On Bulk Singularities in the Random Normal Matrix Model

We extend the method of rescaled Ward identities of Ameur, Kang, and Makarov to study the distribution of eigenvalues close to a bulk singularity, i.e., a point in the interior of the droplet where the density of the classical equilibrium measure vanishes. We prove results to the effect that a certain “dominant part” of the Taylor expansion determines the microscopic properties near a bulk singularity. A description of the distribution is given in terms of the Bergman kernel of a certain Fock-type space of entire functions.


Introduction and Main Results
1.1. Consider a system {ζ j } n 1 of identical point-charges in the complex plane in the presence of an external field n Q, where Q is a suitable function. The system is assumed to be picked randomly with respect to the Boltzmann-Gibbs probability law at inverse temperature β = 1, dP n (ζ ) = 1 Z n e −H n (ζ ) d 2n ζ, (1.1) where H n is the weighted energy of the system, The constant Z n in (1.1) is chosen so that the total mass is 1.
It is well known that (with natural conditions on Q) the normalized counting measures μ n = 1 n n j=1 δ ζ j converge to Frostman's equilibrium measure as n → ∞. This is a probability measure of the form dσ (ζ ) = χ S (ζ ) Q(ζ ) dA(ζ ), (1.2) where χ S is the indicator function of a certain compact set S called the droplet. We necessarily have Q ≥ 0 on S. In the papers [4,5], the method of rescaled Ward identities was introduced and applied to study microscopic properties of the system {ζ j } n 1 close to a (moving) point p ∈ S. The situation in those papers is however restricted by the condition that the point p be "regular" in the sense that Q( p) ≥ const. > 0. In this paper, we extend the method to allow for a "bulk singularity," i.e., an isolated point p in the interior of S at which Q = 0.
A bulk singularity tends to repel particles away, which means that one must use a relatively coarse scale in order to capture the relevant structure. We prove results to the effect that (in many cases) the dominant terms in the Taylor expansion of Q about p determine the microscopic properties of the system in the vicinity of p. Our characterization uses the Bergman kernel for a certain space of entire functions, associated with these dominant terms. In particular, we obtain quite different distributions depending on the degree to which Q vanishes at p. 1 2 (∂/∂ x + i∂/∂y) are the usual complex derivatives. We writez (or occasionally z * ) for the complex conjugate of a number z. A continuous function h(z, w) will be called Hermitian if h(z, w) = h(w, z) * . h is called Hermitian-analytic (Hermitian-entire) if h is Hermitian and analytic (entire) in z andw. A Hermitian function c(z, w) is called a cocycle if there is a unimodular function g such that c(z, w) = g(z)ḡ(w), where for functions we use the notationf (z) = f (z) * . We write D( p, r ) for the open disk with center p and radius r .

Potential and Equilibrium Measure
The function Q is usually called the "external potential." This function is assumed to be lower semi-continuous and real-valued, except that it may assume the value +∞ in portions of the plane. We also assume: (i) the set 0 = {Q < ∞} has dense interior, (ii) Q is real-analytic in Int 0 , and (iii) Q satisfies the growth condition For a suitable measure on C, we define its Q-energy by The equilibrium measure σ = σ Q is defined as the probability measure that minimizes I Q [μ] over all compactly supported Borel probability measures μ. Existence and uniqueness of such a minimizer is well known, see, e.g., [28], where also the explicit expression (1.2) is derived, with S = supp σ .
which gives the familiar scaling factor used in papers such as [1,4].
Since the Laplacian Q vanishes at 0 and is real-analytic and nonnegative in a neighborhood, there is an integer k ≥ 1 such that the Taylor expansion of Q about 0 takes the form Q(ζ ) =P(ζ ) + O(| ζ | 2k−1 ), whereP(x + iy) = 2k−2 j=0 a j x j y 2k−2− j is a positive semi-definite polynomial, homogeneous of degree 2k − 2.
We refer to the number 2k − 2 = degreeP as the type of the bulk-singularity at the origin. We shall say that the singularity is non-degenerate ifP is positive definite, i.e., if there is a positive constant c such thatP(ζ ) ≥ c | ζ | 2k−2 . Hereafter, we tacitly assume that this condition is satisfied.
It will be important to have a good grasp of the size of r n = r n (0) as n → ∞. For this, we note that (1.4) We will call τ 0 the modulus of the bulk singularity at 0. We have the following lemma; the simple verification is omitted here.
Let p be an integer, 1 ≤ p ≤ n. The p-point function of the point-process {ζ j } n 1 is the function of p complex variables η 1 , . . . , η p defined by The p-point function R n, p should really be understood as the density in the measure R n, p (η 1 , . . . , η p ) d A(η 1 ) · · · d A(η p ). This should be kept in mind when we subject the η j to various transformations.
A well-known algebraic fact ("Dyson's determinant formula," see, e.g., [26] or [28, p. 249]) states that the p-point function takes the form of a determinant, where K n is a certain Hermitian function called a correlation kernel of the process (Cf. Sect. 2). Of particular importance is the one-point function R n = R n, 1 .
We now rescale about the origin on the microscopic scale r n about the bulk singularity at 0. The rescaled system {z j } n 1 is taken to be with the law given by the image of the Boltzmann-Gibbs distribution (1.1) under the scaling (1.5). It follows that the rescaled system {z j } n 1 is determinantal with p-point function where the correlation kernel K n for the rescaled system is given by In particular, the one-point function of the process {z j } n 1 is R n (z) = K n (z, z). Clearly, a correlation kernel K n (z, w) is only determined up to multiplication by a cocycle c n (z, w).

Main Structural Lemma
Now suppose that Q has a bulk-singularity of type 2k −2 at the origin. It will be useful to single out a canonical "dominant part" of Q near 0. To this end, let P(x + iy) be the Taylor polynomial of Q of degree 2k about the origin. Let H be the holomorphic polynomial We will write We then have the basic decomposition where The following lemma gives the basic structure of limiting kernels at a singular point (not necessarily in the bulk).

Lemma 1.2
There exists a sequence c n of cocycles such that every subsequence of the sequence c n K n has a subsequence converging uniformly on compact subsets to some Hermitian function K . Every limit point K has the structure where L is a Hermitian-entire function.
Following [4], we refer to a limit point K in Lemma 1.2 as a limiting kernel, whereas L is a limiting holomorphic kernel. We also speak of the limiting 1-point function Note that R determines K and L by polarization.
Remark Each limiting one-point function gives rise to a unique limiting point field (or "infinite particle system") {z j } ∞ 1 with intensity functions (This follows from Lenard's theory, see [29] or [4].) It is possible that a limiting point field is trivial in the sense that K = 0.

Universality Results
We will prove universality for two kinds of bulk singularities. Referring to the canonical decomposition Q = Q 0 + Re H + Q 1 with Q 0 of degree 2k, we say a singularity at 0 is: We remark that a homogeneous singularity is necessarily located in the bulk of the droplet; for other types of singularities this must be postulated. In the following, we denote by L 0 the Bergman kernel of the space of entire functions L 2 a (μ 0 ) associated with the measure

Theorem 1.3
If there is a homogeneous singularity at 0, we have L = L 0 for each limiting holomorphic kernel L.
The next result concerns limiting holomorphic kernels L(z, w) which are rotationally symmetric in the sense that L(z, w) = L(ze it , we it ) for all real t. Equivalently, L is rotationally symmetric if there is an entire function E such that L(z, w) = E(zw). (We leave the simple verification of this to the reader.)  We do not know whether or not each limiting kernel at a dominant radial bulk singularity is rotationally symmetric. This question has some similarity to that of deciding the translation invariance of limiting kernels at regular boundary points. See [4] for several comments about this, notably the interpretation in terms of a twisted convolution equation in Section 7.1.
It is natural to conjecture that the kernel in Theorem 1.3 be equal to the limiting kernel in general, regardless of the nature of a (nondegenerate) bulk singularity.
Remark Note that, as a consequence of the reproducing property of the kernel L 0 , we have in the situation of the above theorems the mass-one equation for a limiting kernel K , Example For the Mittag-Leffler potential Q = | ζ | 2k , it is possible to calculate the limiting kernel L explicitly, using orthogonal polynomials (see [4,Section 7.3] or [30]). The result is that where The function M k can be expressed as M k (z) = τ 2 0 k E 1/k,1/k (τ 2 0 z), where E a,b is the Mittag-Leffler function (see [18]) Using Theorem 1.3, we can now see that the kernel in (1.7) is universal for potentials of the form Q = | ζ | 2k + Re c ζ 2k . (We must insist that | c | < 1 to insure that the growth assumption of Q at infinity is satisfied, see (1.3).) By Theorem 1.4, the universality holds also for all rotationally symmetric limiting kernels L(z, w) = E(zw) for more general potentials of the form Remark For k = 1 (i.e., when 0 is a "regular" bulk point) the space L 2 a (μ 0 ) becomes the standard Fock space, normed by f 2 = C | f (z) | 2 e − | z | 2 d A(z). In this case, we have R = 1 for the limiting 1-point function, by the well-known Ginibre(∞)-limit. (See, e.g., [4].)

Further Results
In the following, we consider a potential with canonical decomposition Q = Q 0 + Re H + Q 1 . Following [4], we shall prove auxiliary results that fall into three categories.
We call B(z, w) a limiting Berezin kernel rooted at z; C(z) is its Cauchy transform.
As n → ∞, it may well happen that R n → 0 locally uniformly (if the singularity at 0 is in the exterior of the droplet). Apriori Estimates To rule out the possibility of trivial limiting kernels, we shall use the following result. Theorem 1.6 Let R be any limiting kernel, and let R 0 (z) = L 0 (z, z)e −Q 0 (τ 0 z) , where L 0 is the Bergman kernel of the space L 2 a (μ 0 ). Then Part (i) depends on an estimate of the Bergman kernel for the space L 2 a (μ 0 ). Related estimates valid when Q 0 is a function satisfying uniform estimates of the type 0 < c ≤ Q 0 ≤ C are found in Lindholm's paper [24].
In our situation, the function Q 0 takes on all values between 0 and +∞, which means that the results from [24] are not directly applicable. It is convenient to include an elementary discussion for the case at hand, following the method of "approximate Bergman projections" in the spirit of [4,Section 5]. This has the advantage that proof of part (ii) follows after relatively simple modifications.
Remark Part (i) of Theorem 1.6 seems to be of some relevance for the investigation of density conditions for sampling and interpolation in Fock-type spaces L 2 a (μ 0 ); see the recent paper [17,Remark 5.6], cf. [31] for the classical Fock space case. (A very general result of this sort was obtained by different methods in the paper [25], where the hypothesis on the "weight" Q 0 is merely that the Laplacian Q 0 be a doubling measure.) Remark In the case Q = | z | 2λ , the asymptotic formula in Theorem 1.6 (i) has an alternative proof by more classical methods, using an asymptotic expansion for the function M λ (z) as z → ∞ ([18, Section 4.7]). The formula (i) can be recognized as giving the leading term in that expansion.

Positivity Recall that a Hermitian function
for all points z j ∈ C and all complex scalars α j . It is clear that each limiting (holomorphic) kernel is a positive matrix.

Theorem 1.7 Let L be a limiting holomorphic kernel. Then L is the Bergman kernel for a Hilbert space H * of entire functions that sits contractively in L
Here L 0 is the Bergman kernel of L 2 a (μ 0 ). It may well happen that the space H * degenerates to {0}. This is the case when the singularity at 0 is located in the exterior of the droplet.
Comments An interesting generalization of our situation is obtained by allowing for a suitably scaled logarithmic singularity at a (regular or singular) bulk point. More precisely, ifQ is smooth in a neighborhood of 0, we consider a potential of the form for suitable constants λ and μ. The potential (1.11) was introduced in the paper [3], where all rotationally symmetric solutions to the corresponding Ward equation (1.10) were found. Recently, certain potentials of this form were studied in a context of Riemann surfaces, in a scaling limit about certain types of singular points (conical singularities and branch points), see [22]. We will return to this issue in a forthcoming paper [6]. An interesting occurrence of Mittag-Leffler potentials with logarithmic singularities is found in the paper [11]; the dominant part Q 0 = |z| 2 − (2/M) log |z| is there connected to the eigenvalue density of the product of M complex Gaussian matrices. We are grateful to one of the referees for bringing this to our attention.
As in [4, Section 7.7], we note that it is possible to introduce an "inverse temperature" β into the setting; the case at hand then corresponds to β = 1. For general β, the rescaled process {z j } n 1 is no longer determinantal, but the rescaled intensity functions R β n, p make perfect sense. As n → ∞, we formally obtain a "Ward's equation at a bulk singularity" of the form Here C β (z) should be understood as the Cauchy transform of the β-Berezin ker- The objects in (1.12) are so far understood mostly on a physical level. We now give a few remarks in this spirit.
First, if 0 is a regular bulk-point, i.e., if Q(0) > 0, then it is believed that R β = 1 identically, i.e., the right-hand side in (1.12) should vanish. The equation (1.12) then reflects the fact that the Berezin kernel B β (z, w) = b β (r ) depends only on the distance r = | z − w |. When β = 1, one has the well-known identity b 1 (r ) = e − r 2 . For other β we do not know of an explicit expression, but it was shown by Jancovici in [20] where f is a certain explicit function. In the bulk-singular case, the kernel B β (z, w) will not just depend on | z − w |, but still it seems natural to expect that we have an expansion of the form . A natural problem, which will not be taken up here, is to determine the function f (z, w) in (1.13). (A similar investigation at regular boundary points was made recently in the paper [12].) For boundary points, the term "singular" has a different meaning than for bulk points. Indeed, the singular points p (cusps or double points) studied in the paper [5] all satisfy Q( p) > 0. An example of a situation at which Q = 0 at a boundary point (at 0) is provided by the potential . (The boundary of S is here a "figure 8" with 0 at the point of self-intersection, see [9].) A natural question is whether it is possible to define nontrivial scaling limits at (or near) these kinds of singular points, in the spirit of [5].
There is a parallel theory for scaling limits for Hermitian random matrix ensembles. In this situation, the droplet is a union of compact intervals. It is well known that the sine-kernel appears in the scaling limit about a "regular bulk point," i.e., an interior point where the density of the equilibrium measure is strictly positive. In a generic case, all points are regular, see [21]. Special bulk points where the equilibrium density vanishes may be called "singular"; at such points other types of universality classes appear, see [10,14,27]; cf. [15] for a corresponding nonbulk situation.
Finally, we wish to mention that the investigations in this paper were partly motivated by applications to the distribution of Fekete points close to a bulk singularity (see [1]). This issue will be taken up in a later publication.
Addendum After the completion of this paper, at least two relevant papers have appeared. The note [7] generalizes and improves some of our above results, notably Theorem 1.6, in a context allowing for the logarithmic singularities discussed above. The paper [23] investigates logarithmic singularities with respect to properties of associated orthogonal polynomials.

Plan of the Paper
In Sect. 2, we prove the general structure formula for limiting kernels (Lemma 1.2). We also prove the positivity theorem (Theorem 1.7). In Sect. 3, we prove Ward's equation and the zero-one law (Theorem 1.5). In Sect. 4, we prove the universality results (Theorems 1.3 and 1.4). Our proof of Theorem 1.4 depends on the apriori estimate from Theorem 1.6, part (ii). In the last two sections, we prove the asymptotics for the functions R 0 and R in Theorem 1.6. For R 0 (part (i)), see Sect. 5; for R (part (ii)), see Sect. 6.

Convention
Multiplying the potential Q by a suitable constant, we can in the following assume that the modulus τ 0 = 1. In fact, the slightly more general assumption that τ 0 = 1 + O(n −1/2k ) as n → ∞ will do equally well. This means the microscopic scale about 0 can be taken as r n = n −1/2k , where 2k − 2 is the type of the singularity. This will be assumed throughout the rest of this paper.

Structure of Limiting Kernels
In this section, we prove Lemma 1.2 on the general structure of limiting kernels and the positivity Theorem 1.7. We shall actually prove a little more: a limiting holomorphic kernel can be written as a subsequential limit of kernels for certain specific Hilbert spaces of entire functions. In later sections, we will use this additional information for our analysis of homogeneous bulk singularities.

Spaces of Weighted Polynomials
It is well known that we can take for correlation kernel for the process {ζ j } n 1 the reproducing kernel for a suitable space of weighted polynomials. Here the "weight" can either be incorporated into the polynomials themselves or into the norm of the polynomials. We will use both these possibilities. In the following, we shall use the symbol "Pol(n)" for the linear space of holomorphic polynomials of degree at most n − 1 (without any topology). We write μ n for the measure dμ n = e −n Q d A.
We let P n denote the space Pol(n) regarded as a subspace of L 2 (μ n ). The symbol W n will denote the set of weighted polynomials f = p e −n Q/2 , (p ∈ Pol(n)) regarded as a subspace of L 2 = L 2 (d A). We write k n and K n for the reproducing kernels of P n and W n , respectively, and we note that Now suppose that Q has a bulk singularity at the origin, of type 2k − 2 and rescale at the microscopic scale by

Limiting Holomorphic Kernels
Suppose that there is a bulk singularity of type 2k − 2 at the origin. Consider the canonical decomposition Q = Q 0 + Re H + Q 1 and write h = Re H . Thus h is of degree at most 2k, Q 0 is a positive definite homogeneous polynomial of degree 2k, and Proof LetW n denote the space of all "rescaled" weighted polynomials p · e −Q n /2 , where p ∈ Pol(n) andQ n (z) = n Q(r n z). RegardingW n as a subspace of L 2 , we recognize that K n is the reproducing kernel ofW n . Hence We also pick a number α > sup{ Q 0 (z) ; z ∈ V δ }. Now let u be an analytic function in a neighborhood of V δ , and consider the function We obtain where M V = sup z∈V | z |. By (2.1) and (2.2), K n (z, z) is bounded for z ∈ V .

Lemma 2.2
Each subsequence of the kernels L n has a further subsequence converging locally uniformly to a Hermitian-entire limit L. Furthermore, L n is the reproducing kernel of the space H n , and L satisfies the "mass-one inequality," Finally, there exists a sequence of cocycles c n such that each subsequence of c n K n converges locally uniformly to a Hermitian function K of the type Proof Define a function E n (z, w) by We have shown that where o(1) → 0 locally uniformly on C 2 and c n is a cocycle: On the other hand, for each compact subset V of C 2 , there is a constant C such that for sufficiently large n. By Lemma 2.1, the functions L n have a uniform bound on V . We have shown that {L n } is a normal family. We can hence extract a subsequence {L n }, converging locally uniformly to a Hermitian-entire function L(z, w).
Choosing cocycles c n such that c n E n → e −Q 0 (z)/2−Q 0 (w)/2 uniformly on compact subsets as n → ∞, we now obtain that c n K n = c n E n L n → e −Q 0 (z)/2−Q 0 (w)/2 L(z, w) = K (z, w).
There remains to prove that L n is the reproducing kernel for the space H n . For this, we write L n,w (z) = L n (z, w) and note that for an element f = q · e −H n /2 of H n , we have Noting that k n is the reproducing kernel for the spaceP n of polynomials of degree at most n − 1 normed by p 2 = C | p(z) | 2 e −n Q(r n z) d A(z), we now see that The proof of the lemma is complete.

The Positivity Theorem
Let μ 0 be the measure dμ 0 = e −Q 0 d A, and define L 0 (z, w) to be the Bergman kernel for the Bergman space L 2 a (μ 0 ). Let L = lim L n be a limiting holomorphic kernel at 0.
Recall that the kernel L n is the reproducing kernel for a certain subspace H n of L 2 a (μ n ), whereμ n → μ 0 in the sense that the densities converge uniformly on compact sets, as n → ∞. See Lemma 2.2.
For L = lim L n , the assignment L z , L w * = L(w, z) defines a positive semidefinite inner product on the linear span M of the L z 's. In fact, the inner product is either trivial (L(z, z) = 0 for all z), or else it is positive definite: this holds by the zero-one law in Theorem 1.5, which will be proved in the next section.
By Fatou's lemma, we now see that, for all choices of points z j and scalars α j , This shows that M is contained in L 2 (μ 0 ) and that the inclusion I : M → L 2 (μ 0 ) is a contraction. Hence the completion H * of M can be regarded as a contractively embedded subspace of L 2 a (μ 0 ). Since the space L 2 a (μ 0 ) has reproducing kernel L 0 (z, w), it follows from a theorem of Aronszajn ([8, p. 355]) that the difference L 0 − L is a positive matrix. The proof of Theorem 1.7 is complete.

Ward's Equation
Given a limiting kernel K in Lemma 2.2, we recall the definitions The goal of this section is to prove Theorem 1.5, which we here restate in the following form (the case τ 0 = 1).

Lemma 3.1 If R does not vanish identically, then R > 0 everywhere and we havē
For the proof of Lemma 3.1, we recall the setting of Ward's identity from [4]. For a test function ψ ∈ C ∞ 0 (C), we define a function W ∂ Q(ζ j ) · ψ(ζ j ), and We now regard ζ as picked randomly with respect to the Boltzmann-Gibbs distribution (1.1). W + n [ψ] is then a random variable; the Ward identity proved in [4, Section 4.1] states that its expectation vanishes: We shall now rescale in Ward's identity about 0 at the microscopic scale r n = n −1/2k , given that the basic decomposition Q = Q 0 + ReH + Q 1 in (1.6) holds. (We do not need to assume that 0 is in the bulk at this stage.) To facilitate the calculations, it is convenient to recall a simple algebraic fact (see, e.g., [26]): if f is a function of p complex variables, and if f (ζ 1 , . . . , ζ p ) is regarded as a random variable on the sample space {ζ j } n 1 with respect to the Boltzmann-Gibbs law, then the expectation is We rescale about 0 via z = r −1 n ζ , w = r −1 n η, recalling that the p-point functions transform as densities. We recall that R n, p (z) = r 2 p n R n, p (ζ ) denotes the rescaled p-point function and use the abbreviation R n = R n,1 for the one-point function. We also write

Lemma 3.2 We have that
where o(1) → 0 uniformly on compact subsets of C as n → ∞.
Proof We fix a test function ψ ∈ C ∞ 0 (C) and let ψ n (ζ ) = ψ(r −1 n ζ ). The change of variables z = r −1 n ζ and w = r −1 n η gives that and Likewise, changing variables and integrating by parts, we obtain Hence, by the Ward identity in (3.1), we have Since ψ is an arbitrary test function, we have in the sense of distributions, C R n,2 (z, w) z − w d A(w) = nr n ∂ Q(r n z) R n,1 (z) + ∂ R n,1 (z).
Dividing through by R n,1 (z) and using the fact that Differentiating with respect toz, we get R n,1 (z) −∂C n (z) = nr 2 n Q(r n z) + log R n,1 (z).

The Proof of Theorem 1.5
We will need a few lemmas.

If R does not vanish identically, then all zeros of R are isolated.
Proof The assumption gives that the holomorphic kernel L corresponding to R satisfies L(z 0 , z 0 ) = 0. Hence e −Q 0 (w) | L(z 0 , w) | 2 d A(w) ≤ 0 by the mass-one inequality (2.5). Thus L(z 0 , w) = 0 for all w ∈ C. Since L is Hermitian-entire, we can thus write for some Hermitian-entire functionL. We now have R(z) = | z − z 0 | 2L (z, z)e −Q 0 (z) .
For the second statement, we assume that R does not vanish identically and there exists a zero z 0 of R which is not isolated. Then, we can take a sequence {z j } ∞ 1 of distinct zeros of R which converges to z 0 , whence by the above argument, for each j we obtain L(z j , w) = 0 for all w ∈ C. If we fix w, then L(z, w) = 0 for all z ∈ C since L(z, w) is holomorphic in z. Hence L = 0 identically.

Lemma 3.4 L(z, w) is a positive matrix and z → L(z, z) is logarithmically subharmonic.
Proof It is clear that L is a positive matrix. Now write L z (w) := L(w, z) and define a semi-definite inner product by L z , L w * := L(w, z) on the linear span of the functions L z for z ∈ C. The completion of this span forms a (perhaps semi-normed) Hilbert space H * , and L is a reproducing kernel of the space. Now when L(z, z) > 0, Since L(z, w) is Hermitian-entire, we have∂ z L z ∈ H * , ∂ z L z , L z * =∂ z L(z, z), and ∂ z L z ,∂ z L z * = L(z, z). Hence, the numerator of (3.4) can be written as which is nonnegative by the Cauchy-Schwarz inequality. At points where L(z, z) = 0, log L(z, z) satisfies the sub-mean value property since log L(z, z) = −∞. Hence the function log L(z, z) is subharmonic on C.
Proof We choose a small disc D = D(z 0 , ) and consider the function Observing that z log L(z, z) = Q 0 (z) + logR(z) + δ z 0 in the sense of distributions, Lemma 3.4 gives us that S ≥ 0 in the sense of distributions on D\{z 0 }. If R(z 0 ) > 0, we extend S analytically to z 0 . On the other hand, ifR(z 0 ) = 0, we define S(z 0 ) = −∞. In both cases, the extended function S is subharmonic on D.
We now turn to the left-hand side in the rescaled version of Ward's identity, namely the function∂C n , where C n is the Cauchy transform of B n (see (3.3)).

Lemma 3.6 Suppose that R = lim R n is a limiting 1-point function that does not vanish identically. Let Z be the set of isolated zeros of R, and let B(z, w) = lim B n (z, w)
be the corresponding Berezin kernel for z / ∈ Z . Then C n → C locally uniformly on the complement Z c = C \ Z as → ∞, where the function is bounded on Z c ∩ V for each compact subset V of C.
Proof We have that c n K n → K locally uniformly on C 2 , where K (z, z) = R(z) > 0 when z / ∈ Z . Hence, for fixed with 0 < < 1, we can choose N such that if ≥ N , then Here, we have used the mass-one inequality for the third inequality. Thus C n → C uniformly on compact subsets of Z c . Now fix a compact subset V of C. Then, for all z, w with z ∈ V \ Z and dist(w, V ) ≤ 1, Hence we obtain | C(z) | ≤ 2M + 1 for z ∈ V \ Z .

Lemma 3.7 If R does not vanish identically, the Ward's equation
Proof The preceding lemmas show that and that a subsequence C n converges to C boundedly and locally uniformly on C \ Z . Since Z ∩ V is a finite set for each compact set V , it follows that C n → C in the sense of distributions, and hence∂C n →∂C. By Ward's equation and the locally uniform convergence R n → R, it then follows that log R n → log R in the sense of distributions. We can thus pass to the limit as n → ∞ in the rescaled Ward identity (3.5).
Proof of Theorem 1. 5 We follow the strategy in [4,Theorem 4.8]. Suppose that R(z 0 ) = 0. We must prove that R = 0 identically. Let D be a small disk centered at z 0 , and write χ = χ D for the characteristic function. Also write R(z) = | z − z 0 | 2R (z).

Universality Results
In this section, we prove Theorems 1.3 and 1.4. The proof of Theorem 1.4 relies on certain apriori estimates, whose proofs are postponed to Sect. 6.

Homogeneous Singularities
Assume that Q has a homogeneous singularity of type 2k − 2 at the origin, i.e., that the canonical decomposition is of the form Q = Q 0 + Re H, H = c ζ 2k , where Q 0 is positively homogeneous of degree 2k. As always, we write μ 0 for the measure dμ 0 = e −Q 0 d A.
In the present case, L n (z, w) = k n (z, w)e −c z 2k /2−cw 2k /2 . By Lemma 2.2, L n is the reproducing kernel for the space regarded as a subspace of L 2 (μ 0 ). (This is becauseμ n = μ 0 for the measureμ n in (2.4).) Since the spaces H n are increasing, H n ⊂ H n+1 , where the inclusions are isometric, it follows that a unique limiting holomorphic kernel L = lim L n exists. By Theorem 1.7, the kernel L is the reproducing kernel for a contractively embedded subspace H * of L 2 a (μ 0 ), which must contain the dense subset U = H n . Furthermore, by the reproducing property of L n , we have for each element f (z) = q(z) · e −cz 2k /2 ∈ U that f, L n,z L 2 (μ 0 ) = f (z), whenever n > degree q. It follows that Since U is dense in L 2 a (μ 0 ), L must equal to the reproducing kernel L 0 of L 2 a (μ 0 ). The proof of Theorem 1.3 is complete.

Rotational Symmetry
Referring to the canonical decomposition Q = Q 0 + Re H + Q 1 , we now suppose that Q 0 (z) = Q 0 (| z |), and we fix a rotationally symmetric limiting holomorphic kernel Writing E(z) = ∞ 0 a j z j , the mass-one inequality is seen to be equivalent to that To use Ward's equation, we first compute the Cauchy transform C(z) as follows: However, as is shown in [3], we have that where The term A(z) can be written as which gives∂ The term B(z) is computed as follows: Noting that we infer that Ward's equation is equivalent to that∂(B − ∂ z log L(z, z)) = 0. This in turn, is equivalent to that the function It is easy to check that this is the case if and only if all coefficients in the sum vanish, that is, if and only if for each k ≥ 1, we have that We now apply the growth estimate in Theorem 1.6, part (ii), which says that We claim that this implies the second alternative in (4.2). Indeed, (4.3) is clearly not satisfied if E is constant. Next note that the mass-one inequality (4.1) and the zero-one law (Theorem 1.5) imply that 0 < a 0 ≤ 1/ 1 2 L 2 (μ 0 ) . Since E(z) is not a polynomial by (4.3), for any k there exists N ≥ k such that a N = 0. By (4.2), we obtain that if a N = 0 but a j = 0 for all j with 1 ≤ j ≤ N −1, then N = 1 and a 0 = 1/ 1 2 L 2 (μ 0 ) . By a simple induction, we then have a k = 1/ z k 2 L 2 (μ 0 ) for all k ≥ 0. Thus, we have Since the polynomial φ j (z) = z j / z j L 2 (μ 0 ) is the j:th orthonormal polynomial with respect to the measure μ 0 , we have where L 0 is the Bergman kernel for the space L 2 a (μ 0 ). The proof is complete.

Asymptotics for L 0 (z, z)
In this section, we prove part (i) of Theorem 1.6.
To this end, let A 0 (z, w) be the Hermitian polynomial such that A 0 (z, z) = Q 0 (z), and write We write L z (w) for L 0 (w, z) and, for suitable functions u, Below, we fix a z with |z| large enough; we must estimate L 0 (z, z). We also fix a number δ 0 = δ 0 (z) > 0 and write χ z for a fixed C ∞ -smooth test-function with χ z (w) = 1 when | w − z | ≤ δ 0 and χ z (w) = 0 when | w − z | ≥ 2δ 0 .
We will use the following estimate.
(5.1) However, since Q 0 is homogeneous of degree 2k, the derivative ∂ i∂ j Q 0 is homogeneous of degree 2k − i − j. Hence Thus, if i + j ≥ 3 and |1 − w/z| is sufficiently small, then the left-hand side in (5.1) is dominated by an arbitrarily small multiple of |z| 2k−2 |z − w| 2 . On the other hand, by homogeneity and positive definiteness of Q 0 , we have that Q 0 (z)|z − w| 2 ≥ c |z| 2k−2 |z − w| 2 , where c is a positive constant. The lemma thus follows with any positive constant c < c As always, we write dμ 0 = e −Q 0 d A; L 2 a (μ 0 ) denotes the associated Bergman space of entire functions, and L 0 is the Bergman kernel of that space.

Lemma 5.2
Let |z| ≥ 1 and δ 0 be a positive number with δ 0 /|z| sufficiently small. Then there is a constant C = C(δ 0 ) such that, for all functions u ∈ L 2 a (μ 0 ), Here the derivative . We then have that By the form of F it is also clear that An integration by parts in (5.2) gives π [χ z u](z) = u(z) Inserting the estimates (5.4) and (5.5), using also that∂χ z (w) = 0 when | w − z | ≤ δ 0 , we find that To estimate 1 , we use Lemma 5.1 to get This gives To estimate 2 , we note that (again by (5.6)) The proof is complete.
we see that If we now choose u = L z in Lemma 5.2 and recall that L z 2 L 2 (μ 0 ) = L 0 (z, z), we obtain the estimate
Since Q 0 is strictly subharmonic on the support of χ z , we can apply the standard Hörmander estimate (e.g., [19, p. 250]) to obtain where we used homogeneity of Q 0 . By Taylor's formula and the estimate (5.6), we have when δ 0 ≤ | w − z | ≤ 2δ 0 , By the homogeneity of Q 0 , we thus obtain the estimate We now pick another (small) number δ > 0 and invoke the following pointwise-L 2 estimate (see, e.g., [4,Lemma 3.1] or the proof of the inequality (2.2)): Combining with (5.8), this gives Choosing δ 0 a small multiple of |z| 1/2 and then δ small enough, we insure that the right-hand side is dominated by C|z| 2k−2 e Q 0 (z) , as desired.
Proof of Part (i) of Theorem 1.6 By the estimate (5.7) and Lemma 5.3, we have .
We must prove that the left-hand side in (5.10) is dominated by M | z | 1−k Q 0 (z) for all large | z |, where M is a suitable constant. If this is false, there are two possibilities.
for arbitrarily large | z |, then (5.10) implies and we reach a contradiction for large enough M.
In the remaining case, we have R 0 (z) ≥ (1 + M| z | 1−k ) Q 0 (z). Then (5.10) gives the estimate R 0 (z) ≥ cM 2 | z | 2k−2 for some c > 0. Since Q 0 (z) ≤ c | z | 2k−2 for some c > 0, we obtain Choosing M large enough, we obtain R 0 (z) ≥ C 3 M | z | 4k−4 by (5.10) again. Repeating the above argument gives R 0 (z) ≥ C p M | z | 2 p for all sufficiently large | z | for some constant C p > 0. On the other hand, we will show that for all z, which will give the desired contradiction. To see this, note that for functions u ∈ L 2 a (μ 0 ), the estimate (5.9) gives we now obtain the estimate (5.11).

Apriori Estimates for the One-Point Function
In this section, we prove part (ii) of Theorem 1.6. As before, we write Q = Q 0 + Re H + Q 1 for the canonical decomposition of Q at 0, and we write μ 0 for the measure dμ 0 = e −Q 0 d A. In this section, the assumption that 0 is in the bulk of the droplet will become important.
The corresponding "approximate projection" is defined on suitable functions u by where, for convenience, we write L ζ instead of L n,ζ . Lemma 6.1 Suppose that u is holomorphic in a neighborhood of ζ and δ 0 (ζ )·r n /|ζ | ≤ ε 0 (small enough). Then there is a constant C = C(ε 0 ) such that, when r n ≤ | ζ | ≤ r n log n, Proof It will be sufficient to indicate how the proof of Lemma 5.2 is modified in the present setting. We start as earlier, by writing (ζ, ω) .
Proof Let u 0 = χ ζ L ζ − π n χ ζ L ζ be the norm-minimal solution in L 2 (μ n ) to the problem∂u 0 =∂ f , where f = χ ζ L ζ . We will prove that the problem∂u =∂ f has a solution u with u − f ∈ Pol(n) and u L 2 (μ n ) ≤ Cn −1/2 |ζ | −(k−1) ∂ χ ζ L ζ L 2 (μ n ) . (6.6) This is done by a standard device, which now we briefly recall. LetQ be the "obstacle function" pertaining to Q. The main facts about this function to be used here are the following (cf. [28] for details). The obstacle function can be defined asQ = γ − 2U σ , where U σ is the logarithmic potential of the equilibrium measure and γ is a constant chosen so thatQ = Q on S. One has thatQ is harmonic outside S, and that its gradient is Lipschitz continuous on C. Furthermore,Q(ω) grows like 2 log | ω | + O(1) as ω → ∞.
We use the obstacle function to form the strictly subharmonic function φ(ω) = Q(ω) + n −1 log(1 + | ω | 2 ), and we go on to define a measure μ n by dμ n (ω) = e −nφ(ω) d A(ω). Write P n for the subspace of L 2 (μ n ) of holomorphic polynomials of degree at most n − 1, and let π n be the corresponding orthogonal projection. Finally, we write v 0 = f − π n f.
Next we use the estimate nφ ≤ n Q + const. which holds by the growth assumption on Q near infinity. This gives v 0 L 2 (μ n ) ≤ C v 0 L 2 (μ n ) , and so we have shown (6.6) with u = v 0 .
Choosing δ 0 r n as a small multiple of |ζ | and then δ small enough, we can now use (6.7) to deduce that