Rescaling Ward identities in the random normal matrix model

We study existence and universality of scaling limits for the eigenvalues of a random normal matrix, in particular at points on the boundary of the spectrum. Our approach uses Ward's equation, which is an identity satisfied by the 1-point function.

In random normal matrix theory, one studies normal matrices M (i.e. MM * = M * M) of some large order n picked randomly with respect to a probability measure of the form Here dM is the surface measure on normal n×n matrices inherited from n 2 -dimensional Lebesgue measure via the natural embedding into C n 2 , Q(ζ) is a suitable real-valued function defined on C ("large" as ζ → ∞), and Z n is a normalizing constant; tr Q(M) = n 1 Q(ζ j ) is the usual trace of the matrix Q(M).
If Q is just defined on R and dM is surface measure on the Hermitian matrices (i.e. M * = M), one obtains random Hermitian matrices. The study of such eigenvalue ensembles, e.g. using the technique of Riemann-Hilbert problems, has been an active area of research.
The eigenvalues {ζ j } n 1 of a normal matrix M, picked randomly with respect to the measure (0.1), form a random point process in the complex plane C. The same point processes also appear as a special case of OCP ensembles, where {ζ j } n 1 has the interpretation of a system of repelling point charges subjected to the external magnetic field nQ. Here the factor n is needed to ensure that the system stays in a finite portion of the plane as n → ∞.
The joint distribution of the particles/eigenvalues then follows the the Botzmann-Gibbs law, (0.2) dP n (ζ) = 1 Z n e − H n (ζ) dV n (ζ), ζ = (ζ 1 , . . . , ζ n ) ∈ C n , where dV n is Lebesgue measure in C n divided by π n , and Z n = e − H n dV n . We can henceforth treat the system {ζ j } n 1 simply as a random sample from the distribution (0.2), having the interpretations as random eigenvalues or as point charges. An important feature of this process is that it is determinantal.
As n → ∞, the system {ζ j } n 1 will tend to occupy a certain set S called the droplet. More precisely, the system will tend to follow the classical equilibrium distribution given by weighted potential theory. (In addition, the "fluctuations" of the system about the equilibrium tends to the so called Gaussian field on S with free boundary conditions. See [3]. ) In this paper we will study microscopic properties of the system, close to a fixed point p in the droplet; in particular, a boundary point. The figure below shows a random sample from the classical Ginibre ensemble, in which Q = | ζ | 2 and S = { ζ ; | ζ | ≤ 1 }. In this case, the process {ζ j } n 1 can alternatively be interpreted as the eigenvalues of an n × n matrix whose entries are independent complex, centered Gaussian random variables of variance 1/n, see [24].
We rescale about the point p = 1 by letting z j = √ n ζ j − 1 and refer to the rescaled system Θ n = {z j } n 1 as the free boundary Ginibre process. The processes Θ n converge as n → ∞ to a determinantal random point field in C with correlation kernel (0. 3) K(z, w) = G(z, w)F (z +w) , where we call G(z, w) := e zw −| z | 2 /2 −| w | 2 /2 the Ginibre kernel and F(z) := 1 2 erfc z √ 2 the free boundary plasma function. To the best of our knowledge, this formula for the limiting point field first appeared in [21] and a rigorous proof was given in the paper [11]. (An alternative, simple argument depending on normal approximation of the Poisson distribution is given in Section 2.) We shall prove that the kernel (0.3) appears "universally" at regular points of the boundary, at least under an additional assumption of "translation invariance". This condition is satisfied e.g. when Q is radially symmetric. In a certain sense, (0.3) is an analogue to the Airy kernel in Hermitian random matrix theory, i.e., (0. 4) K(x, y) = Ai(x) Ai (y) − Ai (x) Ai(y) x − y , which describes the eigenvalue spacing at the edge of the spectrum. See [7], [19] and the references there for further information.
We will also establish existence and some basic properties (including non-triviality) of sequential limiting point fields pertaining to a quite general Q at a general (perhaps singular) boundary point of the droplet.
Our approach uses a relation between the 1-and 2-point functions, a particular case of Ward identities for Boltzmann-Gibbs ensembles. This relation is well known in field theories [53], [30], and has also been used in the papers [29], [3] to study fluctuations of eigenvalues.
We here fix some point on the boundary and rescale Ward's in identity about that point. It turns out that, if this is done properly, then limiting one-and two-point functions R = R 1 and R 2 can be defined in a way so that the Berezin kernel B(z, w) := R(z)R(w) − R 2 (z, w) R(z) satisfies the equation We refer to the equation (0.5) as Ward's equation. We stress that this equation is valid at all points, including singular boundary points, provided that ∆Q does not vanish there. It is more or less immediate from the computations with the Ginibre ensemble that the correlation kernel (0.3) gives rise to a solution to Ward's equation. However, in order to have uniqueness of solution to Ward's equation, we need to impose certain side-conditions, which depend on the nature of the point we are rescaling about. Our verification of these conditions uses Bergman space techniques.
The method of rescaled Ward identities apply to some other situations as well. We shall for example consider hard edge ensembles, where we completely confine the system {ζ j } n 1 to the droplet by setting Q = +∞ outside of S. In this case, a new kernel arises at regular boundary points: the hard edge plasma kernel  For reasons of length, we here just mention the case of the hard edge Ginibre ensemble, and postpone a more complete treatment, including universality, to the companion paper [5]. We

Introduction and results
1.1. Potential theory and droplets. Fix a suitable function ("external potential") Q : C → R∪{+∞}. Let P denote the class of positive, compactly supported Borel measures on C.
Define the weighted logarithmic energy of µ ∈ P in external field Q by Thinking of µ as the distribution of an electric charge, I Q [µ] is the sum of the Coulomb interaction energy and the energy of interaction of µ with the external field Q. We always assume that Q is lower semi-continuous, and that Q is finite on some set of positive logarithmic capacity. We will also assume that Q is sufficiently large at infinity, in the sense that To be precise, it will suffice to assume that lim inf ζ→∞ Q (ζ) / log | ζ | > 2. A classical theorem of Frostman states that there then exists a unique equilibrium measure σ which minimizes the weighted energy, See [41]. We denote the compact support of the equilibrium measure by We refer to S as the droplet in the external field Q. It is known that if Q is smooth in some neighbourhood of S, then σ is absolutely continuous and takes the explicit form (see [41]) Since σ is a probability measure, we have ∆Q ≥ 0 on S.
Our main assumptions throughout are that there is some neighbourhood Ω of S such that (a) Q is real analytic in Ω, (b) ∆Q > 0 in Ω. With these assumptions, the complement S c has a local Schwarz function at each boundary point, and we can rely on the fundamental theorem of Sakai [42] concerning domains with local Schwarz functions. In particular, we can apply Sakai's regularity theorem, which implies that a boundary point p ∈ ∂S is of one of the following types.
A point p ∈ ∂S is regular if there is a disc D = D(p; ) such that D \ S is a Jordan domain and D ∩ ∂S is a simple real analytic arc. A non-regular point p ∈ ∂S is called a singular boundary point.
There are two kinds of singular boundary points. A point p ∈ ∂S is a (conformal) cusp if there is D = D(p; ) such that D \ S is a Jordan domain and every conformal map φ : D(0; 1) → D \ S with φ(1) = p extends analytically to a neighbourhood of 1 and satisfies φ (1) = 0; p is a double point if there is a disc D about p such that D \ S is a union of two Jordan domains, and p is a regular boundary point of each of them.
One can further classify singular points according to degrees of tangency, but at this point it is not clear that such a classification would be relevant for our study. On the other hand, some cusps, in particular 3/2-cusps, which are generic in Sakai's theory can not appear on a free boundary.  1.2. Rescaling eigenvalue ensembles. Fix a potential Q as above, and let (ζ j ) n 1 denote a point in C n picked randomly with respect to the measure P n in (0.2). We refer to the corresponding (unordered) configuration {ζ j } n 1 as the n-point process (or simply "system") associated to Q. We also speak of {ζ j } n 1 as a "configuration picked randomly with respect to P n ". To each Borel set B in C we associate a random variable N B by N B = # j ; ζ j ∈ B . The system {ζ j } n 1 is then determined by the set of joint intensities (1.2) R n,k η 1 , . . . , η k = lim ε↓0 P n k j=1 N D(η j ;ε) ≥ 1 ε 2k , k = 1, . . . , n, n ∈ Z + .
We sometimes identify the intensity R n,k with the measure R n,k dV k . According to Dyson's determinant formula the joint intensities can be represented in the form R n,k (ζ 1 , . . . , ζ k ) = det K n (ζ i , ζ j ) where K n is a Hermitian function known as a correlation kernel of the process.
We are interested in microscopic properties of the system {ζ j } n 1 near a point p ∈ ∂S. It is natural to magnify distances about p by a factor n∆Q(p) and fix an angle θ ∈ R. We shall consider rescaled point processes of the form Θ n = Θ n (p, θ) = {z j } n 1 , where (1.4) z j = e −iθ n∆Q(p) ζ j − p , j = 1, . . . , n.
The law of Θ n is defined as the image of the Boltzmann-Gibbs measure (0.2) under the map (1.4).
Convention. (Choice of θ.) Let p be a boundary point of S. We rescale about p according to (1.4), where (i) if p is a regular point we take e iθ as the outer normal, (ii) if p is a cusp we take e iθ = −iν, where ν is the direction of the cusp, and, (iii) if p is a double point, we choose e iθ as a normal direction to ∂S at p. For the image of the droplet after rescaling, the tangent to the boundary is then vertical at z = 0, at regular as well as at singular points. Moreover, cusps will always point upwards.
With this choice of θ, we denote by Θ n = {z j } n 1 the rescaled process. This is a determinantal process with kernel K n given by The fundamental problem is existence and uniqueness of a limiting determinantal point field of the processes Θ n , as n → ∞. For our purposes, convergence will mean locally uniform convergence of all intensites R n,k to some limits R k as n → ∞. Whenever this is the case, R k can be interpreted in terms of Lenard's theory (see [48]) as a k-point function for a "point field" in C, meaning a probability law on a suitable space of (perhaps) infinite configurations {z i } ∞ 1 ⊂ C. A precise definition, and a discussion of relevant convergence results, is given in the appendix.
It here suffices to note that the desired convergence of the processes Θ n will hold if if the correlation kernels K n converge to a limit K locally uniformly on C 2 . Moreover, the limiting point field is uniquely determined by K if the functions K n (z, z) are uniformly bounded. In this case, the limiting point field is determinantal with intensity functions More generally, if K n is a correlation kernel and (c n ) ∞ 1 is a sequence of cocycles, then c n K n is another kernel giving rise to the same joint intensities R n,k . The problem is thus to show that there exists a sequence of cocycles such that c n K n converges locally uniformly to a non-trivial limit K with bounded convergence on the diagonal in C 2 .
The best understood case is when p ∈ Int S. Then, under the much weaker assumption that Q is C 2 -smooth near p, the rescaled processes Θ n converge to the Ginibre(∞) point field. The correlation kernel of this field is the Ginibre kernel, If p ∈ Int S, and if we rescale about p according to (1.4) with arbitrary angle θ, then we have as n → ∞ with locally uniform convergence on C k . (Cf. [2]; a new proof is given in Section 8.6.) We will henceforth consider the case p ∈ ∂S. In the following three sections, we will state the main results of the paper.

Compactness, non-triviality, and Ward's equation.
Our first theorem states the existence of sequential limits of the rescaled point processes Θ n and specifies the form of limiting correlation kernels.
Theorem A. Let p be a boundary point of S, and rescale about p as above.
(i) There is a sequence c n of cocycles such that every subsequence of (c n K n ) ∞ 1 has a subsequence converging uniformly on compact subsets. ("Compactness.") (ii) Each limit point in (i) satisfies K = GΨ where Ψ is a Hermitian entire function. ("Analyticity.") A limit point K in Theorem A will be called a limiting kernel. By the general theory mentioned in the previous section, such a K is the correlation kernel of a some point field in the plane, which we call a limiting point field. The 1-point function of this point field is denoted by R(z) = K(z, z). If R 0 on C we can define the Berezin kernel and consider its "Cauchy transform" Theorem B. Let K be a limiting kernel in Theorem A and write R(z) = K(z, z). Let p ∈ ∂S. If p is a singular boundary point, then we assume that p is strongly laminar. We then have (i) Non-triviality: R > 0 everywhere.
(ii) Ward's equation: The integral C(z) in (1.6) converges and defines a smooth function such that See Section 7.3 for the definition of strong laminarity. Here we only mention that all singular points for algebraic potentials (again, see Section 7.3) are strongly laminar, and it is believed that all singular points of real-analytic potentials (satisfying the assumptions (a), (b)) are strongly laminar, so this additional assumption would not be needed.
In view of the foregoing theorems, it is natural to try to find all (or at least some) limiting kernels K giving rise to a solution to Ward's equation. In order to fix a solution uniquely, we need to know that certain additional conditions are satisfied, which depend on the nature of the point we are rescaling about. The following theorem provides conditions of this type.
Theorem C. Fix a boundary point p and let K be a corresponding limiting kernel in Theorem A. Also fix an arbitrary number < 1 and write x = Re z. Then there is a constant C such that (i) If p is a regular boundary point, then (ii) If p is a cusp or a double point, then The estimates (i) and (ii) are mutually inconsistent, so limiting point fields at singular points are certainly different from those at regular points. Moreover, as R is bounded, we can conclude that all limiting point fields are uniquely determined by the corresponding limiting kernel.
Remark. Our proof of Theorem B gives the locally uniform lower bound R(w) ≥ c K > 0 for w ∈ K, where K is a given compact subset of C. By the same token, for each fixed number δ > 0, there is c = c(δ) > 0 such that R n (ζ) ≥ cn whenever dist (ζ, S) ≤ δ/ √ n. (All assuming strong laminarity.)

Translation invariant solutions to Ward's equation.
Let K = GΨ be a limiting kernel in Theorem A. We say that In this case, Ψ can be represented in the form for some entire function Φ. Our next theorem describes translation invariant solutions to Ward's equation. We will write Let g be a bounded measurable function on R. We shall use the symbol " * " to denote the convolution operation Thus γ * g has the meaning of ordinary convolution in R followed by analytic continuation. The function γ * g is a version of the Bargmann transform of g, but the application is not quite standard, since g does not necessarily belong to L 2 (R). (See [19] for more on the Bargmann transform.) Theorem D. Let Ψ(z, w) be a translation invariant Hermitian entire function, Ψ(z, w) = Φ(z +w). Denote K = GΨ and suppose that Then K gives rise to a solution to Ward's equation if and only if there is a connected set I ⊂ R of positive measure such that Let K be an arbitrary limiting kernel in Theorem A. Note that the condition (1.7) is satisfied at all boundary points (both at regular and singular points) by Theorem C. However, we do not know whether all limiting kernels necessarily are translation invariant, except when the potential is radially symmetric, i.e., Q(z) = Q(|z|). See Section 6.1. We shall see that the corresponding interval I in (1.8) is typically I = (−∞, 0) so that Φ = F where F is the free boundary plasma function, Note that for real x we have F(x) = P(X ≥ x) where X is a standard normal random variable.
We have the following universality result.
Theorem E. Suppose that Q is radially symmetric. For a boundary point p we let Θ n be the free boundary process rescaled about p in the outer normal direction. Then Θ n converges to the unique point field BG in C with the correlation kernel We will call the kernel K in (1.10), resp. the point field BG, the "boundary Ginibre" kernel/point field. It is a natural conjecture that BG is universal at regular boundary points for all (not necessarily symmetric) potentials.
At the same time, we do not have enough evidence to make a similar conjecture for singular boundary points. We know that the limiting kernels are non-trivial, satisfy Ward's equation and the mass-one equation, and that they satisfy the estimate R(iy) ≤ Ce −2 x 2 . A family of kernels which satisfy these conditions is provided by K(z, w) = G(z, w) · γ * 1 I (z +w), where I ⊂ R is a bounded interval. We shall moreover prove in Section 8 that this K is the correlation kernel of some random point field. It would be interesting to know if there are examples of cusps or double points where the limiting point field has such a kernel. If so, we do not know whether the limiting distribution depends on the type of the singular point.
1.5. Berezin kernel and mass-one equation. Let B n be the Berezin kernels of the rescaled processes Θ n , i.e., We have (e.g. by (1.3)) Let us write R n = R n,1 and denote by R (a) n−1 the 1-point function for the conditional (n − 1)-point process Θ n | {a ∈ Θ n }. For each fixed a ∈ C, we then have n−1 (w). (For details, see [2], Section 7.6.) It is natural to ask whether the relation (1.11) also holds for the limiting kernel B = lim n B n . Figure 4 shows the Berezin kernel corresponding to the boundary Ginibre ensembles. In this case, direct computations show that B(a, w) dA(w) = 1 and so we have The corresponding conditional 1-point function rescaled about a bulk point is Elementary calculations now show that if we rescale about the point p and insert a point charge at a = 0, then the repulsion caused if p is in the bulk is stronger, by a factor π/(π−2) ≈ 3, compared to if the point p is on the boundary.
Let us say that a limiting kernel K = GΨ satisfies the mass-one equation if the corresponding Berezin kernel B satisfies C B(z, w) dA(w) = 1 for all z, i.e., This equation is technically similar to Ward's equation and it has a simple spectral interpretation (see Section 8.4). Every kernel K giving rise to a solution to the mass-one equation is furthermore a correlation kernel of some point field.
The following theorem describes the translation invariant solutions to the mass-one equation. In particular, it shows that Ward's equation implies the mass-one equation, in the translation invariant case.
Theorem F. Let Ψ(z, w) be a translation invariant Hermitian entire function, Ψ(z, w) = Φ(z +w). Denote K = GΨ and suppose that Then K gives rise to a solution to the mass-one equation if and only if there is a Borel set e ⊂ R of positive measure such that 1.6. Organization of the paper. In Section 2 we consider the boundary Ginibre ensembles, both for the free boundary and the hard edge. We give a short proof of the convergence of rescaled ensembles to the boundary Ginibre point fields with kernels (0.3) and (0.6), respectively.
In Section 3, we prove Theorem A (compactness and analyticity).
In Section 5, we establish some a priori bounds for regular points. We also prove part (i) of Theorem C and establish non-triviality (part (i) of Theorem A) in the case of regular boundary points and translation invariant kernels. This is what is needed for the analysis in Section 6.
In Section 6 we find all translation invariant solutions to Ward's equation and prove theorems D, E, and F. In Section 7 we prove a priori estimates at singular boundary points, thus completing the proof of Theorem C (part (ii)) and the non-triviality part of Theorem B.
The last section, Section 8, contains various concluding remarks. In particular, we comment on the nature of the mass-one and Ward equations, we show that these equations take the form of twisted convolution equations, we discuss Hilbert spaces of entire functions associated to limiting kernels, and write down versions of Ward's equation in some settings which are different from the free boundary case studied in this paper (hard edge, bulk singularities, β-ensembles).

Example: The Ginibre ensembles
2.1. Principles of notation. Consider first a general potential Q. By a weighted polynomial of order n we mean a function of the form f = q · e −nQ/2 where q is an (analytic) polynomial of degree at most n − 1. Let W n denote the space of all weighted polynomials of order n, considered as a subspace of L 2 = L 2 (C, dA). It is well-known that the reproducing kernel K n (ζ, η) for the space W n is a correlation kernel for the process {ζ j } n 1 corresponding to Q. See e.g. [26]. This implies that one has the formula where q j is the j:th orthonormal polynomial with respect to the measure e −nQ(ζ) dA(ζ).
Recall the Ginibre potential Q = | ζ | 2 . The corresponding droplet is S = { ζ ; | ζ | ≤ 1 }. We shall give an elementary proof for BG-convergence using Poisson approximation of the normal distribution. Our proof is quite similar to the argument in the paper [40], where the spectral radius of a Ginibre matrix is studied. Let us mention also that there are several other proofs of BG convergence for the free boundary Ginibre ensemble.

Free boundary Ginibre ensemble.
Let {ζ j } n 1 denote a random configuration for the free boundary Ginibre process. We rescale about the boundary point p = 1 in the outer normal direction, via z j = √ n ζ j − 1 , writing Θ n = {z j } n 1 for the rescaled process. Let G and F be the Ginibre kernel and the free boundary plasma function, respectively. See (1.5) and (1.9). We shall prove the following theorem, found in [21] (cf. [11]). Theorem 2.1. There exists a unique point field BG with correlation kernel K(z, w) = G(z, w)F(z +w), and the processes Θ n converge to BG as n → ∞ with locally uniform convergence of intensity functions.
Since K(z, z) < 1, it suffices to prove the statement about convergence of intensity functions. By (2.1), a correlation kernel for the process {ζ j } n 1 is computed to Now rescale according to and note that the rescaled process Θ n has correlation kernel K n (z, w) = 1 n K n ζ, η . Using (2.2), we write K n in the form We next let X n be a Poisson distributed random variable with intensity λ = λ(n) (in short: X n ∼ Po(λ)), i.e., We then have the identity By the central limit theorem, Y n converges in distribution to the standard normal as n → ∞; the convergence is moreover uniform. (This is the well known "normal approximation of the Poisson distribution"; uniform convergence follows e.g. by the Berry-Esseen theorem.) Now factorize K n (z, w) in the following way, Note that α n → − Re(z + w) as n → ∞.

Lemma 2.2.
We have the convergence where b = b(z, w) = Im(z +w) and F is the free boundary kernel (1.9). Moreover, where G is the Ginibre kernel and o(1) → 0 uniformly on compact sets as n → ∞.
Proof. By a straightforward calculation we have Inserting these expressions into B n (see (2.4)) using the fact that the Y n are asymptotically normal, we now approximate as follows (the symbol "∼" stands for asymptotic equality as n → ∞) We now turn to the factor A n in (2.4). To deal with it, we denote c = Re(z + w). By (2.5), we then have (1) .
Noting that we finish the proof of the lemma.
By the lemma and relation (2.4), we have where K is the free boundary kernel defined in (1.10). Since the factor c n (z, w) = e i √ n Im(z−w) is a cocycle, this factor can be dropped when computing intensity functions R n,k (z) = det(K n (z i , z j )). This proves the desired convergence of intensity functions, at the same time establishing existence and uniqueness of the field BG. The proof of Theorem 2.1 is complete.
2.3. Hard edge Ginibre ensemble. Let Q S (z) = | z | 2 when | z | ≤ 1 and Q S = +∞ otherwise. Let {ζ i } n 1 denote a random configuration from the corresponding ensemble. Rescaling about p = 1 via z j = √ n (ζ j − 1) we obtain a process Θ n . Let H be the hard edge plasma function (0.6).

Theorem 2.3.
There exists a unique point field BG h with correlation kernel The processes Θ n converge to BG h in the sense that all intensity functions converge locally boundedly almost everywhere, and locally uniformly in L 2 , to the intensity functions of BG h .
Note that K(z, z) < 2 for all z. We prove in the appendix that the convergence of intensity functions in the theorem implies the existence and uniqueness of a field BG h with correlation kernel K. It thus suffices to prove convergence. By (2.1) and a calculation, a correlation kernel for the hard edge Ginibre process is given by where γ j + 1, n = n 0 s j e −s ds is the lower incomplete Gamma function. Now rescale by where λ = n(| ζ | 2 + | η | 2 )/2 is as in (2.3). We shall use a rough estimate for γ( j + 1, n). Observe that, by a well-known fact, we have where U n ∼ Po(n). By normal approximation of the Poisson distribution ∞ ξ e −t 2 /2 dt, ξ j,n = ( j − n)/ √ n, and and o(1) → 0 as n → ∞ uniformly in j. We have shown that Finally, if X n ∼ Po(λ), we can write the last sum in the form Defining Y n by X n = λ + √ λY n , we now get a relation of the form Using the asymptotic identities α n ∼ − Re(z + w) and λ/n ∼ 1, we approximate the factorB n as follows, using the central limit theorem, Using the asymptotics for A n in Lemma 2.2 we now conclude that where o(1) → 0 uniformly on compacts. The first factor is a cocycle. We conclude that if K(z, w) = G(z, w)H(z +w)1 L (z)1 L (w) is the hard edge kernel, then K n → K almost everywhere with locally bounded convergence, finishing the proof of the theorem. q.e.d.

Analyticity and compactness
In this section, we prove Theorem A.
3.1. General notation. For a measurable function φ : C → R ∪ {+∞} we define L 2 φ to be the space of functions normed by When φ = 0 we just write L 2 for L 2 φ and denote the norm by · . We denote by C (large) and c (small) various positive unspecified constants (independent of n) whose exact value can change meaning from time to time.
3.2. Potentials and reproducing kernels. We shall henceforth assume that the potential Q is real analytic and strictly subharmonic in some neighbourhood of the reduced droplet S. More precisely, we fix a neighbourhood Ω of S and a number δ 0 > 0 such that Q is real-analytic and strictly subharmonic in the 2δ 0 -neighbourhood of Ω.
We give the definitions of some reproducing kernels related to a potential Q. Let P n be the space of analytic polynomials of degree at most n − 1; we equip P n with the norm of L 2 nQ . The corresponding space W n of weighted polynomials is defined to consist of all functions of the form f = qe − nQ/2 where q ∈ P n ; we regard W n as a subspace of L 2 and denote the corresponding orthogonal projections by π n : L 2 nQ → P n and Π n : We write k n and K n for the reproducing kernels of P n and W n respectively. Then It is easy to see that the assignment is unitary, maps P n onto W n , and satisfies U n π n = Π n U n .

Analytic continuation and bulk approximations.
Let A(ζ, η) be a Hermitian-analytic function defined in a neighbourhood in C 2 of the "diagonal" X := { (ζ, ζ) ; ζ ∈ Ω }, such that We can choose δ 0 > 0 small enough that A(ζ, η) is defined and Hermitian-analytic in the set of points (ζ, η) ∈ C 2 whose distance to X is < 2δ 0 . Call this set Λ. We now define "bulk approximations" k # n and K # n , defined in the domain of A(ζ, η) via

Elementary estimates for the one-point function.
We write R n (ζ) = K n (ζ, ζ) for the onepoint function. By a basic fact for reproducing kernels we have the identity We shall also use the following simple pointwise-L 2 estimate (c.f. [1], Section 2 for a proof).
is the logarithmic potential of the equilibrium measure σ = ∆Q · 1 S dA. By the obstacle function corresponding to Q, we mean the subharmonic functionQ It is known (see [41]) thatQ = Q on S whileQ is harmonic on S c and is of logarithmic increasě FurthermoreQ has a Lipschitz continuous gradient on C. We remind of the following basic result; the "maximum principle of weighted potential theory" (see e.g. [41] for proof).

Lemma 3.2.
If f ∈ W n and f ≤ 1 on S, then f ≤ e − n(Q−Q) on C.
Combining the preceding lemmas with the identity (3.2) gives the following bound for the 1-point function.

Lemma 3.3.
There is a constant C independent of n and ζ such that 3.5. Rescaled kernels. Let p ∈ C and fix a real parameter θ. Let {ζ j } n 1 denote the point process corresponding to Q. Recall that rescaled point process at p in the direction e iθ is the process . A kernel K n for the rescaled process is given by We define the rescaled bulk approximation K # n by Here K # n is the bulk approximation to K n defined in (3.1).
3.6. Convergence of the approximate kernels.
Let V n denote the set of points (z, w) such that (ζ, η) ∈ Λ and (3.3) holds. Here Λ is the 2δ 0neighbourhood of the diagonal, see Section 3.3. It is clear from (3.3) that the sets V n eventually contains each compact subset of C 2 . Indeed, there is a constant ρ > 0 depending only on ∆Q(p) such that We have the following lemma. (1)) as n → ∞ where c n are cocycles on V n and o(1) → 0 as n → ∞, uniformly on compact subsets of C 2 .
Proof. We can assume that p = 0 and θ = 0. Put ∆Q(0) = δ. Recall that As n → ∞, our rescaling means that ∂ 1∂2 A(ζ, η) → δ. Moreover, by Taylor's formula, the expression in the exponent is and this equals The first two terms in correspond to cocycles.
Remark. The proof of the lemma shows that if | ζ | < δ n where nδ 3 n → 0 as n → ∞. Then Hence (replacing "0" by "p") we see that there is a number C such that

Compactness.
Recall that Ω denotes some sufficiently small neighbourhood of S. Fix a point p in Ω and rescale about p as in (1.4).

Theorem 3.5. Let p be an arbitrary point of Ω.
There is a sequence of cocycles c n such that every subsequence of c n K n has a subsequence converging uniformly on compact subsets of C 2 . Furthermore, every limit point has the form K = GΨ where Ψ(z, w) is an Hermitian entire function (or "holomorphic kernel").
The theorem implies Theorem A; this is but the special case when p is a boundary point of S.
In the proof, we will use the functions Ψ n defined on V n by the equation We will need two lemmas. Recall the definition of the set V n from the beginning of Section 3.6.
Lemma 3.6. The function Ψ n is Hermitian-analytic in the set V n .
Proof. For (z, w) ∈ V n we have (ζ, η) ∈ Λ and The statement follows since ζ and η depend analytically on z and w.
when (z, w) ∈ K and n is large enough.
Proof. Choose n 0 large enough that K ⊂ V n 0 . Since K n is a positive kernel, and since (by Lemma 3.4) K # n (z, w) → | G(z, w) | uniformly on compact subsets, we have uniformly on K, for all n ≥ n 0 . By Lemma 3.3, we have a uniform bound R n ≤ C, which finishes the proof of the lemma.
Proof of Theorem 3.5. Lemma 3.7 shows that the family {Ψ n } is locally bounded on C 2 , viz. is a normal family. Pick a locally uniformly convergent subsequence converging to a limit Ψ. Also fix z and recall that K n k (z, w) 2 dA(w) = K n k (z, z). In terms of the functions Ψ n , where ρ > 0 is the constant in (3.4). Letting k → ∞ we get, by Fatou's lemma, that Finally we use Lemma 3.4 to select cocycles c n such that c n K # n → G uniformly on compact subsets of C 2 . Then c n k K n k = Ψ n k · c n k K # n k → ΨG, finishing the proof of the theorem. Definition. A Hermitian-entire function Ψ is said to satisfy the mass-one inequality if (3.5) holds.

Ward's equation and the mass-one equation
In this section, we prove Ward's equation, i.e., part (ii) of Theorem B. We also also introduce the "mass-one inequality" for holomorphic kernels. Our discussion presupposes non-triviality, i.e., part (i) of Theorem B. We remind that the proof of non-triviality is postponed to sections 5 (translation invariant solutions at regular points) and 7 (the general case).

Ward's identity.
We shall here prove a slightly modified (or "localized") form of the Ward identity used in [3]. This modification is necessary when dealing with hard edge processes, and is in general quite convenient.
To set things up, fix a test-function ψ ∈ C ∞ 0 (C). Define a function W + n [ψ] of n variables by Here we think of ζ j n 1 as being randomized with respect to the Boltzmann-Gibbs law P n , see (0.2).
We assume only that Q be smooth in a neighbourhood of the support of ψ. We can then make sense of W + n [ψ] even though ∂Q may be undefined in portions of the plane. Indeed, we define We then have the following form of Ward's identity.

Now (4.1) and (4.3) imply that the Hamiltonian
It follows that the partition function Z n := C n e −H n( η) dV n η satisfies Since the integral is independent of ε, the coefficient of ε in the right hand side must vanish, which means that or Re E n W + n [ψ] = 0. Replacing ψ by iψ in the preceding argument gives Im E n W + n [ψ] = 0 and the theorem follows.

Fundamental relation.
We now fix a point p and rescale the system ζ j n 1 about p in the usual way, obtaining the rescaled system The value of θ is here irrelevant. Recall that the rescaled intensity functions are defined via R n,k (z 1 , . . . , z k ) = R n,k (ζ 1 , . . . , ζ k ), while the Berezin kernel rooted at p is defined by The following result is a rescaled form of Ward's identity; we call it the fundamental relation.

Theorem 4.2.
Suppose that Q is C 2 -smooth and strictly subharmonic in the closure of a neighbourhood Ω of S and rescale about a sequence p in Ω. Then for the free boundary process we havē and o(1) → 0 uniformly on compact subsets of C as n → ∞.
Proof. Fix a point p such that Q is C 2 -smooth and strictly subharmonic in a neighbourhood U of p. We can without loss of generality assume that p = 0 and θ = 0. Fix a test-function ψ supported in the dilated set and let ψ n (ζ) = ψ (z). Thus supp ψ n ⊂ U. The change of variables (ζ, η) → (z, w) gives Similarly, since supp ψ n ⊂ U, Finally, in the sense of distributions, After dividing by √ nδ in Ward's identity (Theorem 4.1), we deduce from eq.'s (4.4)-(4.6) that Since ψ is arbitrary, we get the following identity, in the sense of distributions, so we can write (4.7) as Here C n (z) = B n (z,w) z−w dA(w). Taking a∂-derivative now gives As n → ∞ we have that ∆Q(ζ/ √ nδ)/δ → 1 uniformly on compact subsets of C. We have shown where o(1) → 0 uniformly on compact sets as n → ∞. Recalling that R n,1 (z) = B n (z, z) we conclude the proof of Theorem 4.2.

Ward's equation.
Let K = ΨG denote any limiting kernel in Theorem 3.5 (or Theorem A).
Referring to a suitable subsequence, we write for the one-point function, and for the corresponding Berezin kernel, which is well-defined for all z ∈ C since R is non-trivial (part (i) of Theorem B). Note that the mass-one inequality (3.5) is equivalent to the condition This inequality implies that the Cauchy transform is a well-defined, continuous function.
We are now ready to prove Ward's equation: To this end, we put As before, fix a subsequence n k such that c n k K n k converges locally uniformly to K = GΨ. Then B n k → B locally uniformly in C 2 . Fix an arbitrary ε in the interval 0 < ε < 1, and pick N such that if k > N, | z | < 1/ε, and | w | < 2/ε, then For such k, We next use the relation (see Theorem 4.2) where o(1) → 0 uniformly on compact subsets of C. This implies that uniformly on compact subsets. It follows that a limit C = lim C n k exists in the sense of distributions and that∂ C = R − 1 − ∆ log R in that sense. Since the right hand side is smooth and C is continuous, the equation (4.8) holds pointwise, by Weyl's lemma.
It is convenient to somewhat reformulate Ward's equation. Given a Hermitian-entire function Ψ (positive on the diagonal in C 2 ) we define the functions Thus D = RC. Proof. The equation (4.8) means that Let P 0 be an arbitrary solution to the equation∂P 0 = R − 1. Then (4.12) becomes This last identity is fulfilled if and only if there is an entire function E such that Setting P = P 0 + E, we see that (4.10) and (4.11) are satisfied. Conversely, if (4.10) and (4.11) hold, then∂ i.e. (4.12) holds.
4.4. The mass-one inequality. Consider a Hermitian-entire function Ψ(z, w) and write R(z) := Ψ(z, z). Recall (cf. (3.5)) that Ψ is said to satisfy the mass-one inequality if We now reformulate this inequality. Likewise, It follows that The proof of the lemma is finished.
It follows from the lemma that the mass-one equation is equivalent to that One can regard this as a differential equation of infinite order.
We also note the following theorem. Proof 2. We here give an alternative, direct verification that the function R(z) = F(2 Re z) satisfies the mass-one equation (4.14). To this end, note that Using this, we obtain by differentiating in (4.13) the equivalent equation Dividing by F , and using the Rodrigues formula for the Hermite polynomial h n , one can rewrite (4.15) in the form But both sides of (4.16) have a zero at the origin, so we need only verify that the derivatives are equal. Using the recursion h n (z) = nh n−1 (z), one realizes that our assertion is equivalent to that so the sum in the right hand side of (4.17) equals and this equals 1 by the recursive definition of Hermite polynomials: h 0 = 1, h 1 = z, and h n = zh n−1 − (n − 1)h n−2 for n ≥ 2.

Regular boundary points
In this section we prove asymptotic estimates for the rescaled 1-point function at a regular boundary point p ∈ ∂S.
To set things up, we rescale about p in the outer normal direction and let K = GΨ denote an arbitrary limiting kernel in Theorem A. Our goal is to prove that the function R(z) = K(z, z) is everywhere strictly positive. By Lemma 4.5, if it is known that K is translation invariant, i.e. if Ψ(z, w) = Φ(z +w), then it suffices to prove that R does not vanish identically. We will now prove that R 0; in fact, we shall prove the estimate in Theorem C, part (i), which in particular says that R(x) → 1 as x → −∞. In this way, we will obtain a proof of Theorem B for the important case of translation invariant solutions at regular boundary points. This is precisely the situation for our applications in the next section.
We next fix a sequence (δ n ) of positive numbers in the interval (2γ/ √ n, δ 0 /2), where γ is a sufficiently small positive number independent of n, and nδ 3 n → 0 as n → ∞. Below we will fix a point ζ in a small neighbourhood Ω of S. We shall also use the Hermitian We will use the kernels K n and K # n , where we recall that (cf. Section 3.3) For a fixed η we will use abbreviations such as Theorem 5.1. There is a constant C independent of the positive integer n, the point ζ ∈ Ω, and the number δ in the interval 0 < δ < δ n such that In particular, We shall below choose ζ ∈ Int S and δ = dist(ζ, ∂S). However, for applications in Section 7, it is convenient to prove the slightly more general result above.
We shall use the following lemma.
Proof. Assume that ζ = 0 and write χ = χ ζ . Then Π # n χ f (ζ) equals to the integral which means that Since the holomorphic function H(ξ) as ω → 0. Our assumptions imply that the O-constants can be chose independent of ζ.
Integrating by parts in (5.2) one obtains It follows that there is a constant C (independent of ζ, n and δ) such that By the remark after Lemma 3.4 and the assumption nδ 3 n → 0, we have the estimate Using this and the Cauchy-Schwarz inequality, we now find (since | ω | ≥ ϑδ when∂χ(ω) 0) The proof of the lemma is complete.

First BP estimate.
Recall that L 2 φ denotes the space of functions f normed by f 2 φ = | f | 2 e −φ . We shall let A 2 φ denote the subspace of L 2 φ consisting of entire functions. We write π φ for the orthogonal (Bergman) projection L 2 φ → A 2 φ . When π is the orthogonal projection of a Hilbert space onto a closed subspace, we denote by π ⊥ = I − π the complementary projection.
Our starting point is a simple estimate of "Hörmander type" (cf. [28], p. 250). The result states that if φ is smooth and strictly subharmonic in C, and if u ∈ C ∞ 0 (C), then Lemma 5.3. Fix ζ ∈ Int S. Put δ = dist(ζ, ∂S) and 2γ/ √ n < δ < δ n . Then is a constant C such that and observe that u is a norm-minimal solution in L 2 nQ to the problem∂u =∂ f where f = χ ζ · g ζ . We shall prove that To do this, we introduce the strictly subharmonic function and consider v 0 = π ⊥ nφ χ ζ g ζ . HereQ is the equilibrium potential, cf. Section 3.4. By the estimate (5.5) we have v 0 2 nφ ≤ ∂ χ ζ · g ζ 2 e − nφ n∆φ dA.
Next note the estimate nφ ≤ nQ + const. on C, which is obvious in view of the growth assumption on Q near infinity. This gives v 0 nQ ≤ C v 0 nφ , and we have shown (5.7) with u = v 0 . Since nφ(ω) = (n + 1) log | ω | 2 + O(1) as ω → ∞ (see Section 3.4), we have the equality A 2 nφ = P n in the sense of sets. Hence u = v 0 solves, in addition to (5.7), the problem ∂u =∂ χ ζ g ζ and u − χ ζ g ζ ∈ P n .
Applying the estimate (5.7), one obtains We shall now finally use the assumption that δ ≥ γ/ √ n. This gives that the function u is analytic in the disc D ζ; γ/ √ n , so that Lemma 3.1 applies. We obtain that | u(ζ) | 2 e − nQ(ζ) ≤ Cn u 2 nQ , where C depends on γ and ∆Q(ζ). Combining with (5.8), we have shown (with a new C) The proof of the lemma is complete.
Theorem 5.4. Fix a constant < 1. There is then a constant C such that if ζ ∈ S and δ = dist(ζ, ∂S), then Proof. By Lemma 3.3 we have K ζ 2 ≤ Cn. This gives that (5.9) holds trivially when δ < γ/ √ n (because K n (ζ, ζ) ≤ Cn and K # n (ζ, ζ) ≤ Cn for sufficiently large C). We can thus assume that δ > γ/ √ n. To this end, we will apply the preceding results with = ϑ 2 . Then where we have used Theorem 5.1 to estimate the first term and the estimate (5.6) to estimate the second one.

5.
3. An exterior estimate. We shall in the following consider only the free boundary process. Recall from Lemma 3.3 that there is a constant C such that An analysis of the exponent in (5.10) gives the following well known result.
Lemma 5.5. Suppose that Q be real analytic and strictly subharmonic in a neighbourhood of S. Let Γ be a sub-arc of a regular, real-analytic arc of ∂S. Assume also that the end-points of Γ are regular boundary points of S. Pick p ∈ Γ and let e iθ be the outwards unit normal to ∂S at p. There is then a positive number C depending only on Γ and Q such that where c = inf Γ ∆Q and the O-constant is uniform on Γ.
Proof. Our assumptions imply that the obstacle functionQ (see Section 3.4) extends harmonically from the exterior of Γ to a neighbourhood of Γ. Hence (Q−Q)| S c extends to a real-analytic function. Now fix p as above and consider the function f (h) = (Q −Q)(h) where h > 0 is small. By the foregoing remarks, we can expand f (h) in a Taylor series about h = 0. But f (0) = f (0) = 0 and sinceQ is harmonic outside of S while ∂ 2 ∂s 2 (Q −Q)(p) = 0 where ∂/∂s denotes differentiation in the direction tangential to ∂S at p, it is seen that lim h↓0 f (h) = 4 lim z→p, z S ∆(Q −Q)(z) = 4δ where δ = ∆Q(p) > 0. It now follows from Taylor's formula that f (h) = δh 2 /2 + O(h 3 ) for small h > 0. The assertion follows, since R n (p + he iθ ) ≤ Cne − f (h) by (5.10).

5.4.
A priori estimates and non-triviality at regular points. Fix a number < 1 and a regular boundary point p ∈ ∂S. Let e iθ be the outwards unit normal at p. By convention, we rescale at p in the direction of e iθ , The estimate | R n (z) − 1 L (z) | ≤ Ce − x 2 /2 where z = x + iy is now immediate from Theorem 5.4 and Lemma 5.5. This shows that any limit point R of the R n 's must satisfy Next suppose that Re z ≥ 0 and | z | ≤ log n. Write z = n∆Q(p)(ζ − p), δ n = log 2 n/ √ n, and D n = D(p; δ n ). By Lemma 5.5 we then have the estimate R n (z) ≤ Ce − 2c n x 2 where where we take M to be an upper bound for the Lipschitz constant of ∆Q in a neighbourhood of S. We have shown that, for all n sufficiently large, We have thus completely proved part (i) of Theorem C. Now write R(z) = K(z, z) where K = GΨ is a limiting kernel in Theorem A. We remind that K is called translation invariant if Ψ(z, w) = Φ(z +w) for some entire function Φ. The following special case of Theorem B will be useful in the following. Proof. The estimate (5.12) shows that R does not vanish identically. This implies that R > 0 everywhere, by Lemma 4.5.

Translation invariant solutions
In this section, we prove our main results: theorems D, E, and F. 6.1. Basic observations. An arbitrary Hermitian entire function Ψ will be called translation in- Proof. If Ψ is translation invariant, we define Φ(z) = Ψ(z, 0). We must prove that Ψ(z, w) = Ψ(z +w, 0).
However, for fixed z both functions are analytic inw and they coincide on the imaginary axis.

Lemma 6.2.
Assume that Q is radially symmetric. Let p be a boundary point of the droplet and rescale in the outwards normal direction (see (1.4)). Then every limiting rescaled function Ψ given by Theorem 3.5 is translation invariant.
We can suppose that p = 1 ∈ ∂S and we rescale about p = 1. Set δ = ∆Q(1) and z = This means that Ψ n (z, w) = Ψ n z + it 6.2. Setup. In the translation invariant case Ψ(z, w) = Φ(z +w) we will consider the restriction J := Φ| R , i.e., Observe that a function V(z), which is translation invariant in the sense that V(z + it) = V(z) for all z ∈ C and t ∈ R, will satisfy that ∂V = 1 2 ∂ x V. Since R(z) = Φ(z +z) is translation invariant, the mass-one inequality (4.13) can be written It is an easy matter to reformulate Ward's equation in terms of J: Proof. Set G(z) = P(Re z/2) and L(z) = D(Re z/2) in Proposition 4.3, where we recall that D(z) is defined by the integral (4.9).

Fourier inversion as analytic continuation.
We shall presently see that the mass-one inequality (6.1) implies a special structure of the function J. To this end, we shall use Fourier analysis. We shall use the restriction of Bargmann's kernel to R, We shall use the Fourier transform with normalization This means that F f * g = √ 2πfĝ where "*" is the usual convolution product.

Lemma 6.4. Suppose that a function V(x) satisfies the mass-one inequality (6.1) and the integrability condition
Then there is a function f ∈ L 2 + L ∞ (R) such that V = γ * f . Proof.
The assumption (6.4) and the mass-one inequality imply that In particular V ∈ L 2 (R). Letf be the function on R defined by the equation We claim that Indeed, by Plancherel's theorem, we have that Now denote χ = 1 (−1,1) , χ * = 1 − χ, and let f be defined by Note that the definition (6.5) off and the estimate (6.6) shows that the functionsf χ * and t → tf (t)χ(t) are both in L 2 (R). We can thus consider f as a tempered distribution. We now contend that, in the sense of distributions, Indeed, (6.8) is equivalent to that It now follows that where we used (6.5) for the last equality. We have thus verified (6.8). From (6.8) we now conclude that there is a constant C such that V(x) = γ * f (x) + C. Absorbing this constant into f , we get a representation V = γ * f. It remains to prove that f ∈ L 2 + L ∞ (R).
But by the definition (6.7), it is clear that we have a decomposition f (x) = l(x) + h(x) + C where l and h are in L 2 (R) and C is a constant. This implies that V = γ * f = γ * l + γ * h + C.
Since V and γ * l are bounded, it follows that γ * h ∈ L ∞ (R). This implies that h ∈ L ∞ (R), because there is a constant C such that The proof of the lemma is finished.
Remark. The proof shows that a representation V = γ * f where f = f 2 + f ∞ with f p ∈ L p for p = 2, ∞, can be accomplished where furthermoref ∞ is compactly supported.
Remark. There is a shorter way to prove Lemma 6.4 in the case when We know that F 0 satisfies the mass-one equation by Theorem 4.6, and V satisfies the mass-one inequality by assumption. Since moreover W ∈ L 1 (R) this implies In view of Plancherel's theorem, this gives This shows that the function h such thatĥ(t) =Ŵ(t) e t 2 /2 belongs to L 2 (R). Finally, we have W = γ * h and V = W + F 0 = γ * h + 1 (−∞,0) .
Let V be a function satisfying the mass-one inequality and the condition V − V 2 ∈ L 1 (R). For example, V = J will do. Using Lemma 6.4 we can interpret integrals like I(z) = R e iztV (t) dt. To do this, we use Lemma 6.4 to represent V = γ * f where f ∈ L 2 + L ∞ (R) is regarded as a tempered distribution. We then interpret I(z) to mean the value of the distributionf applied to the (Schwarz) test function t → e izt e − t 2 /2 . The Fourier inversion theorem then implies that the functions Ψ and J, defined in Section 6.2, obey (6.9) Φ(z) = 1 √ 2π R e iztĴ (t) dt.

The Fourier transform and Ward's equation.
Having dealt with the mass-one inequality, we shall now formulate Ward's equation in terms of Fourier transforms. We start with a couple of elementary lemmas. Proof. A Taylor expansion of r → e ir(te iθ +se −iθ ) around r = 0 gives If n is odd, the zeroth Fourier coefficient of (te iθ + se −iθ ) n vanishes, while if n is even, then 1 2π 2π 0 te iθ + se −iθ n dθ = (st) n/2 n n/2 .
We have shown that finishing the proof of the lemma. Lemma 6.6. For all s, t ∈ C we have Proof. Fix s and write I(t) for the left hand side in (6.10). Then I(0) = 0 and Lemma 6.5 shows that I (t) = ie −st . It follows that The proof of the lemma is complete.
Applying the identity (6.9) and Lemma 6.6, we conclude that Multiplying these identities together, we find that Now recall the expression for the function L(x) in (6.3). Using (6.11) and Lemma 6.6 we have Next note that the relation J = G + 1 (see (6.2)) means that where δ s is the Dirac measure at s. Inserting this in the last expression for L(x) we get where we have used that 1 2π and also that In view of Lemma 6.3, Ward's equation is equivalent to that L = GJ − J . Comparing with the last expression for L(x) we have arrived at the following result. R (see Section 6.2). By Lemma 6.4 we can represent J = γ * f where f ∈ L 2 + L ∞ (R). Let g be a continuous function on R such that g = f − 1; this determines g up to a constant. Let us define G = g * γ. Then By Lemma 6.7, Ward's equation is equivalent to the identity (6.12) for a suitable choice of integration constant for g. We can rewrite (6.12) in the form This means that g f = 0 in the sense of distributions and hence as measurable functions. Let Then E is a closed set, and the complement E * = R \ E can be written as a countable union of disjoint open intervals I j . On each I j we have f = 0 and g = −1 almost everywhere. Since g = 0 at the endpoints, none of the intervals can be finite. Hence E is connected. Differentiating the relation f g = 0 and using g = f − 1 we obtain that f = f 2 when f 0. Hence f = 1 E almost everywhere. We have shown that Φ is representable in the form The proof of Theorem D is finished. q.e.d.
6.6. Proof of Theorem E. Assume that Q is radially symmetric and let {ζ j } n 1 be a random sample from the corresponding point process. The droplet S is then bounded by a family of concentric circles. We can assume that T = { ζ ; | ζ | = 1 } is such a circle, and that S is on the inside of T. We can also assume that ∆Q(p) = 1 for all p ∈ T. For an arbitrary point p = e iθ ∈ T we rescale in the outer normal direction, z = e −iθ √ n ζ − p .
By radial symmetry, the law of the rescaled process Φ n = {z j } n 1 is independent of p, so we can unambiguously denote by R n (z) = R n (ζ) the one-point function rescaled about p. Take C > 0 (large) and define the C/ √ n-neighbourhood of T by Γ n := T + D 0; C/ √ n . Now suppose that f is smooth and f = 0 on T. Consider the integral By Theorem A and B (and our assumption that ∆Q = 1 on T) there is a subsequence n k such that for some function R > 0. (∂/∂n is differentiation in the exterior normal direction.) Moreover, by Theorem D and Lemma 6.2, we know that R satisfies (6.14) R = 1 I * γ, R(−∞) = 1, R(+∞) = 0.
Here I ⊂ R is a connected set; the conditions (6.14) readily imply that I = (−∞, a) for some a ∈ (0, ∞]. Hence Clearly, we must prove that a = 0. Let us first prove that (6.16) Indeed, let X be a standard normal and note that F 0 (t) = 1 2 P (| X | ≥ t) where P is the law of X. It follows that for differentiable functions ϕ the expectation of ϕ It is now seen that ∞ 0 tF 0 (t) dt = 1 4 E X 2 = 1/4, proving (6.16).
By (6.16), we have R t(F 0 (t) − 1 (−∞,0) (t)) dt = 1 2 , and consequently, by (6.13) and (6.15), We shall now apply a result from [3], concerning the random variable fluct n f given by Here f is a smooth test-function and dσ = ∆Q·1 S dA is the equilibrium measure; the point ζ = (ζ j ) n 1 is picked randomly with respect to P n . Write ν n ( f ) for the expectation of fluct n f , i.e., Now suppose that f = 0 on T and that f vanishes in a neighbourhood of all other boundary components of S. The result from [3] then implies that Put δ(ζ) = | | ζ | − 1 | and write δ n = log n/ √ n. We choose C large and define regions S j = S j,n by Let g be any smooth function supported sufficiently close to T and put f (ζ) = g(ζ) ( | ζ | − 1 ). Then For ζ ∈ S 2 , Theorem 5.4 gives the estimate R n (ζ) = n∆Q(ζ) + O ne − nδ(ζ) 2 /2 , giving The functional ν (3) n ( f ) coincides precisely with µ n,C ( f ) introduced above, so, by (6.17), Finally, for ζ ∈ S 4 , we have R n (ζ) ≤ Cne − nδ(ζ) 2 for δ(ζ) < δ n and R n (ζ) ≤ Cne − n(Q−Q)(ζ) for δ(ζ) ≥ δ n . See Lemma 5.5 and the estimate (5.10). This gives Summing and passing to limits first as k → ∞, then as C → ∞ we obtain the estimate On the other hand, by (6.18) we have the estimate It follows that T(a) ≤ 2π(C 1 + C 2 ) S | g | dA for all g such that g = 1 on T. This implies that T(a) = 0 and hence a = 0. q.e.d. 6.7. Proof of Theorem F. Using Lemma 6.5 and the assumption that Ψ(z, w) = Φ(z +w), we can rewrite the mass-one equation (equality in (3.5)) in terms of the function J = Φ| R , as follows in the sense of distributions. Passing to Fourier transforms we find that the mass-one equation is equivalent to that (with δ the Dirac delta function) This identity is equivalent to 1 E = 1 2 E , proving that the function J = γ * 1 E satisfies the mass-one equation. It is also clear that J(1 − J) ∈ L 1 (R).
(II) If Ψ(z, w) = Φ(z +w) satisfies the mass-one equation and the integrability condition J − J 2 ∈ L 1 (R), then by Lemma 6.4 we can represent J = γ * f where f ∈ L 2 + L ∞ (R). The same calculations as above with 1 E replaced by f now lead to the equation Passing to Fourier transforms the last condition becomes equivalent to that f (x) = f (x) 2 almost everywhere. Hence f = 1 E almost everywhere where E is some measurable set of positive measure, and Φ = 1 E * γ. The proof of Theorem F is finished. q.e.d.

Singular boundary points
In this section, we fix a boundary point p ∈ ∂S. We rescale according to the convention in Section 1.2 and let K = GΨ denote a limiting kernel in Theorem A. Write R(z) = K(z, z) and R n (z) = K n (z, z).
We shall prove that if p is singular (i.e. a cusp or a double point), and if < 1 is given, then we have the asymptotic estimate which is part (ii) of Theorem C.
We shall also prove non-triviality, i.e., that if p is "strongly laminar", then R(z) > 0 everywhere. In this way, we will in particular complete the proof Theorem B.
We remark that the proof of non-triviality will be conducted for points p on the outer boundary of S. (I.e., the boundary of the unbounded component of S c .) However, this is just a matter of convenience; any other boundary component can be treated likewise. 7.1. Exterior estimates near singular boundary points. We now prove the estimate (7.1), i.e., part (ii) of Theorem C. We shall use the estimate (5.10), viz.
If p is a double point (or a regular point), then the argument in Lemma 5.5 shows that the estimate (7.1) holds. It remains to treat the case of a cusp on the outer boundary of the droplet.
Let H be the upper half plane in C and assume that p = 0 is a cusp on the outer boundary, the cusp pointing in the positive real direction. The rescaling is then simply given by and in the z-plane, the droplet appears as a narrow neighbourhood of the negative imaginary axis (−i∞, 0]. Write U for the unbounded component of C\S and let Φ : H → U be a conformal map such that Φ(0) = 0 and Φ(i) = ∞. Since 0 is a conformal cusp, Φ extends analytically to some neighbourhood N of the origin. Likewise, the harmonic functionQ • Φ : H → R extends harmonically across R to a harmonic function V. Referring to (7.3), we shall put We can assume that Φ has the Taylor expansion Φ (λ) = λ + a 2 λ 2 + a 3 λ 3 . . . , (λ ∈ N).
We form the functions The functionQ Φ is harmonic in H and extends across R to a harmonic function V. Write Lemma 7.1. For λ = σ + iτ, we have Before we prove the lemma, we proceed to prove the estimate (7.1). To this end, note that the estimate (7.2) gives (with a new C depending on ∆Q(p)) The estimates (7.4) and (7.6) now give that nM (λ n (z)) = 2x 2 + O n | λ n (z) | 3 , (n → ∞).
Choosing, for example, | z | ≤ log n, we see via (7.5) that the estimate (7.1) holds, thus proving part (ii) of Theorem C.
It remains to prove Lemma 7.1.
Recalling thatQ(λ) ∼ log | λ | 2 as λ → ∞, it is now seen that the Poisson representation of the harmonic functionQ Φ | H takes the form where G(λ) = log λ+i λ−i 2 is (twice) the Green's function for H with pole at i, and Let us write C ω for the real-analytic class. For a function f ∈ C ω (R) we write P f (λ) = R f (t)P(λ, t) dt for the Poisson integral. Noting that, for λ = σ + iτ ∈ H, we compute Note that the last integral approaches 1 2 f (σ) as τ ↓ 0. Let us denote by S σ : C ω (R) → C ω (R) the backward shift by σ: A repetition of the calculation in (7.8) gives that P f has the asymptotic expansion Choosing f = Q Φ and using identity (7.7), we find More generally, it is easy to verify by induction that Since ∂Q Φ is continuous on cl H, while Q Φ =Q Φ in the lower half plane, we have that Inserting the expansions (7.9) and (7.10), we thus find that The last line vanishes because all coefficients are derivatives of ∆G evaluated at σ. This implies that The proof of Lemma 7.1 is complete, and so is the proof of Theorem C.

Second HK estimate.
We reformulate the first HK estimate (Theorem 5.1) in terms of the polynomial kernel k n and its approximation k # n which is given by (see Section 3.3).
We will apply these remarks in the proof of the following result.
Proof. We can assume that T = 1 so δ = 1/ √ n. By (7.11) there is a constant C such that k m,η (η) − π m χ η · k # m,η (η) ≤ C mk m,η (η) e mQ(η)/2 , so, by (7.12), we need only to verify that However, the left hand side in (7.13) equals to which by the Cauchy-Schwarz inequality is no larger than The proof will thus be finished when we can verify that (7.14) To prove this, we suppose that η = Q(η) = 0 and drop subscripts. Recalling that k # n (ζ) = n ∂ 1∂2 A (ζ, 0)e nA(ζ,0) , we estimate as follows: where we have used (7.12) to get the last inequality. By our choice of δ we have | ζ | ≤ 4/ √ n when χ(ζ) 0. For such ζ we also have where we have used (7.12) and A(0, 0) = 0. Using that | z | ≤ C implies | e z − 1 | ≤ C | z | for suitable C , we infer from the last two estimates that Since The right hand side can be estimated using the remark after Lemma 3.4, giving The last integral is dominated by a constant multiple of n, proving (7.14) in the case Q(η) = 0. Replacing Q(ζ) with Q(ζ) − Q(η) gives the general case.

Hele-Shaw flows and the second BP estimate.
For a fixed positive number t (close to 1), we let σ t be the unique measure of total mass t such that I Q [σ t ] = min µ(C)=t I Q [µ] (over µ ∈ P). We denote S t := supp σ t and note that if Q is smooth in a neighbourhood of S t , then S t = S [Q/t] and dσ t = ∆Q · 1 S t dA.
The droplets S t form an increasing chain known as (weighted) Hele-Shaw evolution; cf. [26], [31]. Let U t be the component ofĈ \ S t containing ∞. We will denote by ω t the harmonic measure of U t evaluated at ∞. This means that ω t is the measure on ∂U t such that ∂U t h dω t = h(∞) where h is harmonic in U t and continuous up to the boundary.
Definition. Let p ∈ ∂S be a point on the outer boundary.
(i) We say that p is laminar if there exists a neighbourhood N of p and a number > 0 such that the set γ t := ∂S t ∩ N is either a regular, real-analytic arc, or, if p is a double-point, the union of two disjoint regular, real-analytic arcs, whenever 1 < t < 1 + . (ii) We say that a laminar boundary point p is strongly laminar if there is a constant C such that, when 1 < t < 1 + , the arc-length measure s t on γ t satisfies (7.15) s t ≤ Cω t . Strong laminarity is used to avoid fjord-like shapes and turbulence. It is a case of folk-lore that the boundary of a droplet corresponding to an algebraic potential is strongly laminar. Here, a potential is called algebraic, if there is a neighbourhood Ω of the droplet where Q is strictly subharmonic and of the form where and ∂H are rational functions.
Remark. The strong laminarity of algebraic potentials has been mentioned in various forms, see e.g. Chapter 6 in [17]. The reason for it is that, in the algebraic situation, Hele-Shaw flow takes place in a finite-dimensional space of real algebraic curves, whereas turbulence is an infinite-dimensional phenomenon. A formal way to organize a proof, e.g. under the simplifying assumptions that (ζ) = ζ and the droplets are simply connected, is as follows. Our assumption means that U t =Ĉ \ S t is a quadrature domain with quadrature function h = ∂H, see [31]. One can then determine a rational conformal map ϕ t of ∆ = {|ζ| > 1} onto U t . Combining this with the Polubarinova-Galin equation (see [25]) for the evolution of the maps ϕ t , one can show that the flow t → ϕ t is a continuous map [1, 1 + ) → C 1 (T) for some > 0.
The next result is known as Richardson's formula. We include a proof for completeness.

Lemma 7.3. If t < t , then for all functions h harmonic in U t and continuous up to the boundary, we have
Proof. By subtracting a constant from h, we can assume that h(∞) = 0. Using the properties of equilibrium potentials in Section 3.4, we obtain by means of Green's identity that (HereQ t is the obstacle function corresponding to Q t ; see Section 3.4.) Subtracting the corresponding identity with t replaced by t , we find that By Richardson's formula, we have for a continuous function f on C that d dt C f dσ t = h(∞), where h is the harmonic function on U t which equals f on ∂S t . In other words, where ω t is harmonic measure of U t evaluated at ∞. Our next theorem combines Hörmander estimates with Laplacian growth. Fix a strongly laminar point p ∈ ∂S on the outer boundary. Replacing Q by a constant multiple if necessary, we can assume that ∆Q(p) = 1.
We also fix a point w ∈ C and put We shall need the following lemma concerning Hele-Shaw flows.
Lemma 7.5. Let p be a strongly laminar point on the outer boundary of S. Then for any given number ρ > 0 there is a number M > 0 such that Proof. Fix positive numbers ρ and M. We can assume that the sets S t and D p; 4ρ/ √ n are contained in a small fixed neighbourhood Ω of S. Fix a number K > 0 such that ∆Q ≤ K in Ω. Now suppose that Then D p; ρ/ √ n S τ for any τ ∈ [1, t). This implies that the curve γ τ := ∂S τ ∩ D p; 2ρ/ √ n has length at least ρ/ √ n. By (7.15) there is a constant C such that Let f be a continuous function on C with 0 ≤ f ≤ 1, such that f = 1 in D p; 2ρ/ √ n and f = 0 outside of D p; 4ρ/ √ n . By the identity (7.16) and the inequality (7.18), we then have This implies that M ≤ 16CKρ. The inclusion (7.17) thus holds whenever M > 16CKρ.
Proof of Theorem 7.4. Let M and T be positive constants to be fixed later. Put m = n + M √ n (the integer part) and define a strictly subharmonic function by Now put ρ = | w |+2T. By Lemma 7.5, u is then supported in S t , where t = 1+M/ √ n, provided that the constant M = M(ρ) is chosen sufficiently large. Furthermore, u is holomorphic in D η; 2δ .
By the estimate (5.5), and since n∆φ ∼ n∆Q on S t , we then have The estimate in Lemma 5.3 means that there are constants C and c > 0 such that π ⊥ nφ u 2 nφ ≤ Cne − cnδ 2 e nQ(η) = Ce − cT 2 ne nQ(η) .
Combining these inequalities, we find that Subtracting a constant from Q does not affect the one-point function R m (η) = k m,η (η)e − mQ(η) , so we can here assume that Q η = 0. It follows that, if m is large enough, then R m η ≥ cn with c equal to the square-root of can be chosen positive and independent of w in a given compact subset of C. We have shown that each limit point R of the rescaled 1-point functions R m satisfies R > 0 on C. The proof of Theorem B is thereby completely finished. q.e.d.

Concluding remarks
In this section, we comment on the nature of Ward's equation and the mass-one equation in the general (non translation invariant) case, relating those equations to harmonic analysis on the Heisenberg group. We also explain how the technique of rescaling in Ward identities can be applied in several settings different from the one we have studied hitherto. Namely, we will derive Ward's equation for the random normal matrix model with a hard edge spectrum, for certain types of bulk singularities, and for so called β-ensembles. Finally, we will mention some connections to the theory of Hilbert spaces of entire functions, and to the theories of certain special functions.

Twisted convolutions.
For two functions f, g defined on C, the twisted convolution f g is defined by See the book [18]. We will show that Ward's equation and the mass-one equation have precisely the form of twisted convolution equations. In the translation invariant case, the equations reduce to ordinary convolution equations, which is how we were able to solve them. However, the general twisted case is certainly more interesting.
Consider the following transform Letting F be two-dimensional Fourier transform with normalization we then havef (t) = 2F [ f ](2t), and the inverse Fourier transform takes the form Let K = GΨ denote a limiting kernel in Theorem A and write R(z) = Ψ(z, z). Using the transform in a proper generalized sense, we expect (compare Section 6.3) that there is a function f , such that Heref is understood in the sense of tempered distributions. Under these conditions, we can assert the following analogue of the identity (6.9).
Now define R 0 = 1 − R, and assume that we, can represent R 0 in a similar way to (8.1) where g is a suitable function. Lemma 8.1 then allows us to rewrite the mass-one and Ward equations as follows.
Mass-one equation. Compare with Section 6.7 for the translation invariant analogue.
Ward's equation. There exists a smooth function P 0 such that∂P 0 = R 0 and Compare with Lemma 6.7.
Note that both equations take the form Using the assumptions thatR andR 0 can be represented as in (8.1) resp. (8.2), we can write them in the form of twisted convolution equations, f ĝ = 0.

Ward's equation at the hard edge of the spectrum.
For simplicity, we shall restrict our discussion to the hard edge Ginibre ensemble; we refer to [5] for a discussion of more general hard edge ensembles. Let {ζ j } n 1 be the hard-edge Ginibre process and rescale about the boundary point p = 1 to obtain the boundary process Θ n = {z j } n 1 , where z j = √ n ζ j − 1 . As before, we let R n (z) = K n (z, z) denote the 1-point function of the rescaled process. The hard edge Berezin kernel and Cauchy transform are defined, respectively, by with the understanding that B n (z, w) = 0 when the point ζ = 1 + z/ √ n satisfies |ζ| > 1. We recall that the hard edge kernel is defined by where H is the hard edge plasma function, H = γ * . In terms of this kernel, we put Observe that R(z) = B(z, w) = 0 when Re z > 0.
Proof. We claim first that we have the asymptotic relation where the error term o(1) converges to zero uniformly on compact subsets of the left half plane L.
In order to prove this, it is convenient to consider the Ginibre potential Q(ζ) = | ζ + 1 | 2 which has the droplet S = { | ζ + 1 | ≤ 1 }. We rescale about the boundary point p = 0 according to fix a number ε > 0. Write U := S ∩ D(0; ε) and consider test functions ψ supported in the dilated set √ n · U. As in the free case we define ψ n (ζ) := ψ (z). Since Q S = Q in the set U where ψ n is supported, the same arguments used in the free boundary case remain valid (cf. Section 4.2). The only difference is that the dilated domains √ n · U will, in our present case, increase to the open left half plane L. Hence we deduce the Ward's equation (8.4) for z ∈ L precisely as before.
By Theorem 2.3, we have convergence R n → R and C n → C locally uniformly in L and boundedly almost everywhere in C. It follows that we can pass to the limit in (8.4). The proof is complete. Corollary 8.3. H satisfies the following "hard edge mass-one equation", Proof. The approximate Berezin kernels B n satisfy B n (z, w) dA(w) = 1 for z ∈ L. The identity (8.5) now follows from the convergence B n → B in Theorem 2.3 and the argument used in the foregoing proof.

Ward's equation at bulk singularities and Mittag-Leffler fields.
Let us weaken our standing assumptions on the potential Q. We still require real-analyticity in a neighbourhood of S, but now allow that ∆Q = 0 at isolated points in the bulk of S. A point p ∈ Int S such that ∆Q(p) = 0 will be called a bulk singularity. Assume that p = 0 is a bulk singularity and let {ζ j } n 1 be the point process corresponding to Q. The effect of the bulk singularity is to repel the particles away from it.
There are various types of bulk singularities depending on the local behaviour of ∆Q near p.
where a and b are positive constants, then the local behaviour of the system {ζ j } near 0 will depend on a as well as b. We will here only consider the simplest case when a = b, and more generally, that there is a number λ ≥ 1 such that where the dots represent negligible terms. If we wish Q to be real-analytic, we should of course assume that λ be an integer. However, the condition of real-analyticity is important only in a neighbourhood of the boundary, e.g. in connection with Sakai's theory. In the bulk it suffices to assume C 2 -smoothness. Thus we can in fact we can choose λ as an arbitrary real constant ≥ 1.
Note that λ = 1 is the well-known case of an ordinary "regular" bulk point, in which case we know that the usual Ginibre point field arises. We may thus assume that λ > 1. It turns out that the proper scaling in the case at hand is We write Θ n = {z j } n 1 for the rescaled system, equipped with the law which is the image of the Boltzmann-Gibbs measure P n under the map (8.6).
Theorem 8.4. The point process Θ n converges as n → ∞ to the unique point-field ML λ in C with kernel The convergence holds in the sense of locally uniform convergence of intensity functions.
Proof. It is easy to see that M λ is of exponential type λ. This implies that the kernel K is uniformly bounded. Existence and uniqueness of a point field ML λ with the given properties now follows, via Lenard's theory, from the convergence of intensities in the preceding example (cf. the appendix).
We next consider Ward's equation at p = 0 for the the potential Q λ (ζ) = | ζ | 2λ . To this end, we introduce the Berezin kernel rescaled about 0 on the scale (8.6), i.e.
Ward's equation takes the following form.
Theorem 8.5. In the above situation, B n → B uniformly on compact subsets of C 2 where B is a solution to the "Ward equation of type λ" Proof. We shall first establish the asymptotic relation where o(1) → 0 as n → ∞, uniformly on compact subsets of C.
To this end, fix a test-function ψ and let ψ n (ζ) = ψ (z) where z = n 1/2λ ζ. We shall use Ward's identity; we therefore recalculate the expectations of the terms I n [ψ n ], II n [ψ n ] and III n [ψ n ] used in the free boundary case, in Section 4.2. As customary, we use the symbol R n,k to denote the k-point function of the system {ζ j } n 1 . The rescaling z j = n 1/2λ ζ j then implies that the k-point function of the rescaled system {z j } n 1 is R n,k (z 1 , . . . , z k ) = n −k/λ R n,k (ζ 1 , . . . , ζ k ) .
In view of Ward's identity (Section 4.1) we now infer that, in the sense of distributions, Dividing by R n,1 and applying∂, we conclude the proof of the formula (8.8). To pass to the limit as n → ∞, we now use the convergence in the example preceding Theorem 8.4 and the argument in Section 4.3.
Remark. Consider a potential Q(x + iy) = p(x, y) where p is a polynomial in x and y, positive definite and homogeneous of degree 2k where k is a positive integer. Write ζ = x + iy and rescale by z = n 1/2k ζ. As in the above proof, one deduces without difficulty the asymptotic relation Another equation of the type (8.9) was studied in the paper [4].
We now consider "kernels" of the form where E is an entire function, real and positive on (0, ∞). We refer to B E as a (generalized) Berezin kernel of type λ (or of the "second kind" as in [4]). We say that the entire function E satisfies the mass-one equation of type λ if The following theorem appears in the paper [4]. Finally, let Q be a potential of the form Q(ζ) = ζ − p 2λ + . . . where p ∈ Int S and the dots represent negligible terms near ζ = p. Let Θ n be the corresponding process rescaled about p by a factor n 1/(2λ) about p. We conjecture that lim n→∞ Θ n = ML λ in the sense of point fields.

The mass-one equation and Hilbert spaces of entire functions.
In this section, we shall interpret the mass-one equation as the reproducing property in a suitable space of entire functions. As a consequence, we shall find non-trivial relations for the functions F, H, and M λ .
It has been observed (e.g. [35], [36], [9] and the references there) that universality laws in the theory of random Hermitian matrices are related to certain specific de Branges spaces B(E) of entire functions. See [12] for the definition of these spaces. In particular, the sine-kernel describing the spacing of eigenvalues in the bulk is the restriction to R 2 ⊂ C 2 of the reproducing kernel of the Paley-Wiener space, i.e., the space B(E) where E(z) = e −iπz . Moreover, the Airy kernel (see (0.4)) which describes the spacing at the edge of the spectrum is the restriction to R 2 of the reproducing kernel of B(E) where E = Ai −i Ai, and the Bessel kernel (hard edge) is the restriction to R 2 − of the reproducing kernel of the de Branges space corresponding to the function .) The appearance of de Branges spaces in the context of Hermitian random matrices is quite natural given the fact that orthogonal polynomials on the real line can be related to a second order one-dimensional self-adjoint spectral problem.
The Hilbert spaces H of entire functions arising in the random normal matrix theory are not of de Branges type, and we are not sure about their spectral interpretation. Nevertheless, we will use the term "spectral measure": µ is a spectral measure for H if H sits isometrically in L 2 (µ).
Lemma 8.7. Let Ψ be a Hermitian entire function and G = G(z, w) the Ginibre kernel. The following conditions are equivalent.
(i) The kernel K = GΨ satisfies the mass-one equation, i.e., (ii) The function L(z, w) = e zw Ψ(z, w) is the reproducing kernel of some Hilbert space H with spectral measure dµ(z) := e − | z | 2 dA(z). If this is the case, then there is a unique point field with correlation kernel K.
Proof. Write L w (z) = L(z, w) = e zw Ψ(z, w). The function L is the reproducing kernel for a Hilbert space with spectral measure dµ(z) = e − | z | 2 dA(z) if and only if (8.12) L w , L z L 2 (µ) = L(z, w), z, w ∈ C.
For z = w, the identity (8.12) means that which is precisely the mass-one equation (8.11). On the other hand, if the last equation holds, then (8.12) follows for z w by analytic continuation. This proves the equivalence of (i) and (ii). Next note that the kernel K = GΨ can be written From this we conclude that if L gives rise to a reproducing kernel as in (ii), then K is the reproducing kernel of the subspace W = f ; f (z) = g(z)e − | z | 2 /2 of L 2 .
Consider the integral operator T on L 2 with kernel K. It is easy to check that this operator satisfies the following conditions: T is a Hermitian operator which satisfies 0 ≤ T ≤ 1, and is locally trace class. (That an operator T on L 2 is "locally trace class" means that the operator T B on L 2 defined by T B ( f ) = T(1 B f ) is trace class for every compact set B ⊂ C.) By a theorem of Soshnikov (Theorem 3 in [48]), the conditions above guarantee that K is the correlation kernel of a unique random point field in C.
In the following, we write A 2 (µ) for the space of all entire functions of class L 2 (µ). It follows from general facts for reproducing kernels that the Hilbert space H in (ii) is the following closed linear span Let us look at some examples. For the (bulk) Ginibre process we have L(z, w) = e zw , and hence H = A 2 (µ) is Fock space. The free boundary Ginibre process corresponds to the kernel L(z, w) = e zw F(z +w), and hence where F is the free boundary plasma function. One can similarly interpret the hard edge mass-one equation (8.5) as a reproducing property in a suitable space of entire functions. In fact, this space is where H is the hard edge plasma function. The fact that the last span consists of entire functions requires a compactness property in the hard edge situation, which will be established in the paper [5].
It would be interesting to describe the above spaces in more constructive terms (e.g. similar to de Branges theory). It would also be interesting to know the meaning of Ward's equation for the spaces H. (By Lemma 8.7, the mass-one equation is a statement about spectral measures.) We finally describe the Hilbert spaces corresponding to the Mittag-Leffler processes. To this end, recall (Theorem 8.6) that the mass-one equation for the function M λ says that This gives, by polarization (This formula has an alternative, elementary proof: insert M λ (ζ) = λ ζ j /Γ j+1 λ in the left hand side and integrate termwise.) Let dµ λ (z) = e − | z | 2λ dA(z). The Hilbert space pertaining to the process ML λ is thus It is not hard to show that polynomials are dense in H, and consequently H = A 2 (µ λ ).
Remark. For a Borel set e ⊂ R of positive measure, let Thus F (−∞,0) is the free boundary plasma function F. By Theorem F we know that the kernel K e := GΨ e , Ψ e (z, w) = F e (z +w) satisfies the mass-one equation (8.11). The corresponding Hilbert space is H = span L 2 (µ) z → e zw F e (z +w); w ∈ C , (dµ(z) = e − | z | 2 dA(z)), and the weighted version W, is the closed linear span in L 2 of the kernels K w (z) = G(z, w)F e (z +w).
Then K w is the reproducing kernel in W, i.e., This can be regarded as a polarized version of the mass-one equation. At the same time, (8.13) gives a quite non-trivial relation for the function F e . The positivity property of the kernel K(z, w) = G(z, w)F e (z +w) also implies non-trivial inequalities. We give a few examples in the case e = (−∞, 0).
Letting y = 0 this gives the inequality F(x) − F(x) 2 ≥ e − x 2 /4, (x ∈ R). We also mention the following polarized form of the hard edge mass-one equation L G(t, z)H (t +z) G(w, t)H (w +t) dA(t) = G(w, z)H(w +z), z, w ∈ C.
In a similar way as for F, one can explore some consequences of the inequality | H(z +w) | 2 ≤ e |z−w | 2 H(z +z)H(w +w) for z, w ∈ L, which follows from the positivity of the hard edge correlation kernel. We note first that Setting w = 0 in the preceding inequality now gives that Letting z = 0 shows that the estimate is sharp. As a final example, we note that the positivity of the Mittag-Leffler kernel implies that Remark. Consider the "Hele-Shaw" case when Q is of the form where H is holomorphic in a neighbourhood of S. The Hermitian analytic continuation A(ζ, η) of the function A(ζ, ζ) = Q(ζ) is then obviously A(ζ, η) = ζη + H(ζ) +H(η). Assume that p = 0 is a regular boundary point of S and that the exterior normal to ∂S at 0 is directed towards the positive real axis. Let k n be the reproducing kernel for the space P n of polynomials of degree ≤ n − 1 with the weighted norm of L 2 nQ . Let us write P n e −nQ for this subspace. The corresponding rescaled kernel k n is given by This is the reproducing kernel for the space P n e −Q n where Now recall that the function Ψ provided by Theorem A is a limit point of the normal family See Section 3.7. Put L n (z, w) := Ψ n (z, w)e zw = k n (z, w)e −H n (z) e −H n (w) .
If f = q · e −H n for some q ∈ W n , then The kernel L n is therefore the reproducing kernel for the subspace H n := f = qe −H n ; q ∈ P n of L 2 Q n −H n −H n = L 2 (µ) (where dµ(z) = e − | z | 2 dA(z)). It is natural to expect that a subsequential limit Ψ = lim Ψ n k is the reproducing kernel of some space with spectral measure µ.
Finally, consider the Ginibre potential Q(ζ) = | ζ − 1 | 2 . The droplet is S = { | ζ − 1 | ≤ 1 }, so p = 0 is a boundary point. Moreover, Q is a Hele-Shaw potential, and the corresponding functions H and H n from (8.15) resp. (8.16) become H(ζ) = −ζ and H n (z) = − √ nz. In this case, H n is entire, so the space H n becomes the subspace H n = f ; f (z) = q(z)e √ nz , q ∈ P n ⊂ L 2 (µ). 8.5. Sections of power series. It seems that the type of asymptotics one encounters for the free boundary was first observed in connection with sections of power series of the exponential function. By a section of an entire function a j ζ j we here simply mean a partial sum s n (ζ) = n−1 j=0 a j ζ j .
Szegő's original study in [49] concerns the distribution of zeros of the blow-up sections s n (w) := s n (nw) pertaining to the exponential function f (ζ) = e ζ . In the course of the investigation, Szegő proves asymptotic results for the function s n (w) valid for all w except for w in a fixed neighbourhood of 1. This gap was later closed, and the following result ensued. Consider the rescaled sectioñ s n (z) := s n (n ζ) , z = √ n (ζ − 1) .
One then has the following (locally uniform) convergence where F is the free boundary plasma function. We are unsure concerning whom should be credited for the convergence in (8.17) when f (ζ) = e ζ . However, the book [15] (Theorem 1) contains a statement valid for more general f , and the appendix in [8] contains a very detailed convergence result for the case at hand. To see the connection with the scaling limits of the present paper, we remind of the expression for the correlation kernel for the Ginibre ensemble from Section 2, Noting that and rescaling via we now recognize that K n (z, w) =s n (z +w + o(1)) e −n e − √ n Re(z+w) · e − | z | 2 /2 − | w | 2 /2 .
By analytic continuation, one can now recover the limit in (8.17). The convergence in (8.17) has been proved for the sections corresponding to more general entire functions. In the monograph [15], the authors consider the Mittag-Leffler function E 1/λ as well as a class denoted "L-functions", while Edrei has supplied even further examples, see e.g. [14]. In each case the authors prove a suitably rescaled version of the limit in (8.17).
To interpret the above results in terms of our Theorem E, one chooses a suitable radially symmetric potential Q. For example, one chooses Q(ζ) = E 1/λ (| ζ | 2 ) in case of the Mittag-Leffler function alluded to above. Expressing the kernel K n in terms of the orthogonal polynomials (as in (2.1)) and rescaling about a boundary point of the droplet, one can apply Theorem E and recover the asymptotic behaviour of the sections. 8.6. The Ginibre-limit in the bulk. We have hitherto been occupied with scaling limits at the boundary, and at bulk singularities. Our methods do however also apply equally well to the more familiar case of a "regular" point p in the bulk, i.e., a point where ∆Q(p) > 0.
For certain applications (e.g. in [6]) it advantageous to allow the point p to vary with n. We shall thus in general work with sequences p = (p n ) ∞ 1 rather than points. Let us write δ n = δ n (p) = dist p n , ∂S .
We say that p belongs to the bulk regime if all p n are in S and lim inf n→∞ √ nδ n = ∞. As customary, let {ζ j } n 1 denote a random sample from the ensemble associated with Q and let Θ n = {z j } n 1 , where (for any fixed θ ∈ R) z j = e −iθ n∆Q(p n ) ζ j − p n , j = 1, . . . , n.
We write K n for the correlation kernel of the rescaled process. By Lemma 4.5 this implies that R n → 1 uniformly on compact sets, which means that Ψ ≡ 1. We have shown that every limiting kernel equals to G, as desired.
Remark. The theorem holds also when Q is replaced by a smooth perturbationQ n = Q + h/n where h is a fixed smooth function. Indeed, it suffices to prove that the estimate (8.18) holds in this more general case. To see that this is the case, one can use the method in the appendix of [3]. where H n (ζ) = − j k log ζ j − ζ k + n Q(ζ j ) is the Hamiltonian in the external electric field Q.
(The parameter β is sometimes interpreted as an inverse temperature; the case β = 1 studied hitherto can be interpreted as the statement that the external field is magnetic.) Denote by (ζ j ) n 1 a point picked randomly with respect to C n and write {ζ j } n 1 for the corresponding unordered configuration. We denote the k-point function of this process by the superscript "β". Thus The function R β n,k can not be written as a determinant when β 1. However, we have the following version of Ward's identity. The proof in Section 4.1 works equally well for the present, more general situation. Theorem 8.9. Put, for a test-function ψ ∈ C ∞ 0 (C), W n [ψ] = β I n [ψ] − II n [ψ] + III n [ψ], where I n , II n , and III n are as in Section 4.1. If Q is C 2 -smooth near supp ψ, then E β n W + n [ψ] = 0. Now fix a point p ∈ S and a real parameter θ and rescale about p according to z j = e −iθ n∆Q(p) ζ j − p , j = 1, . . . , n.
We denote by Θ We can also rescale in the Ward identity in Theorem 8.9. The proof of the following theorem is merely a repetition of the argument in Section 4.2.
Theorem 8.10. If p belongs to some neighbourhood of S in which Q is strictly subharmonic and C 2 -smooth, then We do not know whether it is possible to pass to the limit as n → ∞ in (8.19), but for the sake of argument, let us temporarily assume that we can define a limiting Berezin kernel B β . Letting n → ∞ in (8.19), one then formally obtains the following generalization of Ward's equation This more general equation can easily be transformed to the case β = 1 by the linear scaling The result is the following. We do not know whether the presumptive kernels B β would be non-negative, so speaking about "mass-one" could possibly be misleading. However, if we assume that C B β (z, w) dA(w) = 1, then the corresponding kernel B in (8.21) satisfies the "mass-β equation": C B(u, v) dA(w) = β. Note also that our solution of Ward's equation in the case β = 1 depends on analyticity of solutions. We do not know if this is a valid assumption in general.
Remark. Consider the potential Q λ (ζ) = |ζ| 2λ where λ > 1 and rescale about the bulk singularity p = 0. The argument in Section 8.3 shows that Ward's equation for the system Θ β n takes the form

Appendix A. Convergence of point processes
A configuration {z i } is a finite or countably infinite "set" of points ζ i ∈ C with repetitions allowed. The configuration is said to be locally finite if for any compact set K ⊂ C there are at most finitely many i such that z i ∈ K.
Let X denote the set of all locally finite configurations {z j } in C. Let B be the σ-algebra generated by all "cylinder sets" C B n = { z ∈ X; N B (z) = n } where B ⊂ C is a Borel set and N B (z) = # j; z j ∈ B .
By a random point field, we mean a probability measure P on the measure space (X, B). Now suppose that Θ = (X, B, P) is a random point field. We write Θ = {z j } for a random sample from this distribution.
We say that a locally integrable function ρ k : C k → [0, +∞) is a k-point intensity function for Θ if . . , k m with k 1 + · · · + k m = k. Note that a point field does not necessarily possess intensity functions. Now let Θ n = {z j } n 1 be a sequence of point processes in C with intensity functions R n,k (see (1.2)). We shall say that Θ n converges to a point field Θ as n → ∞ if Θ has intensity functions R k of all orders and R n,k → R k as n → ∞ locally uniformly as n → ∞. (Exception: In the case of a sequence of hard edge processes Θ n , we say that they converge to Θ if the corresponding intensities converge with bounded almost everywhere convergence.) A.1. Existence of limiting point fields. We will use the following result due ro Lenard ([33], [34]). See also [48], Theorem 1, p. 926.
Theorem A.1. (Lenard) A sequence of locally integrable functions ρ k : C k → [0, +∞) can be represented as the intensity functions of some random point field Θ in C if and only if (i) ρ k z π(1) , . . . , z π(k) = ρ k (z 1 , . . . , z k ) for all permutations π of { 1 , . . . , k }, (ii) For any sequence ϕ k : C k → R of compactly supported bounded Borel functions and any N ≥ 0 the following implication holds: The n-point process {ζ j } n 1 associated with a potential Q is not a random point field, since the sample space is not X. However, for fixed n the functions n,k := R n,k (or even R β n,k ) satisfy condition (i) of Lenard's existence theorem (which is immediate) and also that {ζ j } has property P ( N ) whenever n ≥ N (take E N of the random variable in the left hand side of (A.3) and use the non-negativity of the n,k ).
Rescaling about some point z 0 , we find that the functions ρ n,k := R n,k satisfy the condition (i) in Theorem A.1, and also has property P (N) whenever n ≥ N. From this we conclude that if a locally integrable limit ρ k = lim n→∞ ρ n,k exists, with bounded almost everywhere convergence on compact subsets of C, then the limiting functions ρ k must satisfy conditions (i) and (ii) of Theorem A.1. Consequently they will be intensity functions of some random point field.
A.2. Uniqueness of limiting point fields. The following theorem gives a convenient condition for checking the uniqueness of a point field in C.
Theorem A.2. (Lenard) Let ρ k : C k → [0, +∞) be a sequence of intensity functions with respect to some random point field. Suppose that there is a number c such that We then have uniqueness of the random point field in C having ρ k for correlation functions.
For a proof, see [32], p. 42. The condition (A.4) is certainly satisfied for a determinantal point field if we can prove that its correlation kernel K is uniformly bounded. For if |K| ≤ C on C 2 , then the corresponding intensity functions satisfy R k ≤ C k k k/2 by the Hadamard inequality for determinants. Since |K(z, w)| 2 ≤ K(z, z)K(w, w) this gives the following simple uniqueness criterion.
Corollary A.3. Let K n be correlation kernels of the processes Θ n . If K n → K locally uniformly on C 2 and if R(z) := K(z, z) is bounded on C, then K is the correlation kernel of a unique point field Θ in C, and Θ n converges to Θ as point fields. The same conclusion holds if K n → K almost everywhere with bounded convergence on the diagonal in C 2 .