_n = max_1≤ j≤ n|ζ_j|.

We also write

    ρ=max{|ζ|; ζ∈ S}

and introduce the constants
 
    γ_n = log (n/2π) -2loglog n + 2log C_,

where C_ = ρ√( Q(ρ))/Φ_(0) = 2ρ√( Q(ρ))/(√()+1).

Finally, we define a random variable ω_n by rescaling ||_n about ρ in the following way:

    ω_n = √(4nγ_n Q(ρ))(|ζ|_n - ρ - √(γ_n/4n Q(ρ))).


Given these proviso, we have the following theorem, which generalizes earlier results due to Rider <cit.> and Chafaï and Péché <cit.> in the case =1.

 The random variable ω_n converges in distribution to the standard Gumbel distribution: for x∈

    lim_n→∞_n (ω_n ≤ x)= e^-e^-x.



 The Gumbel distribution is used to describe the distribution of extreme values. For example, the fluctuation of the maximum of i.i.d. gaussian random variables is expressed by a Gumbel distribution.
In the present context, it was shown by Rider <cit.> that the scaled maximal modulus of the free boundary Ginibre ensemble converges to the Gumbel distribution. This result was later generalized
to arbitrary radially symmetric potentials by Chafaï and Péché <cit.>.

 For the hard edge ensemble, the situation is very different. In microscopic scale, the eigenvalues are distributed densely near the boundary inside the droplet. With a proper scaling, the limit law for the maximal modulus follows an exponential distribution. See <cit.>. The exponential-type distribution and the Gumbel distribution can be found in the classification of extremal distribution functions. See <cit.>.

On the other hand, it is shown by Butez and García-Zelada in <cit.> that for an n-dependent potential of the form q_n:=(1+1/n)Q̌ (where Q is a suitable radially symmetric potential for which the corresponding droplet is the unit disk) the maximal modulus |ζ|_n converges in distribution (without scaling) to a random variable “|ζ|_∞” with distribution function

    (|ζ|_∞ < x) = ∏_k=1^∞(1-x^-2k),   x>1.

The potential q_n is very weakly confining, and is of another type from the ones studied in the present paper.




 §.§ Plan of this paper; further results

The outline of this paper is as follows. In Section <ref>, we give an elementary proof of Theorem <ref> for the generalized Ginibre ensemble (the potential Q(ζ)=|ζ|^2)
with an arbitrary value of the confinement coefficient .

In Section <ref>, we prove Theorem <ref> about the structure of limiting kernels.

In Section <ref> we analyze the modified Ward equation (<ref>) under some natural assumptions, most importantly we assume apriori
translation invariance of a scaling limit, i.e., R(x+iy)=R(x). Under this assumption,
it turns out that Ward's equation determines the solution R=R^() up to constant. (See Theorem <ref>.) This argument applies
for any reasonable potential, and gives a possible “abstract” approach to universality.

In Section <ref> and <ref> we present a more “concrete” approach, by adapting the method of quasipolynomials of Hedenmalm and Wennman <cit.>.
This method is then applied to prove universality of scaling limits (Theorem <ref>) in Section <ref> and universality of maximum modulus (Theorem <ref>) in Section <ref>.

In the concluding remarks section (Section <ref>) we summarize our results and mention some natural problems going forward. 





 §.§ Index of notation
  We will write L^2(e^-ϕ)=L^2(e^-ϕ,) for the usual L^2-space normed by f_ϕ^2:=∫_|f|^2 e^-ϕ  dA; when ϕ=0 we drop the  “ϕ” and write L^2 and f respectively. The characteristic function of a set E is denoted
_E, and we always reserve the symbol “δ_n” for the number δ_n=n^-1/2log n. For convenience of the reader, we now list a number of other frequently occurring symbols.

=∪{∞}; ={|ζ|<1}; _e(ρ)={|ζ|>ρ}∪{∞}; _e=_e(1); Ł={ z<0};

“Area”: dA(ζ)=1/π  d^2 ζ; “Arclength”: ds(ζ)=1/2π |dζ|; “Laplacian”: =;

S_τ: droplet in external potential Q/τ; U_τ=∖ S_τ; Γ_τ=Ụ_τ;

_τ: exterior unit normal on Γ_τ;

Q̌_τ: obstacle function in external potential Q/τ;

V_τ: harmonic continuation of Q̌_τ|_U_τ across Γ_τ;

ϕ_τ: normalized conformal map ϕ_τ : U_τ→_e with ϕ_τ(∞)=∞ and ϕ_τ'(∞)>0.







§ CONFINED GINIBRE ENSEMBLES


In this section, we consider the Ginibre case, i.e., the random normal matrix model associated with the external potential

    Q^() = Q + (1-)Q̌,   (>0)

where Q() = ||^2.
In this example, the droplet S is the unit disk ={ζ∈: |ζ| ≤ 1} and
the solution to the obstacle problem is given by

    Q̌() =  ||^2      if  ||≤ 1, 
      1 + log ||^2     otherwise.

Let {ζ_j}_1^n denote the eigenvalues of random normal matrices with Q^(). Define a rescaled system {z_j}_1^n at the boundary point p=1 by
z_j = √(n)(ζ_j -1).

We shall give a short proof for the convergence of the rescaled ensemble to the point field in Theorem <ref>. We here use the normal approximation to the Poisson distribution. This approach can be found in <cit.> for the Ginibre ensemble with a free boundary as well as with a hard edge.


Recall that a correlation kernel for the process {ζ_j}_1^n is obtained by

    𝐊_n(ζ,η) = ∑_j=0^n-1()^j/^j^2_nQ^() e^-n(Q^()(ζ)+Q^()(η))/2

since Q^() is radially symmetric.
Here, by direct computation we have

    ζ^j^2_nQ^()= n^-j-1γ(j+1,n) + e^-n(1-)(n)^-j-1 + n(1-)Γ(j+1-n(1-),n),

where γ(j+1,n) = ∫_0^n s^j e^-s ds is the lower incomplete Gamma function and Γ(j+1,n)=∫_n^∞ s^j e^-s ds is the upper incomplete Gamma function.
It follows that

    𝐊_n(ζ,η) = n ∑_j=0^n-1(nζη̅)^j e^-nQ^()(ζ)/2-nQ^()(η)/2/P(j,n,),

where

    P(j,n,)    =  j!  (U_n>j) 
       + e^-n(1-)^-j-1 (n)^n(1-) Γ(j+1-n(1-)) (U_n≤ j+1-n(1-))

and U_s ∼Po(s), i.e., U_s is a Poisson distributed random variable with intensity s.

By normal approximation of the Poisson distribution we obtain

    (U_n>j)  =  φ(ξ_j,n) (1+o(1)),   (U_n≤ j+1-n(1-))= 1 - φ(ξ_j,n/√()) (1+o(1)),

where (ξ)=1/√(2π)∫_ξ^∞ e^-t^2/2  dt, ξ_j,n=(j-n)/√(n), and o(1)→ 0 as n→∞ uniformly in j.

Now rescale by

    ζ=1+z/√(n),    η=1+w/√(n),     z,w∈

and write

    K_n(z,w)=1/n𝐊_n(ζ,η)=∑_j=0^n-1( nζη̅/λ)^jλ^j/P(j,n,) e^-λ,

where λ=n(Q^()(ζ)+Q^()(η))/2.
Note that λ has the asymptotic identity:

    λ   = n + √(n)(z+w) + |z|^2+|w|^2/2 - (1-)( ( z)_+^2 + ( w)_+^2) + o(1).

We have shown that

    K_n(z,w)=(1+o(1))∑_j=0^n-1( nζη̅/λ)^jλ^j/j!e^-λ1/φ(ξ_j,n) + Γ(j,n,)(1-φ(ξ_j,n/√())),

where Γ(s,n,):=e^-n(1-)^-s-1 (n)^n(1-)Γ(s+1-n(1-))/ Γ(s+1).
Finally, if X_n∼Po(λ), we can write the last sum in the form

    ∑_j=0^n-1(nζη̅/λ)^j (X_n=j)/φ(ξ_j,n) + Γ(j,n,)(1-φ(ξ_j,n/√()))
       =𝔼 (nζη̅/λ)^X_n_{X_n<n}/φ(X_n-n/√(n))+Γ(X_n,n,)(1-φ(X_n-n/√(n))).

Defining Y_n by X_n=λ+√(λ)Y_n and α_n=(n-λ)/√(n), we now get a relation of the form

    K_n(z,w)=(1+o(1))A_n  B_n,

where

    A_n = (nζη̅/λ)^λ

and

    B_n =
    𝔼 (nζη̅/λ)^√(λ) Y_n_{Y_n<α_n}/φ(√(λ/n) Y_n -α_n)+Γ(λ+√(λ)Y_n,n,)(1-φ(√(λ/n) Y_n - α_n/√())).


 
Let x =  z, u =  w, and b = b(z,w) = (z+w̅).
We have

    A_n    = e^ib√(n)  e^b^2/2 G(z,w)(1+o(1))   e^(1-)(x_+^2 + u_+^2), 
    
    B_n    = e^-b^2/2(1+o(1)) S(z+w̅),

where G(z,w) = e^zw̅ -|z|^2/2-|w|^2/2 and

    S(z) = 1/√(2π)∫_-∞^0
    e^-(ξ-z)^2/21/Φ_(ξ)  dξ,
        Φ_(ξ) = φ(ξ) +  1/√() e^1-/2ξ^2(1-φ(ξ/√())).



We first analyze that case ζ = η = 1 + x/√(n) is real.
We find

    λ =
     
     n + 2x√(n) + x^2      if x≤0, 
    
     n + 2x√(n) + (2-1)x^2 + o(1)    otherwise,

and

    A_n =
    
    1      if x≤0, 
    
     e^2(1-)x^2(1+o(1))    otherwise.

Using α_n =  -2x + o(1), λ/n = 1 + o(1), and Stirling's formula, we obtain the asymptotic identity for Γ(λ+√(λ)s,n,):

    Γ(λ+√(λ)s,n,) =  1/√() e^1-/2 (s+2x)^2+ o(1).

By means of the central limit theorem and the above asymptotic identities, we now approximate the factor B_n as follows,

    B_n   = 1/√(2π)∫_-∞^-2xe^-s^2/2  ds/φ(s+2x) +  1/√() e^1-/2 (s+2x)^2(1-φ((s+2x)/√()))  + o(1).

Now we consider A_n and B_n for 
    ζ=1+z/√(n),    η=1+w/√(n),     z,w∈

and recall that

    λ = n + √(n)(z+w) + |z|^2+|w|^2/2
    - (1-)( x_+^2  + u_+^2) + o(1).

We use the same argument as in the previous case and deduce (<ref>) and (<ref>).


By the above lemma, we approximate the kernel K_n(z,w) by

    K_n(z,w) =  G(z,w) e^(1-)(x_+^2 + u_+^2)S(z+w̅)(1+o(1))

up to cocycles and the proof is complete.





§ STRUCTURE OF LIMITING CORRELATION KERNELS
 

In this section we prove Theorem <ref>. Our proof closely follows the derivation of similar results in <cit.>; we shall
here indicate the necessary modifications.

We start by proving some generally useful estimates for functions of the form

    w(ζ)=p(ζ)e^-n Q^()(ζ)/2,

where p is a holomorphic function (in practice, a polynomial or a “quasi-polynomial” - see below). 

 It is convenient to introduce the numbers

    M=M_n,:= Q^()_L^∞(S_n),   (S_n:={ζ; (ζ,S)≤ 1/√(n)}).
 The
numbers M_n, clearly do not increase
as n increases; in the sequel we may fix M to be M_n_0, with a sufficiently large n_0.



  Suppose that ζ∈ S and n is large enough and suppose that p is holomorphic in the disc D(ζ;1/√(n)). Then there is a constant C=C_ depending only on M such that 
    |w(ζ)|^2≤ Cn∫_D(ζ;1/√(n))|w|^2  dA.



 This follows from a standard argument, using that z ↦ |w(ζ+z/√(n))|^2 e^M|z|^2 is
logarithmically subharmonic in D(0;1).
(See for instance <cit.>.)





  Let K be a compact subset of the interior of S and let w=qe^-nQ^()/2 where q is holomorphic
on ∖ K and satisfying q(ζ)=O(ζ^j) as ζ→∞. Write τ=j/n and suppose that τ is in the range τ≤ 1. Then there is a constant C_>0 such that

    |w(ζ)|≤ C_√(n)we^-n(Q^()-_τ)(ζ)/2,     ((ζ,K)≥ 1/√(n)).





By Lemma <ref>
there is a constant C such that |w(ζ)|≤√(C n)w when ζ∈ S_τ.

Consider the subharmonic function

    u_n(ζ)=1/nlog(|q(ζ)|^2/C nw^2).

Note that u_n(ζ)≤τlog|ζ|^2+O(1) as ζ→∞ and u_n≤ Q on S_τ, and u_n is subharmonic on . Hence u_n≤_τ on , proving the lemma
(with C_:=√(C)).



 Following <cit.>, we call q above a quasipolynomial of degree j; w is a weighted quasipolynomial.




  The 1-point function _n in external potential Q^() satisfies

    _n(ζ)≤ C_ ne^-n(Q-)(ζ),     (ζ∈).



 Let _n be the reproducing kernel of the space of holomorphic polynomials of degree at most n-1 with the norm of
L^2(e^-nQ^()). Now fix ζ∈ S and put q(η)=_n(η,ζ)/√(_n(ζ,ζ)) and w=qe^-nQ^()/2. Then w=1, and so by Lemma <ref>
(with τ=1) we have

    _n(ζ)=|w(ζ)|^2≤ C_ ne^-n(Q^()-)(ζ)=C_ ne^-n(Q-)(ζ).




Fix a neighborhood Ω of S where Q is real-analytic and strictly subharmonic. Let A be a Hermitian-analytic function defined in a neighborhood in ^2 of the set {(ζ,ζ):ζ∈Ω} such that A(ζ,ζ)=Q(ζ).
As in <cit.>, we fix an outer boundary point 0∈Γ=S such that the outwards normal to Γ at 0 points in the positive real direction.

We define a kernel by

    𝐤_n^#(ζ,η)=n(_1_2A)(ζ,η)  e^nA(ζ,η).

We also introduce rescaled kernels k_n^# and K_n^# by

    k_n^#(z,w)=1/n Q(0)𝐤_n^#(ζ,η);   K_n^#(z,w)=k_n^#(z,w)e^-n(Q^()(ζ)+Q^()(η))/2

where ζ = z/√(n Q(0)) and
η = w/√(n Q(0)).

 For z in a given compact subset of , we have as n→∞

    (Q-Q̌)(ζ) = 2n^-1( z)_+^2(1+O(n^-1/2)).

Moreover, if R_n(z)=1/n Q(0)_n(ζ) denotes the rescaled 1-point function, then

    R_n(z)≤ C_ e^-2( z)_+^2.



 Let V denote the harmonic continuation of Q̌|_S^c across the outer boundary curve Γ=S to a neighborhood N of 0.
Let _ and _ denote the exterior normal and tangential derivatives on Γ∩ N, respectively.
Since Q=V and ∇ Q = ∇ V on N∩Γ, we have

    _(Q-V)= _(Q-V) = _^2(Q-V) =0.

This implies _^2(Q-V)=4(Q-V) = 4 Q on Γ, proving (<ref>).
To prove (<ref>) it now suffices to appeal to Lemma <ref>.




Define a function Ψ_n by

    Ψ_n(z,w) = K_n(z,w)/K_n^#(z,w).

Then Ψ_n is Hermitian-analytic and the rescaled kernel K_n can be expressed as

    K_n(z,w)=k_n^#(z,w)Ψ_n(z,w) e^-n(Q^()(ζ)+Q^()(η))/2.

Using the argument in <cit.>, we see that there exist cocycles c_n such that

    c_n(z,w)  k_n^#(z,w) e^-n(Q(ζ)+Q(η))/2=G(z,w)(1+o(1))

where o(1)→ 0 locally uniformly on ^2 as n→∞. Indeed, (<ref>) is obtained by the Taylor expansion (assuming =1 for simplicity)

    n(A(ζ,η)-Q(ζ)/2-Q(η)/2) = n(A(ζ,η)-A(η,η))/2 - n(A(ζ,ζ)-A(η,ζ))/2
       =i[√(n) _1A(0,0)/√( Q(0))(z-w)+_1^2A(0,0)/2 Q(0)(z^2-w^2)]+ zw̅ -|z|^2/2 -|w|^2/2.

Since Q^()(ζ) = Q̌(ζ) + (Q-Q̌)(ζ),
we obtain the convergence

    c_n(z,w)  K_n^#(z,w)=G(z,w)e^(1-)(( z)_+^2+( w)_+^2)(1+o(1))

by (<ref>) and (<ref>). Here, o(1) → 0 in L^1_(^2) and locally uniformly on (∖ i)^2.

On the other hand, observe that

    |Ψ_n(z,w)|^2 = |K_n(z,w)/K_n^#(z,w)|^2 ≤R_n(z)R_n(w)/|c_n(z,w)K_n^#(z,w)|^2.

Combining with (<ref>),
we obtain that the family {Ψ_n} is locally bounded on ^2. By the normal family argument in <cit.>, for each subsequence of {Ψ_n} there exists a further subsequence which converges locally uniformly on ^2 and every limit Ψ = lim_k Ψ_n_k is Hermitian-entire. This proves part <ref> and <ref> of Theorem <ref>.


 For the remaining part, we observe that the modified Ward equation (<ref>) follows by applying the rescaling procedure in <cit.> to the potential Q^() in a straightforward way. Moreover, to prove non-triviality using the method in <cit.> we need only to prove that R(z)>0 at some point z. For z somewhat inside the (rescaled) droplet
this can be done exactly as in <cit.>, since Q=Q^() in S.










§ AN APPROACH TO UNIVERSALITY USING WARD'S EQUATION

In this section, we shall analyze solutions
R(z) to the (modified) Ward equation (<ref>) which satisfy the “physically reasonable” condition of translation invariance. This leads to an approach to
universality of the 1-point function R^() based only on Ward's equation.

Given the nature of our rescaling (with the rescaled droplet occupying the left half plane) it is natural to assume that each limiting
1-point function R in Theorem <ref> should be vertical translation invariant, in the sense that R(z)=R(z+it) for each t∈.
This is equivalent to that the holomorphic kernel Ψ should satisfy Ψ(z,w)=S(z+w̅) for some entire function S. (In view of the computations in the preceding section, we know that
this holds for the Ginibre ensemble, in which case S=_.)

In the following, we shall work within the class of abstract translation invariant “limiting kernels”
where
S has the “error function-type” (cf. <cit.>)

    S(z) = 1/√(2π)∫_-∞^+∞e^-(ξ-z)^2/2s(ξ)  dξ

for some Borel measurable, non-negative function s on .


To each such function we define a corresponding function B=B_S
(the “Berezin kernel”) by

    B(z,w) = e^-|z-w|^2 e^2(1-)( w)_+^2S(z+w̅)S(z̅ + w)/S(z+z̅).

We will describe all the functions s such that the Berezin kernel B satisfies two conditions called the mass-one condition and Ward's equation.

We say that B satisfies the mass-one equation if for all z

    ∫_ℂ B(z,w)  dA(w) = 1.


In the case of a regular point on the boundary, Ward's equation has the form

    C= R -1 - log R + (1-)_ z>0,

where
R(z) = B(z,z) and

    C(z)=∫_ℂB(z,w)/z-w dA(w).


Throughout this section, we write z=x+iy and w=u+iv with x,y,u,v∈.

The mass-one condition (<ref>) holds if and only if

    s(ξ) = 1_E/Φ_(ξ)

almost everywhere for some Borel subset E⊂, where Φ_:→ (0,∞) is the function defined in (<ref>).



A straightforward computation shows that

    √(2/π)∫_-∞^∞  e^-2u^2 + 2ξ u+2(1-)u_+^2 du = e^1/2ξ^2φ(ξ)+1/√() e^1/2ξ^2(1-φ(ξ/√())).

It follows immediately that

    ∫_ℂ   e^-|z-w|^2 e^2(1-)u_+^2 S(z+w̅) S(z̅ + w)  dA(w) 
       = 1/π∫_-∞^∞∫_-∞^∞e^-2x^2-2u^2-ξ^2 + 2ξ(x+u)+2(1-)u_+^2s(ξ)^2 du dξ
       = 1/√(2π)∫_-∞^∞e^-1/2(ξ-2x)^2s(ξ)^2(φ(ξ) +  1/√() e^1-/2ξ^2(1-φ(ξ/√()))) dξ.

Here, the expression in the large parenthesis is recognized as the function Φ_ defined in (<ref>).

Equating the last expression with S(2x), we see that the mass-one equation is equivalent to that (for a.e.-ξ)

    s(ξ) = s(ξ)^2 Φ_(ξ).

The last condition means that (up to sets of measure 0) s(ξ)=1/Φ_(ξ) whenever s(ξ) 0.


We will now analyze Ward's equation (<ref>). We will require a few initial remarks about the Cauchy transform (<ref>).

By a change of variables, the Cauchy transform C takes the form

    C(z) = - ∫_-∞^∞∫_-∞^∞e^-u^2-v^2+2(1-)(u+x)_+^2/u+ivS(2x+u-iv) S(2x+u+iv)/S(2x) dv du/π.

Let

    L(x) = ∫_-∞^∞∫_-∞^∞e^-u^2-v^2+2(1-)(u+x/2)_+^2/u+iv S(x+u-iv) S(x+u+iv)  dv du/π.

Then

    C(z) = -L(2x)/S(2x),     C(z) = - (L/S)'(2x).

Now we describe all solutions to Ward's equation (<ref>).

Ward's equation holds if and only if

    s(ξ) = _I/Φ_(ξ)

almost everywhere, where Φ_ is the function (<ref>) and
I is an interval in . Furthermore, if R satisfies 
    lim inf_x→ -∞ R(x) > 0,

then I = (-∞, c) for some constant c.



Let

    ℓ(x;ξ,η)
    =∫_-∞^∞∫_-∞^∞  l(x;ξ,η;u,v) e^-iv(ξ-η)/u+iv dv du/π,

where

    l(x;ξ,η;u,v)=e^-2u^2 -1/2(ξ-x)^2-1/2(η-x)^2+u(ξ+η-2x)+2(1-)(u+x/2)_+^2

so that

    L(x) = ∫_-∞^∞∫_-∞^∞ℓ(x;ξ,η)s(ξ)s(η) dξ dη/2π.

First, observe that

    ∫_-∞^∞e^-iv(ξ-η)/u+iv dv/π=
    
    -2  e^u·(ξ-η)1_ξ>η,     if  u < 0;
    -2 e^u·(ξ-η)1_ξ<η,    if  u > 0.

Let

    ℓ_1(x;ξ,η)=- _ξ>η∫_-∞^∞ 2 e^-2u^2 -1/2(ξ-x)^2-1/2(η-x)^2+2u(ξ-x)+2(1-)(u+x/2)_+^2 du

and

    L_1(x) = ∫_-∞^∞∫_-∞^∞ℓ_1(x;ξ,η)s(ξ)s(η) dξ dη/2π.

By direct computation, we have

    ℓ_1(x;ξ,η) = -e^-1/2(η-x)^2√(2π)_ξ>ηΦ_(ξ)

and

    L_1(x) =  -∫_-∞^∞∫_-∞^∞_ξ>ηe^-1/2(η-x)^2s(ξ)s(η)Φ_(ξ)  dξ dη/√(2π).

Next, let  L_2(x)=L(x)-L_1(x).
Then by (<ref>), we have 
    L_2(x)=∫_-∞^∞∫_-∞^∞ℓ_2(x;ξ,η)s(ξ)s(η) dξ dη/2π,
 where

    ℓ_2(x;ξ,η)
    =  2∫_0^∞  e^-2u^2 -1/2(ξ-x)^2-1/2(η-x)^2+2u(ξ-x)+2(1-)(u+x/2)_+^2 du.

This implies that

    L_2(x)
    = S(x) ∫_x^∞ S(v)e^1/2(1-)v_+^2 dv.

From the relation

    R(z) = e^2(1-)x_+^2 S(z+z̅),

we have

    log R = (1-)1_x>0 + (S'/S)'(2x).

Thus Ward's equation holds if and only if

    (L/S)'(2x) =   (S'/S)'(2x) +1 - R(z).

From the equation (<ref>), L_2 satisfies

    (L_2/S)'(2x) =  - R(z),
 and hence
Ward's equation holds if and only if

    (L_1/S)' =   (S'/S)' +1.

Equivalently,

    L_1(x) = S'(x) +(x - c)S(x)

for some constant c.
We find

    L_1(x) =  ∫_-∞^∞ (η - c) e^-1/2(η-x)^2s(η) dη/√(2π).

Comparing it to (<ref>), we have 

    η - c  = -∫_η^∞ s(ξ)Φ_(ξ) dξ

for almost everywhere in the set I={η∈ : s(η)  0}.  This implies that s ·Φ_=_I almost everywhere and I is connected. 
For the last statement, assume on the contrary that I is left-bounded. Then it is a direct consequence that R(x)→ 0 as x→ -∞. This implies that if R satisfies the condition (<ref>), then I=(-∞, c) where  
c is determined by 
    η - c  = -∫_η^∞_I.

Our proof of Theorem <ref> is complete.


 
Theorem <ref> shows that for an “abstract” translation invariant kernel, Ward's equation only determines it up to a real constant c. It is worth pointing out that each limiting 1-point function R in Theorem <ref> satisfies the condition (<ref>). More precisely, one can prove that R(x)→ 1 as x→ -∞ from the argument presented in <cit.>.

In the free boundary case when the confinement parameter =1, we know that I=(-∞, 0). Indeed, this is shown in <cit.> depending on the interior estimate of R and the boundary fluctuation theorem from <cit.>. 
Moreover, we have seen above that
I=(-∞, 0) for the Ginibre ensemble (with general ) and we believe that this should be a universal fact. In the
following sections, we will prove the statement for concrete scaling limits in (for example) radially symmetric cases, by applying the method of quasipolynomials.




§ BOUNDARY UNIVERSALITY: SOME PRELIMINARIES

In this section, we collect some preliminary observations which will come in handy in several occasions, when we prove universality of scaling limits
and maximum moduli (Theorem <ref> and Theorem <ref>).
We will also give some background about Laplacian growth.




 §.§ Outline of strategy




 Let p_j,n be the j:th orthonormal polynomial with respect to the weight e^-nQ^(), and write w_j,n=p_j,ne^-nQ^()/2. We start with the basic identity for the 1-point function

    _n(ζ)=_n(ζ,ζ)=∑_j=0^n-1|w_j,n(ζ)|^2.

As a preliminary step, we shall prove that if ζ is very close to the outer boundary
Γ=S, then all terms but the last √(n)log n ones can be neglected. More precisely, if ζ belongs to a belt

    N_Γ={ζ; (ζ,Γ)≤ C/√(n)},

then

    _n(ζ)∼∑_j=n-√(n)log n^n-1|w_j,n(ζ)|^2.


The idea is to prove an asymptotic formula for the remaining w_j,n's, so that universality will follow after a simple-minded summation. (See Figure <ref> and Figure <ref>.)















 §.§ Preliminaries on Laplacian growth




In this subsection we recast some results on Laplacian growth; this is convenient, if not otherwise, to introduce some notation. References and further reading can be found, for instance in <cit.> and in <cit.>.








For a parameter τ with 0<τ≤ 1, we let
_τ be the obstacle function defined by the obstacle Q, subject to the growth condition

    _τ(ζ)=2τlog|ζ|+O(1),  as  ζ→∞.

The precise definition runs as follows: for each η∈, _τ(η) is the supremum of
s(η) where s is a subharmonic function which is everywhere ≤  Q and satisfies s(ζ)≤ 2τlog|ζ|+O(1) as ζ→∞.


Write S_τ for the droplet in external potential Q/τ and note that
σ(S_τ)=τ
where σ is the equilibrium measure (<ref>).
Under our conditions,
S_τ equals to closure of the interior of the coincidence set {Q=_τ} and the measure σ_τ:=_S_τ·σ minimizes the weighted energy (<ref>) amongst positive measures of total mass τ.

Clearly the droplets S_τ increase with τ. The evolution of the S_τ's is known as Laplacian growth.
We will write Γ_τ for the outer boundary,

    Γ_τ=S_τ.

Hence Γ_τ=Ụ_τ where

    U_τ=∖ S_τ.

Finally, we denote by

    ϕ_τ:U_τ→_e

the unique conformal (surjective) map, normalized so that ϕ_τ(∞)=∞ and ϕ_τ'(∞)>0. (_e is the exterior disk {|ζ|>1}∪{∞}.)

It is well known that _τ is C^1,1-smooth on  and harmonic on U_τ. Moreover, since Γ=Γ_1 is everywhere regular, it follows from standard facts about Laplacian growth that Γ_τ is everywhere regular for all τ in some interval τ_0≤τ≤ 1, where τ_0<1. Below we fix, once and for all, such a τ_0.




 (“Richardson's lemma”). If 0<τ<τ'≤ 1 and if h is harmonic in U_τ and smooth up to the boundary, then

    ∫_S_τ'∖ S_τh Q  dA=(τ'-τ)h(∞).



 Since σ(S_τ)=τ,
the asserted identity is true when h is a constant. Hence we can assume that h(∞)=0. Let us fix such an h and extend it to  in a smooth way.
It follows from the properties of the obstacle function _τ that

    ∫_S_τh Q  dA=∫_ h_τ  dA=∫_ h·_S_τQ  dA.

Subtracting the corresponding identity with τ replaced by τ' we find
∫_S_τ'∖ S_τh Q  dA=0,
finishing the proof of the lemma.
 

The lemma says that if dσ_τ= Q·_S_τ  dA is equilibrium measure of mass τ, then near the outer boundary component Γ_τ, we have, in a suitable “weak” sense that
dσ_τ/dτ=ω_τ
where ω_τ is the harmonic measure of U_τ evaluated at ∞. This means: if h is a continuous function on Ṣ_τ, then ω_τ(h)=h̃(∞), where h̃ is the harmonic extension to U_τ of h.


Let us define the Green's function of U_τ with pole at infinity by
G(ζ,∞)=log|ϕ_τ(ζ)|^2.
Then by Green's identity,

    h̃(∞)=∫_U_τh̃(ζ)  G (ζ,∞)  dA(ζ)=-1/2∫_Ṣ_τ hG̣/ṇ  ds_τ,

where ds_τ stands for the arclength measure on Γ_τ divided by 2π.



Now for values of τ such that Γ_τ is everywhere regular
- ̣/ṇ G(ζ,∞)=2|ϕ'_τ(ζ)| when ζ∈Γ_τ
so we conclude that
d/dτ(_S_τ Q  dA)=|ϕ'_τ|ds_τ,
meaning that the outer boundary Γ_τ moves in the direction normal to Γ_τ, at local speed |ϕ'_τ|/2 Q. (The factor 2 comes about because of
the different normalizations of dA and ds_τ.)

The above dynamic of Γ_τ was of course deduced in a “weak” sense, but using the regularity of the curves involved, one can turn it into a pointwise
estimate. More precisely, one has the following result, which is essentially <cit.>.

In the following, we denote by _τ the outwards unit normal on the curve Γ_τ.


  Let ζ be a point of Γ=Γ_1 and fix
τ with τ_0≤τ≤ 1. Let ζ_τ be the point in Γ_τ∩(ζ+_1(ζ)) which is closest
to ζ. Then we have

    ζ_τ=ζ + (τ-1)_τ(ζ_τ)|ϕ_1'(ζ)|/2 Q(ζ)+O((τ-1)^2),  τ→ 1

and 
    _τ(ζ_τ) = _1(ζ)+O(τ-1),   τ→ 1.


In particular there are constants c_1,c_2>0 such that for all ζ∈Γ and all τ_0≤τ≤ 1,

    c_1(1-τ)≤(Γ_τ,ζ)≤ c_2(1-τ).




For τ_0<τ≤ 1, we will denote by V_τ the harmonic continuation of the harmonic function _τ on U_τ across the analytic curve Γ_τ. Considering the growth as ζ→∞, we obtain the basic identity

    V_τ(ζ)=_τ+τlog|ϕ_τ(ζ)|^2,

where _τ is the holomorphic function on U_τ with _τ=Q on Γ_τ and _τ(∞)=0.

The function Q-V_τ, considered in a neighborhood of the curve Γ_τ, plays a central role for the theory.



Fix a number τ∈ [τ_0,1] and a point p_τ∈Γ_τ. Then we have for v∈

    (Q-V_τ)(p_τ + v) = 2 Q(p_τ) · v_^2 + O(|v|^3),   |v|→ 0,

where v_=v·=(v) is the component of v in the normal direction =_τ(p_τ).



Let _ and _ denote the exterior normal and tangential derivatives on Γ_τ, respectively.
Since Q=V_τ and ∇ Q = ∇ V_τ on Γ_τ, we have for all ζ∈Γ_τ

    _(Q-V_τ)= _(Q-V_τ) = _^2(Q-V_τ) =0.

This implies _^2(Q-V_τ)=4(Q-V_τ) = 4 Q on Γ_τ, finishing the proof of the lemma.







In particular, it follows from Lemma <ref> that
there is a number c>0 such that, if ζ∈ N_Γ,

    (Q-_τ)(ζ)=(Q-V_τ)(ζ)≥ c(ζ,S_τ)^2.














 §.§ Discarding lower order terms

Let N_Γ be the belt (<ref>) and recall that Q=Q^() in N_Γ. Below we fix an arbitrary point
ζ∈ N_Γ.
Also fix a number τ_0<1 such that the curves Γ_τ are regular for all τ with τ_0≤τ≤ 1. Now write

    τ=τ(j)=j/n,     (j≤ n-1).

It follows from Lemma <ref> that there is a number C=C_ such that

    τ≤τ_0  ⇒   |w_j,n(ζ)|^2≤
    Cne^-n(Q^()-_τ_0)(ζ)≤ Cne^-cn,

where c=inf_N_Γ{Q^()-_τ_0}>0.
Hence 
    _n(ζ)∼∑_j=τ_0 n^n-1|w_j,n(ζ)|^2.


Next fix j such that τ_0 n≤ j≤ n-√(n)log n, i.e.,
τ_0≤τ≤ 1-δ_n where we put throughout

    δ_n=log n/√(n).








By Lemma <ref>, we have (Γ_τ,Γ)≥ c(1-τ)≥ cδ_n for some constant c>0.
By (<ref>),
n(Q^()-_τ)(ζ)≥ c_1log^2 n where c_1>0, so

    |w_j,n(ζ)|^2≤ Cne^-n(Q^()-_τ)(ζ)≤ Cne^-c_1log^2 n,     (j≤ n-√(n)log n).

We have shown that

    _n(ζ)∼∑_j=n-√(n)log n^n-1|w_j,n(ζ)|^2,    (n→∞)

in the sense that the difference of the left and right sides converges uniformly to zero on N_Γ, as n→∞.






§ APPROXIMATION BY QUASIPOLYNOMIALS
 

In this section, we discuss an asymptotic formula for weighted quasipolynomials w_j,n when j is in the range
n-√(n)log n≤ j≤ n-1. This formula will be used repeatedly, in our proofs of Theorem <ref> and Theorem <ref>.


The asymptotic formula leads to functions behaving “essentially” like
weighted orthogonal polynomials, provided that
certain conditions are satisfied
(e.g., the potential is radially symmetric). However, in the following discussion, we make no assumptions concerning
radial symmetry, except where this is explicitly stated (namely, it is used in the proof of property (P2) below).

Throughout the section, we assume that n is large enough, and we write τ=τ(j)=j/n and δ_n=n^-1/2log n.
Thus 1-δ_n≤τ< 1.



 §.§ Basic definitions

When n is large, Γ_τ is a real-analytic Jordan curve and thus
the map ϕ_τ:U_τ→_e can be continued analytically across Γ_τ to ∖ K where K is a suitable compact set chosen so that ϕ_τ maps K biholomorphically onto _e(ρ_0-δ) for some ρ_0<1 and δ>0. (_e(ρ) denotes the set {|ζ|>ρ}∪{∞}.)




Let _τ be the bounded holomorphic function defined on a neighborhood of U_τ with _τ = Q on Γ_τ and _τ(∞) = 0. (See (<ref>).)

For each j with n-nδ_n≤ j≤ n-1, we write 
    ξ_j,n(ζ) = j-n/√(n)·|ϕ_τ'(ζ)|/√( Q(ζ)),     (ζ∈∖ K).

It is convenient to introduce the notation

    Φ_j,n(ζ):= Φ_(ξ_j,n(ζ)),
 where Φ_ is the function defined in (<ref>).

Finally, we define the approximate quasipolynomial F_j,n=F_j,n, of degree j associated with the external field nQ^() by

    F_j,n = (n/2π)^1/4√(ϕ_τ') ϕ_τ^j e^n_τ/2 e^_j,n/2,

where _j,n is the bounded holomorphic function defined on ∖ K such that

    _j,n = log√( Q) - logΦ_j,n    on  Ṣ_τ

and _j,n(∞)=0.


We shall show that F_j,n satisfies the following properties in Sections <ref> and <ref>. Fix a smooth function χ_0 on  such that χ_0=0 on K and
χ_0=1 on O:=ϕ_τ^-1(_e(ρ_0)).


  (P1) F_j,n is approximately normalized:

    ∫_χ_0^2  |F_j,n|^2   e^-nQ^()  dA = 1+O(n^-1/2log^2 n).



  (P2) F_j,n satisfies the following “approximate orthogonality”: for any polynomial p with p < j,

    ∫_χ_0  F_j,n p̅  e^-nQ^() dA = O(n^-1/2log^2 n p_nQ^()).



Following <cit.> we now define a “positioning operator” Λ_j,n by

    Λ_j,n[f] = ϕ_τ'·ϕ_τ^j· e^n_τ/2· f∘ϕ_τ.

Then the map Λ_j,n is an isometric isomorphism from L^2_nR_,τ(_e(ρ_0)) to L^2_nQ^()(O) where 
    R_,τ := (Q^()  - V_τ)∘ϕ_τ^-1,

and it furthermore preserves holomorphicity. (See <cit.>.)

The isometry property means that for all f, g∈ L^2_nR_,τ(_e(ρ_0))

    ∫_OΛ_j,n[f] Λ_j,n[g]  e^-nQ^() dA = ∫__e(ρ_0)f g̅ e^-nR_,τ dA.

We now define a function f_j,n on _e(ρ_0) by the formula

    n^1/4Λ_j,n[f_j,n]=F_j,n.

This gives 
    f_j,n = (2π)^-1/4((ϕ_τ')^-1/2· e^_j,n/2)∘ϕ_τ^-1.





 §.§ Approximate normalization

In this subsection we prove the property (P1) of the approximate quasipolynomials. Recall that U_τ=∖ S_τ.


We first note that, for points ζ close to the unit circle ,

    R_,τ(ζ)=
    
    (Q-V_τ -(1-)(Q-V_1))∘ϕ_τ^-1(ζ)
       if    ζ∈ϕ_τ(U_1),  
    
    (Q-V_τ)∘ϕ_τ^-1(ζ)
       otherwise.



For each τ∈ [1-δ_n,1], the curve ϕ_τ(Ṣ) is an analytic Jordan curve, a slight perturbation of the unit circle .
It is convenient to represent this curve according to the polar parameterization

    ϕ_τ( S) = {r_τ(η)·η : η∈},

where r_τ is a smooth function →_+. (It is easy to see that such a parameterization is possible when τ is sufficiently close to 1.)

For a fixed η∈ we now consider the local speed ṙ_τ(η)=d r_τ(η)/dτ. Since the conformal map ϕ_τ locally magnifies distances by a
factor |ϕ_τ'| and since the curve Γ_τ moves towards Γ_1 with local speed |ϕ'_τ|/(2 Q) (see discussion before Lemma <ref>), we find that

    ṙ_τ(η)=(|ϕ_τ'|^2/2 Q)∘ϕ_τ^-1(η)+O(1-τ).

We have shown that

    r_τ(η)=1+(1-τ)·(|ϕ_τ'|^2/2 Q)∘ϕ_τ^-1(η)+O((1-τ)^2),  τ→ 1.


In the following we will write 
    R_τ:=R_1,τ.

It is useful to note that (since V_τ is harmonic)

    R_τ=( Q/|ϕ_τ'|^2)∘ϕ_τ^-1.





We shall now analyze the behavior of the function R_,τ close to .


Fix a point η∈. If r<r_τ(η),

    R_,τ(rη) = 2 R_τ(η)(r-1)^2 + O((r-1)^3),     r→ 1.

If r>r_τ(η),

    R_,τ(rη)=2 R_τ(η)(r-1)^2 -2(1-) R_τ(η)(r-r_τ(η))^2 + O((r-1)^3),     r→ 1.




If r<r_τ (η), then R_,τ=(Q-V_τ)∘ϕ_τ^-1=R_τ. From the argument in Lemma <ref>, we obtain the following Taylor series expansion of R_,τ about η:

    R_,τ(rη) =2 R_τ(η)(r-1)^2 + O((r-1)^3),     r→ 1.

Now assume that r>r_τ(η).
By Lemma <ref>, we have

    _1(r_τ(η)η) = η + O(τ-1),  τ→ 1,

where _1(ζ) denotes the outer unit normal to ϕ_τ( S_1) at ζ∈ϕ_τ( S_1).
Hence the second-order Taylor expansion of (Q-V_1)∘ϕ_τ^-1 about the point r_τ (η)η∈ϕ_τ(Ṣ_1) takes the form

    ((Q-V_1)∘ϕ^-1_τ)(r_τ(η)η+ξ) = 2 R_τ (r_τ(η)η) (ξ·_1(r_τ(η)η))^2 + O(|ξ|^3),   |ξ|→ 0.


Now note that ((Q-V_1)∘ϕ_τ^-1)= |(ϕ_τ^-1)'|^2( Q)∘ϕ_τ^-1 =  R_τ since V_1 is harmonic. Also note that if r → 1 with r>r_τ(η), then r_τ(η)- 1 = O(1-r) and 1-τ = O(1-r) by (<ref>).
Now combining (<ref>) and (<ref>), we obtain

    ((Q-V_1)∘ϕ^-1_τ)(rη)    = ((Q-V_1)∘ϕ^-1_τ)(r_τ(η)η + (r-r_τ(η))η)
       =2 R_τ (η)· (r_τ(η) - r)^2 + O((r-1)^3),   r→ 1.

Hence the proof is complete.


Let D_j,n={ζ∈: 1-Mδ_n ≤ |ζ| ≤ 1+Mδ_n} for a large number M. We have the following lemma which will be used to show that our quasipolynomials
are approximately normalized.

For any integer j with τ = j/n ∈ [1-δ_n,1), we have

    ∫_D_j,n n^1/2 |f_j,n(ζ)|^2  e^-nR_,τ(ζ) dA(ζ)=1+O(n^-1/2log ^2n).




For each η∈, we compute the integral

    ∫_1-Mδ_n^1+Mδ_n|f_j,n(rη)|^2 e^-nR_,τ(rη) rdr.

Fix η∈ and consider the function R_,τ,η(r):=R_,τ(rη).
For r<r_τ(η), using the expansion of R_,τ in Lemma <ref>, we have

    ∫_1-Mδ_n^r_τ(η)|f_j,n(rη)|^2 e^-n R_,τ,η(r) rdr
       = ∫_1-Mδ_n^r_τ(η) r|f_j,n(rη)|^2 e^-2n  R_τ(η)(r-1)^2 + O(nδ_n^3) dr.

Note that for all r with |r-1|<Mδ_n and all j with j/n∈ [1-δ_n,1)

    |f_j,n(rη)|^2 = |f_j,n(η)|^2 + O(δ_n  log n).

Recall from the expansion (<ref>) that

    r_τ(η)-1 = (1-τ)· (2 R_τ(η))^-1 + O((1-τ)^2).

Thus the above integral is approximated by

    ∫_1-Mδ_n^r_τ(η)|f_j,n(rη)|^2 e^-n R_,τ,η(r) rdr
    =|f_j,n(η)|^2/√(4n R_τ(η))∫_-∞^-ξ_τ(η) e^-1/2x^2 dx + E_1,

where E_1 is an error term with E_1 = O(n^-1log^2 n) and

    ξ_τ (η)= √(n)(τ-1)/√( R_τ(η))=j-n/√(n)(|ϕ_τ'|/√( Q))∘ϕ_τ^-1(η).

On the other hand, using Lemma <ref> again,  we obtain for r>r_τ(η)

    R_,τ,η(r)= 2 R_τ(η)(r-r^*_)^2 +
    2(-1)/ R_τ(η)(r_τ(η)-1)^2
    +  O((r-1)^3),

where r^*_=(1-(1-)r_τ(η))/. It follows from the expansion (<ref>) that

    R_,τ,η(r) = 2 R_τ(η)(r-r^*_)^2 + -1/2·(1-τ)^2/ R_τ(η) +  O((r-1)^3)

and r^*_ has the asymptotic expansion of the form

    r^*_ = 1- 1-/·1-τ/2 R_τ(η)+O((1-τ)^2).

Estimating the integral by Laplace's method, we thus obtain

    ∫_r_τ(η)^1+Mδ_n |f_j,n(rη)|^2
       e^-n R_,τ,η(r) rdr
    = |f_j,n(η)|^2/√(4n R_τ(η)) e^1-/2ξ_τ(η)^2∫_-ξ_τ(η)/√()^∞ e^-x^2/2dx + E_2

where E_2 is a new error term with E_2 = O(n^-1log^2 n). Combining (<ref>) and (<ref>), we obtain

    n^1/2∫_D_j,n |f_j,n|^2  e^-nR_,τ dA = √(2π)∫_|f_j,n(η)|^2/√( R_τ(η))Φ_j,n(ϕ_τ^-1(η)) ds(η)+E_3 = 1+ E_3

where E_3 =O(n^-1/2log^2 n) and ds(η) = |dη|/2π is the normalized arclength measure.

We now conclude our proof of the approximate normalization property (P1) of our quasipolynomials.

 
For any integer j with τ = j/n∈ [1-δ_n,1), we have

    ∫_χ_0^2|F_j,n|^2 e^-nQ^() dA = 1+O(n^-1/2log^2 n).



By Lemma <ref> we obtain

    ∫_χ_0^2|F_j,n|^2 e^-nQ^() dA = n^1/2∫_∖ D_j,n (χ_0 ∘ϕ^-1_τ)^2|f_j,n|^2 e^-nR_,τ dA + 1+O(n^-1/2log ^2 n).

In order to estimate the first term on the right, we observe that the functions f_j,n are uniformly bounded in _e(ρ_0). By Lemma <ref>, there is an ϵ>0 such that for all η with Mδ_n ≤(η,) ≤ϵ,

    R_,τ(η) ≥ c  δ_n^2

for some constant c. Also, by the growth rates of Q and V_τ near infinity, i.e., 
    lim inf_ζ→∞ Q(ζ)/log |ζ|^2>1,  lim inf_ζ→∞V_τ(ζ)/log |ζ|^2=τ,

there exists a constant C>0 such that

    |η|≥ 1+ϵ    ⇒     R_,τ(η) ≥ C log |η|.

Summing up, we obtain

    n^1/2∫_∖ D_j,n (χ_0 ∘ϕ^-1_τ)^2|f_j,n|^2 e^-nR_,τ dA ≤ O(n^1/2e^-c'log^2 n)
 for some constant c', which completes the proof.





 §.§ Approximate orthogonality
 Up to this point we have not assumed that Q be radially symmetric.
However, in order to prove the following lemma, we shall use this assumption.




Suppose that Q is radially symmetric. Fix an integer j with τ = j/n ∈ [1-δ_n,1).  Let p be a holomorphic polynomial of degree ℓ less than j.
Then 
    ∫_χ_0  p F_j,n e^-nQ^() dA = O(n^1/4e^-c(log n)^2p_nQ^()).




Write q = Λ_j,n^-1[p]. Then q is holomorphic on _e(ρ_0) and satisfies q(η)= O(|η|^ℓ-j) as η→∞.
We first show that

    ∫_D_j,nq  f_j,n  e^-nR_,τdA = 0.

Define a function h by h = q/f_j,n. Then h is holomorphic in _e(ρ_0) and vanishes at infinity since f_j,n does not vanish at infinity. Now we write the above integral as

    ∫_1-Mδ_n^1+Mδ_n∫_ 2h  |f_j,n|^2 e^-nR_,τ r ds dr.

Since |f_j,n|^2 e^-nR_,τ is radially symmetric and ∫_ h  ds = h(∞) = 0 by mean-value theorem, the above integral vanishes and we obtain (<ref>).

For the remaining part, using Cauchy-Schwarz inequality, we have

    |∫_∖ϕ^-1_τ(D_j,n)χ_0   p  F_j,n e^-nQ^() dA | ≤p_nQ^()(∫_∖ϕ^-1_τ(D_j,n)|χ_0|^2 |F_j,n|^2 e^-nQ^() dA)^1/2,

which gives the bound O(n^1/4e^-c(log n)^2p_nQ^()) for some c by (<ref>).


 Lemma <ref> says that the quasipolynomials F_j,n satisfy the approximate orthogonality property (P2)
with a better error bound. In the sequel we will not use radial symmetry of Q, but merely that the property (P2) holds.






 §.§ Pointwise estimates

Going back to the general case, we assume that F_j,n satisfies the properties (P1) and (P2). (But we do not necessarily assume that Q be radially symmetric.)

 Let p_j,n the j:th orthonormal polynomial in L^2(, e^-nQ^()dA). Then for all j with j/n∈ [1-δ_n,1)

    p_j,n - F_j,nχ_0_nQ^() = O(n^-1/2log^2 n).



Let u_0 be the L^2(e^-nQ^()) norm-minimal solution to the -problem

    u = F_j,nχ_0
 which satisfies u_0(ζ)=O(|ζ|^j-1) as ζ→∞.
By a standard Hörmander estimate (cf. <cit.>), we have

    u_0_nQ^()^2 ≤ Cn^-1∫_ |χ_0 F_j,n|^2   e^-nQ^() = O(e^-cn).

Here, the O(e^-cn) bound is obtained from the fact that χ_0 vanishes on a neighborhood of Γ_τ and |F_j,n|^2 e^-nQ^() has an exponential decay (in n) on the support of χ_0.

Define a function p̃_j,n by

    p̃_j,n = F_j,n χ_0 - u_0.

Then p̃_j,n is entire and has the polynomial growth p̃_j,n(ζ) = a |ζ|^j+O(|ζ|^j-1) with some nonzero a near infinity. It follows that p̃_j,n is a polynomial of exact degree j. By (<ref>)

    p̃_j,n-F_j,n χ_0_nQ^()≤ O(e^-cn).

We thus obtain from (P1) and (P2) that

    p̃_j,n_nQ^() = 1+O(n^-1/2log^2 n)

and

    ∫_p̃_j,n p̅  e^-nQ^() dA = O(n^-1/2log^2 n  p_nQ^())

for all polynomials p of degree ≤ j-1.

Now we consider the orthogonal projection π_j,n from L^2(e^-nQ^()) onto the space consisting of all  polynomials in L^2(e^-nQ^()) of degree ≤ j-1.
Then the function

    p^⋆_j,n:=p̃_j,n - π_j,n[p̃_j,n]
 is a polynomial degree j which satisfies by (<ref>)

    p^⋆_j,n-p̃_j,n_nQ^()=O(n^-1/2log^2 n).
 Since p^⋆_j,n is orthogonal to all polynomials of degree at most j-1, it can be written as
p^⋆_j,n=c_j,n p_j,n for some constant c_j,n. Since p^⋆_j,n_nQ^() = 1+O(n^-1/2log^2 n) by (<ref>) and (<ref>), we have
|c_j,n|= 1+O(n^-1/2log^2 n). We can assume that c_j,n is a positive real number. Hence we get 
    p_j,n-F_j,nχ_0_nQ^() = O(n^-1/2log^2 n).



 
Suppose that 
    (ζ, Γ_τ )≤ n^-1/2(loglog n)^1/2.

Then we have the approximation for all j with j/n∈ [1-δ_n,1)

    p_j,n(ζ) = F_j,n(ζ)(1+O(n^-β)),

where β>0 and the O-constant is uniform in ζ.



For ζ with (ζ,Γ_τ) ≤ n^-1/2 (loglog n)^1/2, we have

    (Q̌_τ - V_τ)(ζ) ≤ c (ζ, Γ_τ)^2 ≤ c  n^-1loglog n.

The estimate in Lemma <ref> gives that there exists a constant C>0 such that

    |p_j,n - F_j,n|≤ C√(n) p_j,n-F_j,nχ_0_nQ^() e^nQ̌_τ/2,

which implies there exists a constant c'>0 such that

    |p_j,n()-F_j,n()| = O((log n)^2 e^nQ̌_τ()/2) = O((log n)^c' e^n V_τ()/2).
 Since 
    |F_j,n| = n^1/4|√(ϕ_τ')| e^n V_τ/2 e^_j,n/2
 and _j,n is bounded,
the lemma is proved.





§ ERROR FUNCTION APPROXIMATION

In this section, we prove Theorem <ref>. We assume that conditions (P1) and (P2) of Section <ref> are satisfied. (E.g.,
Q is radially symmetric.)

Take a point p on the outer boundary Γ of S and consider points ζ in the disc |ζ-p|≤ M/√(n) for some
large constant M. As we observed in Section <ref>, the 1-point function rescaled about p can be written as

    R_n(z) =R_n^♯(z)+o(1),     R_n^♯(z)= e^-nQ^()(ζ)/n Q(p)∑_j=m_n^n-1|p_j,n(ζ)|^2

where m_n = n-nδ_n and ζ = p+_1(p) z/√(n Q(p)).  To obtain the asymptotics of R_n^♯(z), we
shall apply the quasipolynomial approximation in Section <ref>.

Consider a point ζ with (ζ,U_τ)≤ n^-1/2(loglog n)^1/2. By Lemma <ref>, we obtain that for all j with n-nδ_n ≤ j ≤ n-1

    |p_j,n(ζ)|^2 e^-nQ^()(ζ)   = |F_j,n(ζ)|^2 e^-nQ^()(ζ)(1+O(n^-β)),
       = (n/2π)^1/2|ϕ_τ'| e^-n(Q^() - V_τ) e^_j,n(1+O(n^-β)).

Here the error term is uniform for all j with n- nδ_n ≤ j ≤ n-1. Recall that _j,n is a bounded holomorphic function on the relevant set of ζ satisfying
 
    _j,n(ζ) = log√( Q(ζ)) - logΦ_(ξ_j,n(ζ)),  ζ∈ S_τ
 where Φ_ is the function defined in (<ref>) and 
ξ_j,n(ζ) = (j-n)|ϕ_τ'(ζ)|/√(n Q(ζ)).

We have that

    lim_n→∞ R_n^♯(z) = S(2 z)   e^2(1-)( z)_+^2,   z∈.

Here the convergence is bounded on  and locally uniform in ∖ i.


We first observe that

    R_n^♯(z) = 1/√(2π)∑_k=1^nδ_n|ϕ_τ'(ζ)|/√(n) Q(p) e^-n(Q^() - V_τ)(ζ) e^_j,n(ζ) (1+ O(n^-β)),

where k = n -j and ζ = p+ _1(p)  z/√(n Q(p)).
For any compact subset ⊂, we have

    |ϕ_τ'(ζ)| e^_j,n(ζ) = |ϕ_τ'(p)| √( Q(p))/Φ_(ξ_j,n(p))+ O(n^-1/2log n),   n→∞,

where the error term is uniform for z∈ and for j with n-nδ_n ≤ j ≤ n-1. To obtain the asymptotic expansion of Q-V_τ, let p_τ denote the closest point to p where the line p+ _1(p) intersects the boundary S_τ. Then in view of Lemma <ref> the following asymptotic formulas hold: as τ→ 1,

    p_τ = p + (τ -1)  _1(p)  |ϕ_τ'(p)|/2 Q(p) + O((1-τ)^2),   _τ(p_τ) = _1(p)+O(1-τ).
 The Taylor series expansion about p_τ in Lemma <ref> gives that for τ∈ [1-δ_n,1)

    (Q-V_τ)(ζ)    = (Q-V_τ) (p_τ +  _τ(p_τ) k/n|ϕ_τ'(p)|/2 Q(p)  + _τ(p_τ) z/√(n Q(p)) +O(n^-1log^2 n) ) 
       = 2 Q(p_τ) (k/n|ϕ_τ'(p)|/2 Q(p) +  z/√(n Q(p)))^2 + O(n^-3/2log^3 n),

where the error is uniform for z ∈ and for τ∈ [1-δ_n,1). We also note that

    n(Q-V_1)(ζ) = 2( z)^2 + O(n^-1/2),   z∈.


Take a compact subset _1 of . Then for sufficiently large n the points 
    ζ = p+ _1(p)  z/√(n Q(p)), z∈_1
 are contained in the droplet S. Thus for z∈_1, we have

    e^-n(Q^() - V_τ)(ζ) = e^-n(Q-V_τ)(ζ) = e^-(-ξ_j,n(p) + 2 z)^2/2(1+o(1)).

On the other hand, for a compact subset _2 of ∖, it holds that for z ∈_2

    e^-n(Q^() - V_τ)(ζ)   = e^-n(Q-V_τ)(ζ) + n(1-)(Q-V_1)(ζ)
        = e^-(-ξ_j,n(p)+2 z)^2/2 e^2(1-)( z)^2(1+o(1)).


Combining the above asymptotics, we obtain

    R_n^♯(z)= 1/√(2π)|ϕ_τ'(p)|/√(n Q(p))∑_k=1^√(n)log n e^-1/2(ξ_j,n(p)-2 z)^2 e^2(1-)( z)_+^2 /Φ_(ξ_j,n(p))
    (1+o(1)).

Recall that ξ_j,n(p) = -k|ϕ_τ'(p)|/√(n Q(p)). The sum can be considered as an approximate Riemann sum with step length |ϕ_τ'(p)|/√(n Q(p)), which implies

    R_n^♯(z) = ∫_-∞^0e^-1/2(ξ - 2 z)^2/Φ_(ξ)dξ/√(2π)· e^2(1-)( z)_+^2 · (1+o(1)),     (n→∞).

Hence the lemma is proved.



Recall that by Theorem <ref> every limit K of correlation kernels c_n K_n is of the form

    K(z,w) = G(z,w) Ψ(z,w) e^(1-)(( z)_+^2 +( w)_+^2),

where Ψ is a Hermitian-entire function. As a result of Lemma <ref> we obtain the convergence 
    lim_n→∞ R_n(z) = S(2  z)  e^2(1-)( z)_+^2 ,    z∈.

This implies Ψ(z,z) = S(2 z).
By analytic continuation, we obtain Ψ(z,w) = S(z+w̅), completing our proof of Theorem <ref>.




§ SCALING LIMIT OF THE MAXIMAL MODULUS

In this section we discuss the asymptotic distribution of the maximal modulus at the edge and prove Theorem <ref>. For this purpose, we
pick a random sample {ζ_j}_1^n (associated with a modified potential Q^()) and recall that the maximal modulus is defined by

    |ζ|_n = max_1≤ j ≤ n |ζ_j|.


Observe that |ζ|_n ≤ r if and only if
none of the points ζ_j belongs to the exterior disk _e(r). Thus we must investigate the gap probability
that no point belongs to _e(r).


It is well known that the distribution function of |ζ|_n is represented by

    _n(|ζ|_n ≤ r) = (δ_j,k - ∫__e(0,r) p_j,n p_k,ne^-nQ^() dA)_j,k=0^n-1,

where p_j,n is an orthonormal polynomial of degree j in L^2 (,e^-nQ^() dA). Cf. <cit.> or <cit.>.

Here and in what follows we use the shorthand _n to denote the Boltzmann-Gibbs measure _n,^1.

In the case that Q is radially symmetric, the above
probability reduces to

    _n(|ζ|_n ≤ r) = ∏_j=0^n-1(1- ∫__e (0,r) |p_j,n|^2 e^-nQ^() dA).


Note that the outer boundary of the droplet is a circle centered at the origin. Let ρ be the radius of that circle.
We rescale |ζ|_n about ρ by

    ω_n = √(4nγ_n  Q(ρ) )(|ζ|_n - ρ - √(γ_n/4n Q(ρ))),

where γ_n = log(n/2π)-2loglog n + log (ρ^2 Q(ρ)/Φ_^2(0)).

Now we compute the distribution function of the random variable ω_n

    _n(ω_n ≤ x) = _n(|ζ|_n ≤ρ + 1/√(4n Q(ρ))(√(γ_n)+x/√(γ_n))).

It is convenient to write, for x∈, 
    h_n(x) =   r_n(,x)/√(4n Q(ρ));      r_n(,x) = 1/√()(√(γ_n) + x/√(γ_n)).

Then by (<ref>) we have

    _n(ω_n ≤ x) = ∏_j=0^n-1(1- ∫_|ζ|> ρ + h_n(x) |p_j,n|^2 e^-nQ^() dA).



We have the convergence

    lim_n→∞∑_j=0^n-1∫_|ζ|>ρ + h_n(x) |p_j,n|^2 e^-nQ^() dA = e^-x.

Here the convergence is uniform in any compact subset of .


For each x, ρ + h_n(x) is located in ∖ S with O(n^-1/2√(log n))-distance from the boundary S.
By the growth assumption on Q and the estimate in Lemma <ref> we have for some c>0  
    Q^()(ζ) - V_τ (ζ) ≥ c min{(ζ, Γ_τ)^ 2, log(1+|ζ|) },  ζ∈ U_τ,

where we remind of the notation U_τ=∖ S_τ and Γ_τ=Ụ_τ.

From this and Lemma <ref> we infer that the sum of lower degree terms

    ∑_j ≤ n-nδ_n∫_|ζ|>ρ + h_n(x) |p_j,n|^2 e^-nQ^() dA

 is negligible. (Here δ_n = n^-1/2log n, as usual.)

Using the quasipolynomial approximation from Section <ref>, we now analyze the limit of the sum of higher degree terms. We will write

    D_n,x = {ζ∈ : ρ + h_n(x)≤ |ζ| ≤ρ + Mδ_n }.

The problem is reduced to computing, for each x∈, the sum

    ∑_j= m_n^n-1∫_D_n,x |F_j,n|^2 e^-nQ^() dA = √(n/2π)∑_j= m_n^n-1∫_ρ + h_n(x)^ρ + Mδ_n|ϕ_τ'| e^-n(Q^() - V_τ) e^_j,n 2r dr,

where m_n = n-nδ_n.
By the Taylor series expansion for Q-V_τ and Q-V_1 about Γ_τ and Γ respectively, we get for all r with r-ρ=O(δ_n) and τ with τ∈ [1-δ_n,1)

    (Q^() -V_τ)(r)   =(Q-V_τ)(r) - (1-)(Q-V_1)(r)
       =2 Q(ρ_τ)(r-ρ_τ)^2-2(1-) Q(ρ) (r-ρ)^2+O(δ_n^3)

where ρ_τ is the radius of the circle Γ_τ.
By the change of variable 
    s =√(4n Q(ρ))(r -ρ)
 and the argument in Section <ref>,
 the above integral (<ref>) is equal to

    ∑_j=m_n^n-1ρ|ϕ_τ'(ρ)|/√(2π)Φ_(ξ_j,n)∫_r_n(,x)^M'log n e^(1-)s^2/2 e^-(s-ξ_j,n)^2/2 ds  (1+o(1))

where M' is a positive constant and ξ_j,n=ξ_j,n(ρ) = (j-n)|ϕ_τ'(ρ)|/√(n Q(ρ)).
By the Riemann sum approximation as in Section <ref>, we obtain for k=n-j and s with r_n(,x)≤ s≤ M'log n

    |ϕ_τ'(ρ)|/√(n Q(ρ))∑_k=1^nδ_ne^-(s-ξ_j,n)^2/2/Φ_(ξ_j,n)=∫_-log n^0e^-(s-ξ)^2/2/Φ_(ξ)dξ + ϵ_n(s).

The error term ϵ_n(s) from the Riemann sum approximation has a bound

    ϵ_n(s) ≤ C   (log n)^2/√(n)sup_-log n ≤ξ≤ 0|d/dξ(e^-(s-ξ)^2/2/Φ_(ξ))|
 so that ϵ_n(s) is negligible in (<ref>). Hence we need to find the limit of the integral

    √(n Q(ρ)/2π) ρ∫_r_n(,x)^M'log n e^(1-)s^2/2∫_-∞^0e^-(s-ξ)^2/2/Φ_(ξ)dξ  ds.

Write

    E_n(x) = ∫_r_n(,x)^∞ e^(1-)s^2/2∫_-∞^0e^-(s-ξ)^2/2/Φ_(ξ)dξ  ds.

By the change of variables u=s-ξ/ and v=u-r_n(,x),
we obtain

    E_n(x)    = ∫_-∞^0e^1-/2ξ^2/Φ_(ξ)∫_r_n(,x)-ξ/^∞ e^-/2u^2 du  dξ
       =∫_r_n(,x)^∞ e^-/2u^2∫_(r_n(,x)-u)^0e^1-/2ξ^2/Φ_(ξ)dξ du = ∫_0^∞ e^-/2(v+r_n(,x))^2 f(v) dv,

where

    f(v) = ∫_-v^0e^1-/2ξ^2/Φ_(ξ)dξ.

Now the integration by parts gives

    ∫_0^∞ e^-/2v^2-v r_n(,x)f(v)dv    =  -1/r_n(,x)[e^-v r_n(,x)e^-v^2/2f(v)]_v=0^∞
        + ∫_0^∞e^-v r_n(,x)/ r_n(,x)e^-v^2/2(-v f(v) + f'(v))dv.

Since f(0)=0 and e^- v^2/2f(v)=o(1) as v→∞, the first term in the right-hand side of (<ref>) vanishes.
By integrating by parts again we obtain

    ∫_0^∞ e^-/2v^2-v r_n(,x)f(v)dv    = - 1/( r_n(,x))^2[e^-v r_n(,x)e^-v^2/2f_1(v)]_v=0^∞
       +∫_0^∞e^-v r_n(,x)/( r_n(,x))^2e^-v^2/2(-v f_1(v) + f_1'(v))dv

where f_1(v) = f'(v) - vf(v). Since f_1(0)=f'(0)=(Φ_(0))^-1,

    ∫_0^∞ e^-/2v^2-v r_n(,x)f(v)dv    = 1/Φ_(0) r_n(,x)^2 + O(r_n(,x)^-3).

It follows that

    E_n(x) = e^-/2r_n(,x)^2( 1/Φ_(0) r_n(,x)^2 + O(r_n(,x))^-3).

Recall that r_n(,x) = 1/√()(√(γ_n)+ x/√(γ_n)) where 
    γ_n = log(n/2π)-2loglog n + log (ρ^2 Q(ρ)/Φ_^2(0)).

Thus we obtain

    √(n Q(ρ)/2π)ρ E_n(x) = e^-x + o(1),

where o(1)→ 0 locally uniformly as n→∞.
Hence we prove the lemma.



It follows from (<ref>) that

    log_n(ω_n ≤ x)=-∑_j=0^n-1∫_|ζ|>ρ + h_n(x) |p_j,n|^2 e^-nQ^() dA + o(1).

By Lemma <ref>, we finally obtain the convergence

    _n(ω_n ≤ x) → e^-e^-x,   n→∞.


Our proof of Theorem <ref> is complete. 



§ SUMMARY AND OUTLOOK


We have introduced a scale of boundary confinements for the Coulomb gas and analyzed them in the planar, determinantal case.
In particular, we have proved existence of a new scale of point fields, and investigated them for universality in two different ways;
(1) using the method of Ward equations and (2) using a quasipolynomial approximation formula.

The first method works well for
any reasonable potential, but has the drawback that we need to impose some apriori conditions on a solution in order to draw
relevant conclusions.

The method of quasipolynomials becomes more complicated than in the free boundary case, depending on that the quasipolynomial approximation
formula has different forms inside and outside of the droplet. This makes it complicated to verify the approximate orthogonality property (P2)
in Section <ref>. An exception occurs in cases when the curves Γ_τ has everywhere a uniform distance to Γ, meaning in practice that we are working with a radially symmetric situation. In this case the method works well, and allows us to conclude universality of scaling limits and to calculate the distribution of the maximum modulus.

We hope that our two methods will contribute to a future resolution of the intriguing question of universality in the above setting.

It is well known that fluctuations about the equilibrium measure of a determinantal Coulomb gas {ζ_j}_1^n converges, e.g., in the sense of distributions, to a Gaussian field with free boundary conditions, see <cit.>. It is also known (see <cit.>) that if one imposes
hard edge conditions, then the fluctuations will converge to another kind of Gaussian field.
The question of convergence of fluctuations with an arbitrary value of the confinement-parameter  presents itself naturally.

In the paper <cit.>, a notion of “distance to the vacuum” was introduced and analyzed for β-ensembles. This gives a natural
generalization of the maximum modulus, which works also for non radial potentials. The paper <cit.> provides some estimates
of the distribution function for this distance; cf. the paper <cit.> for numerical simulations and more precise predictions.

Finally, we wish to point out that our construction can in principle be applied to a Coulomb gas in any dimension, or, say, in a compact
Riemannian manifold.
Indeed, given a suitable external potential, it is possible to define natural notions of interaction kernel, droplet, obstacle function, and so on
(cf. e.g., <cit.> and <cit.>).
By introducing a confinement-parameter as above, one obtains more general ensembles with various restrictions near the
boundary.






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