RANDOM NORMAL MATRICES AND WARD IDENTITIES

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1 RANDOM NORMAL MATRIE AND WARD IDENTITIE YAIN AMEUR, HAAKAN HEDENMALM, AND NIKOLAI MAKAROV Abstract. onsider the random normal matrix ensemble associated with a potential on the plane which is sufficiently strong near infinity. It is known that, to a first approximation, the eigenvalues obey a certain equilibrium distribution, given by Frostman s solution to the minimum energy problem of weighted logarithmic potential theory. On a finer scale, one can consider fluctuations of eigenvalues about the equilibrium. In the present paper, we give the correction to the expectation of fluctuations, and we prove that the potential field of the corrected fluctuations converge on smooth test functions to a Gaussian free field with free boundary conditions on the droplet associated with the potential. Given a suitable real "weight function in the plane, it is well-known how to associate a corresponding (weighted) random normal matrix ensemble (in short: RNM-ensemble). Under reasonable conditions on the weight function, the eigenvalues of matrices picked randomly from the ensemble will condensate on a certain compact subset of the complex plane, as the order of the matrices tends to infinity. The set is known as the droplet corresponding to the ensemble. It is well-known that the droplet can be described using weighted logarithmic potential theory and, in its turn, the droplet determines the classical equilibrium distribution of the eigenvalues (Frostman s equilibrium measure). In this paper we prove a formula for the expectation of fluctuations about the equilibrium distribution, for linear statistics of the eigenvalues of random normal matrices. We also prove the convergence of the potential fields corresponding to corrected fluctuations to a Gaussian free field on with free boundary conditions. Our approach uses Ward identities, that is, identities satisfied by the joint intensities of the point-process of eigenvalues, which follow from the reparametrization invariance of the partition function of the ensemble. Ward identities are well known in field theories. Analogous results in random Hermitian matrix theory are known due to Johansson [13], in the case of a polynomial weight. General notation. By D(a, r) we mean the open Euclidean disk with center a and radius r. By "dist we mean the Euclidean distance in the plane. If A n and B n are expressions depending on a positive integer n, we write A n B n to indicate that A n B n for all n large enough where is independent of n. The notation A n B n means that A n B n and B n A n. When µ is a measure and f a µ-measurable function, we write µ( f ) = f dµ. We write = 1 2 ( / x i / y) and = 1 2 ( / x + i / y) for the complex derivatives. 1

2 2 YAIN AMEUR, HAAKAN HEDENMALM, AND NIKOLAI MAKAROV 1. Random normal matrix ensembles 1.1. The distribution of eigenvalues. Let Q : R {+ } be a suitable lower semi-continuous function subject to the growth condition (1.1) lim inf z Q(z) log z > 1. We refer to Q as the weight function or the potential. Let N n be the set of all n n normal matrices M, i.e., MM = M M. The partition function on N n associated with Q is the function Z n = e 2n trace Q(M) dm n, N n where dm n is the Riemannian volume form on N n inherited from the space n2 of all n n matrices, and where trace Q : N n R {+ } is the random variable trace Q(M) = Q(λ j ), λ j spec(m) i.e., the usual trace of the matrix Q(M). We equip N n with the probability measure dp n = 1 Z n e 2n trace Q(M) dm n, and speak of the random normal matrix ensemble or "RNM-ensemble associated with Q. The measure P n induces a measure P n on the space n of eigenvalues, which is known as the density of states in the external field Q; it is given by Here we have put dp n (λ) = 1 Z n e H n(λ) da n (λ), λ = (λ j ) n 1 n. 1 H n (λ) = log + 2n λ j λ k j k n Q(λ j ), and da n (λ) = d 2 λ 1 d 2 λ n denotes Lebesgue measure in n, while Z n is the normalizing constant giving P n unit mass. By a slight abuse of language, we will refer to Z n as the partition function of the ensemble. Notice that H n is the energy (Hamiltonian) of a system of n identical point charges in the plane located at the points λ j, under influence of the external field 2nQ. In this interpretation, P n is the law of the oulomb gas in the external magnetic field 2nQ (at inverse temperature β = 2). In particular, this explains the repelling nature of the eigenvalues of random normal matrices; they tend to be very spread out in the vicinity of the droplet, just like point charges would. onsider the n-point configuration ("set with possible repeated elements) {λ j } n of eigenvalues 1 of a normal matrix picked randomly with respect to P n. In an obvious manner, the measure P n induces a probability law on the n-point configuration space; this is the law of the n-point process Ψ n = {λ j } n associated to Q. 1 It is well-known that the process Ψ n is determinantal. This means that there exists a Hermitian function K n, called the correlation kernel of the process such that the density of states can be j=1

3 RANDOM NORMAL MATRIE AND WARD IDENTITIE 3 represented in the form One has dp n (λ) = 1 n! det ( K n (λ j, λ k ) ) n j,k=1 da n(λ), λ n. K n (z, w) = K n (z, w)e n(q(z)+q(w)), where K n is the reproducing kernel of the space P n ( e 2nQ ) of analytic polynomials of degree at most n 1 with norm induced from the usual L 2 space on associated with the weight function e 2nQ. Alternatively, we can regard K n as the reproducing kernel for the subspace W n = {pe nq ; p is an analytic polynomial of degree less than n} L 2 (). We have the frequently useful identities f (z) = f (w)k n (z, w) d 2 w, f W n, and K n (z, z) d 2 z = n. Moreover, if E n denotes expectation with respect to P n, and if f is a function depending on k variables, where k n, then [ E n f (λ1,..., λ k ) ] (n k)!k! [ = E k f (λ1,..., λ k ) ]. n! We refer to [7], [18], [9], [10], [3], [14] for more details on point-processes and random matrices The equilibrium measure and the droplet. We are interested in the asymptotic distribution of eigenvalues as n, the size of the matrices, increases indefinitely. Let u n denote the one-point function of P n, i.e., u n (λ) = 1 n K n(λ, λ), λ. With a suitable function f on, we associate the random variable Tr n [ f ] on the probability space ( n, P n ) via n Tr n [ f ](λ) = f (λ i ). The expectation is given by ( E n Trn [ f ] ) = n f u n. According to Johansson (see [10]) we have weak-star convergence of the measures i=1 dσ n (z) = u n (z)d 2 z to some probability measure σ = σ(q) on. In fact, σ is the Frostman equilibrium measure of the logarithmic potential theory with external field Q. We briefly recall the definition and some basic properties of this probability measure, cf. [16] and [10] for proofs and further details. Let = supp σ and assume that Q is 2 -smooth in some neighbourhood of. Then is compact, Q is subharmonic on, and σ is absolutely continuous with density u = 1 2π Q 1.

4 4 YAIN AMEUR, HAAKAN HEDENMALM, AND NIKOLAI MAKAROV We refer to the compact set = Q as the droplet corresponding to the external field Q. Our present goal is to describe the fluctuations of the density field µ n = n j=1 δ λ j around the equilibrium. More precisely, we will study the distribution (linear statistic) f µ n ( f ) nσ( f ) = Tr n [ f ] nσ( f ), We will denote by ν n the measure with density n(u n u), i.e., ν n [ f ] = E n [ Trn [ f ] ] nσ( f ) = n(σ n σ)( f ), f 0 (). f 0 () Assumptions on the potential. To state the main results of the paper we make the following three assumptions: (A1) (smoothness) Q is real analytic (written Q ω ) in some neighborhood of the droplet = Q ; (A2) (regularity) Q 0 in ; (A3) (topology) is a ω -smooth Jordan curve. We will comment on the nature and consequences of these assumptions later. Let us denote L = log Q. This function is well-defined and ω in a neighborhood of the droplet The Neumann jump operator. We will use the following general system of notation. If g is a continuous function defined in a neighborhood of, then we write g for the function on the Riemann sphere Ĉ such that g equals g in while g equals the harmonic extension of g to Ĉ \ on that set. If g is smooth on, then N Ω g := g n, Ω := int(), where n is the (exterior) unit normal of Ω. We define the normal derivative N Ω g for the complementary domain Ω := Ĉ \ similarly. If both normal derivatives exist, then we define (Neumann s jump) N g N := N Ω g + N Ω g. By Green s formula we have the identity (of measures) (1.2) g = g 1 Ω + N(g ) ds, where ds is the arclength measure on. We now verify (1.2). Let φ be a test function. The left hand side in (1.2) applied to φ is φ g = g φ = g φ + g φ, \ and the right hand side is φ g + φn(g )ds. Thus we need to check that (g φ φ g) + (g φ φ g ) = \ φn(g )ds.

5 RANDOM NORMAL MATRIE AND WARD IDENTITIE 5 But the expression in the left hand side is (g n φ φ n g)ds + (g n φ φ n g )ds = and (1.2) is proved Main results. We have the following results. (φ n g + φ n g )ds = φ N(g )ds, Theorem 1.1. For all test functions f (), the limit 0 ν( f ) := lim ν n ( f ) n exists, and ν( f ) = 1 [ ] f + f L + f N(L ) ds. 8π Equivalently, we have in the sense of distributions. ν n ν = 1 8π ( 1 + L ) Theorem 1.2. Let h () be a real-valued test function. Then, as n, 0 ( ) 1 trace n h E n trace n h N 0, h 2. 2π The last statement means convergence in distribution of random variables to a normal law with indicated expectation and variance. As noted in [3], ection 7, the result can be restated in terms of convergence of random fields to a Gaussian field on with free boundary conditions Derivation of Theorem 1.2. We now show, using the variational approach due to Johansson [13], that the Gaussian convergence stated in Theorem 1.2 follows from a generalized version of Theorem 1.1, which we now state. Fix a real-valued test function h and consider the perturbed potentials Q n := Q 1 n h. We denote by ũ n the one-point function of the density of states P n associated with the potential Q n. We write σ n for the measure with density ũ n and ν n for the measure n( σ n σ), i.e., (1.3) ν n [ f ] = n σ n ( f ) nσ( f ) = Ẽ n Tr n [ f ] nσ( f ). Theorem 1.3. For all f () we have 0 ν n ( f ) ν n ( f ) 1 2π A proof of Theorem 1.3 is given in ection 4. f h.

6 6 YAIN AMEUR, HAAKAN HEDENMALM, AND NIKOLAI MAKAROV laim. Theorem 1.2 is a consequence of Theorem 1.3. Proof. Denote X n = Tr n h E n Tr n h and write a n = Ẽ n X n. By Theorem 1.3, a n a where a = 1 h h. 2π More generally, let λ 0 be a parameter, and let Ẽ n,λ denote expectation corresponding to the potential Q (λh)/n. Write ince F n(λ) = Ẽ n,λ X n, Theorem 1.3 implies F n (λ) := log E n e λx n, 0 λ 1. (1.4) F n(λ) = Ẽ n,λ X n λa, and (1.5) log E n e X n = F n (1) = 1 Here we use the convexity of the functions F n, 0 F n(λ) dλ a, as n. 2 F n (λ) = Ẽ n,λ X 2 n ( Ẽ n,λ X n ) 2 0, which implies that the convergence in (1.4) is dominated: 0 = F n(0) F n(λ) F n(1). Replacing h by th where t R, we get E n (e tx n ) e t2a/2 as n, i.e., we have convergence of all moments of X n to the moments of the normal N(0, a) distribution. It is well known that this implies convergence in distribution, viz. Theorem 1.2 follows omments. (a) Related Work. The one-dimensional analog of the weighted RNM theory is the more wellknown random Hermitian matrix theory, which was studied by Johansson in the important paper [13]. Indeed, Johansson obtains results not only random Hermitian matrix ensembles, but for more general (one-dimensional) β-ensembles. The paper [13] was one of our main sources of inspiration for the present work. In [3], it was shown that the convergence in theorems 1.1 and 1.2 holds for test functions supported in the interior of the droplet. ee also [6]. In [3], we also announced theorems 1.1 and 1.2 and proved several consequences of them, e.g. the convergence of Berezin measures, rooted at a point in the exterior of, to a harmonic measure. Rider and Virág [15] proved theorems 1.1 and 1.2 in the special case Q(z) = z 2 (the Ginibre ensemble). The paper [8] contains results in this direction for β-ginibre ensembles for some special values of β. Our main technique, the method of Ward identities, is common practice in field theories. In this method, one uses reparametrization invariance of the partition function to deduce exact relations satisfied by the joint intensity functions of the ensemble. In particular, the method was applied on the physical level by Wiegmann, Zabrodin et al. to study RNM ensembles as well as more general OP ensembles. ee e.g. the papers [19], [20], [21], [22]. A one-dimensional version of Ward s identity was also used by Johansson in [13].

7 RANDOM NORMAL MATRIE AND WARD IDENTITIE 7 Finally, we wish to mention that one of the topics in this paper, the behaviour of fluctuations near the boundary, is analyzed from another perspective in the forthcoming paper [4]. (b) Assumptions on the potential. We here comment on the assumptions (A1) (A3) which we require of the potential Q. The ω assumption (A1) is natural for the study of fluctuation properties near the boundary of the droplet. (For test functions supported in the interior, one can do with less regularity.) Using akai s theory [18], it can be shown that conditions (A1) and (A2) imply that is a union of finitely many ω curves with a finite number of singularities of known types. It is not difficult to complete a proof using arguments from [11], ection 4. We rule out singularities by the regularity assumption in (A3). What happens in the presence of singularities is probably an interesting topic, which we have not approached. Without singularities the boundary of the droplet is a union of finitely many ω Jordan curves. Assumption (A3) means that we only consider the case of a single boundary component. Our methods extend without difficulty to the case of a multiply connected droplet. The disconnected case requires further analysis, and is not considered in this paper. (c) Droplets and potential theory. We here state the properties of droplets that will be needed for our analysis. Proofs for these properties can be found in [16] and [10]. We will write ˇQ for the maximal subharmonic function Q which grows as log z + O(1) when z. We have that ˇQ = Q on while ˇQ is 1,1 -smooth on and ˇQ(z) = Q (z) + G(z, ), z \, where G is the classical Green s function of \. In particular, if U σ 1 (z) = log z ζ dσ(ζ) denotes the logarithmic potential of the equilibrium measure, then (1.6) ˇQ + U σ const. The following proposition sums up some basic properties of the droplet and the function ˇQ. Proposition 1.4. uppose Q satisfies (A1) (A3). Then is a ω Jordan curve, ˇQ W 2, (), and therefore Furthermore, we have ˇQ = ( Q). (1.7) Q(z) ˇQ(z) δ(z) 2, z, δ(z) 0, where δ(z) denotes the distance from z to the droplet. (d) Joint intensities. We will occasionally use the intensity k-point function of the process Ψ n. This is the function defined by { Ψn D(z j, ε) }) In particular, R (1) n = nu n. P R (k) n n (z 1,..., z k ) = lim ε 0 ( k j=1 π k ε 2k = det ( K n (z i, z j ) ) k i,j=1.

8 8 YAIN AMEUR, HAAKAN HEDENMALM, AND NIKOLAI MAKAROV (e) Organization of the paper. We will derive the following statement which combines theorems 1.1 and 1.3 (whence, by Lemma 1.6, it implies Theorem 1.2). Main formula: Let ν n be the measure defined in (1.3). Then (1.8) lim ν n ( f ) = 1 [ ] f + f L + f N(L ) ds + 1 f h. n 8π 2π Our proof of this formula is based on the limit form of Ward s identities which we discuss in the next section. To justify this limit form we need to estimate certain error terms; this is done in ection 3. In the proof, we refer to some basic estimates of polynomial Bergman kernels, which we collect the appendix. The proof of the main theorem is completed in ection Ward identities 2.1. Exact identities. For a suitable function v on we define a random variable W n + [v] on the probability space ( n, P n ) by W n + [v] = 1 v(λ j ) v(λ k ) 2n Tr n [v Q] + Tr n [ v]. 2 λ j λ k j k Proposition 2.1. Let v : be Lipschitz continuous with compact support. Then E n W + n [v] = 0. Proof. We write where (almost everywhere) I n [v] (z) = 1 2 n j k W + n [v] = I n [v] II n [v] + III n [v] v ( z j ) v (zk ) z j z k ; II n [v] (z) = 2 n Q ( ) ( ) z j v zj j=1 ; III n [v](z) = n v ( ) z j. j=1 Let ε be a real parameter and put z j = φ ( ) ζ j = ζj + ε v ( ) ζ j /2, 1 j n. Then ( d 2 z j = φ ( ) 2 ( ) 2 ζ j + φ ) ζj d 2 ζ j = [ 1 + ε Re v ( ) ( ζ ) ] j + O ε 2 d 2 ζ j, so that (with III n = III n [v]) Moreover, so that (2.1) da n (z) = [ 1 + ε Re III n (ζ) + O ( ε 2) ] da n (ζ). log 2 2 zi z j = log ζi ζ j + log 1 + ε v(ζ i ) v(ζ j ) 2 ζ i ζ j n log 1 zj z k = j k = log ζi ζ j 2 + ε Re v(ζ i ) v(ζ j ) ζ i ζ j + O ( ε 2), n log 1 ζj ζ k ε Re In (ζ) + O ( ε 2), as ε 0. j k 2 =

9 RANDOM NORMAL MATRIE AND WARD IDENTITIE 9 Finally, so (2.2) 2n Q ( z j ) = Q (ζ j + ε 2 v ( ζ j ) ) = Q ( ζ j ) + ε Re ( Q ( ζj ) v ( ζj )), n Q ( ) n z j = 2n Q ( ) ζ j + ε Re IIn (ζ) + O ( ε 2). j=1 j=1 Now (2.1) and (2.2) imply that the Hamiltonian H n (z) = j k log zj z k 1 +2n n j=1 Q(z j) satisfies (2.3) H n (z) = H n (ζ) + ε Re ( I n (ζ) + II n (ζ)) + O ( ε 2). It follows that Z n : = e Hn(z) da n (z) = e H n(ζ) ε Re ( I n (ζ)+ii n (ζ))+o(ε2 ) [ 1 + ε Re III n (ζ) + O ( ε 2)] da n (ζ). n n ince the integral is independent of ε, the coefficient of ε in the right hand side must vanish, which means that (2.4) Re (III n (ζ) + I n (ζ) II n (ζ)) e Hn(ζ) da n (ζ) = 0, n or Re E n W n + [v] = 0. Replacing v by iv in the preceding argument gives Im E n W n + [v] = 0 and the proposition follows. Applying Proposition 2.1 to the potential Q n = Q h/n, we get the identity (2.5) Ẽ n W + n [v] = 0, where (2.6) W + n [v] = W + n [v] + 2 Tr n [v h]. If we denote B n [v] = 1 v(λ i ) v(λ j ), 2n λ i λ j i j we can rewrite (2.5) and (2.6) as follows, (2.7) Ẽ n B n [v] = 2Ẽ n Tr n [v Q] σ n ( v + 2v h), where we recall that σ n is the measure with density ũ n auchy kernels. For each z let k z denote the function k z (λ) = 1 z λ, so z σ(k z ) is the auchy transform of the the measure σ. We have (see (1.6)) σ(k z ) = 2 ˇQ(z). We will also consider the auchy integrals σ n (k z ) and σ n (k z ). We have z [σ n (k z )] = πu n (z), z [ σ n (k z )] = πũ n (z), z, and σ n (k z ) σ(k z )

10 10 YAIN AMEUR, HAAKAN HEDENMALM, AND NIKOLAI MAKAROV with uniform convergence on (the uniform convergence follows easily from the one-point function estimates in Lemma 3.1 and Theorem 3.2). Let us now introduce the functions We have D n (z) = ν n (k z ) ; D n (z) = ν n (k z ). (2.8) D n (z) = n[ σ n (k z ) 2 ˇQ(z)], D n = nπ(ũ n u), and if f is a test function, then (2.9) ν n ( f ) = 1 π f D n = 1 π f D n. Let K n denote the correlation kernel with respect to Q n. Using D n, we can rewrite the B n [v] term in the Ward identity as follows. Lemma 2.2. One has that Ẽ n B n [v] = 2 v ˇQ K n + v D n ũ n 1 2n v(z) v(w) z w (In the first integral K n (z) means the 1-point intensity R (1) n (z) = K n (z, z).) K n (z, w) 2. Proof. We have where Ẽ n B n [v] = 1 v(z) v(w) R (2) n (z, w), 2n z w 2 R (2) n (z, w) = K n (z) K n (w) K n (z, w) 2. The integral involving K n (z) K n (w) is 1 n 2 and by (2.8) σ n (k z ) = 1 n D n + 2 ˇQ. v(z) z w K n (z) K n (w) = v(z) K n (z) σ n (k z ), 2.3. Limit form of Ward s identity. The main formula (1.8) will be derived from Theorem 2.3 below. In this theorem we make the following assumptions on the vector field v: (i) v is bounded on ; (ii) v is Lip-continuous in ; (iii) v is uniformly 2 -smooth in \. (The last condition means that the restriction of v to and the restriction to ( \ ) are both 2 -smooth.) Theorem 2.3. If v satisfies (i)-(iii), then as n, 2 v D n π Q + 2 v( ˇQ Q) π D n \ 1 σ( v) 2σ(v h). 2

11 RANDOM NORMAL MATRIE AND WARD IDENTITIE 11 Before we come to the proof, we check that it is possible to integrate by parts in the second integral in Theorem 2.3. To control the boundary term we can use the next lemma. Lemma 2.4. For every fixed n we have Proof. We have ince D n (z) n = we need to show that the integrals D n (z) 1 z 2, (ũn u)d 2 λ z λ = [ 1 z λ 1 z = 1 z 2 (z ). 1 z λ 1 z λ 1 λ/z, λ ũn u d 2 λ 1 λ/z ] (ũ n u)d 2 λ are uniformly bounded. To prove this, we only need the estimate ũ n (λ) 1, which holds (for sufficiently large n) by the growth assumption (1.2) and the simple estimate ũ n (λ) exp( 2n(Q(λ) λ 3 ˇQ(λ))), which is given below in Lemma 3.1. Using that Q = ˇQ in the interior of, we deduce the following corollary of Theorem 2.3. orollary 2.5. ("Limit Ward identity ) uppose that v satisfies conditions (i)-(iii). Then as n we have the convergence 2 [ v Q + v( Q ˇQ) ] D n 1 σ( v) 2σ(v h). π Error terms and the proof of Theorem 2.3. Theorem 2.3 follows if we combine the expressions for Ẽ n B n [v] in (2.7) and Lemma 2.2 and use the following approximations of the last two terms in Lemma 2.2. More precisely, if we introduce the first error term by (2.10) 1 n v(z) v(w) z w K n (z, w) 2 = σ n ( v) + ε 1 n[v], and the second error term by (2.11) ε 2 n[v] = π v D n (ũ n u) = 1 2 v D 2 n n. Using (2.7), Lemma 2.2, and that ˇQ = Q a.e. on, one deduces that 2 (2.12) v D n π Q + 2 v( ˇQ Q) π D n = 1 \ 2 σ( v) 2σ(v h) ε1 n[v] 1 π ε2 n[v] + o(1), where o(1) = (σ σ n )( v/2+2v h) converges to zero as n by the one-point function estimates in Lemma 3.1 and Theorem 3.2. In the next section we will show that for each v satisfying conditions (i)-(iii), the error terms ε j n[v] tend to zero as n, which will finish the proof of Theorem 2.3.

12 12 YAIN AMEUR, HAAKAN HEDENMALM, AND NIKOLAI MAKAROV 3. Estimates of the error terms 3.1. Estimates of the kernel K n. We will use two different estimates for the correlation kernel, one to handle the interior and another for the exterior of the droplet. (a) Exterior estimate. Recall that K n (z, w) is the kernel of the n-point process associated with potential Q n = Q h/n; as usual, we write K n (z) = K n (z, z). We have the following global estimate, which is particularly useful in the exterior of the droplet. Lemma 3.1. For all z we have where the constant is independent of n and z. K n (z) ne 2n(Q ˇQ)(z), This estimate has been recorded (see e.g. [2], ection 3) for the kernels K n, i.e. in the case h = 0. ince obviously p e 2n Qn p e 2nQ, we have K n (z) K n (z) with a constant independent of z. Indeed, K n (z) is the supremum of p(z) 2 e 2nQ(z) where p is an analytic polynomial of degree less than n such that p e 2nQ 1, and we have an analogous supremum characterization of K n (z). Hence the case h 0 does not require any special treatment. In the following we write and δ(z) = dist(z, ) δ n = log2 n n. By our assumption on the droplet (see Proposition 1.4) we have Q(z) ˇQ(z) δ 2 (z), z, δ(z) 0. In view of the growth assumption (1.1), it follows that for any N > 0 there exists N such that K n (z) N n N when z is outside the δ n -neighborhood of. (b) Interior estimate. Recall that we assume that Q is real analytic in some neighbourhood of. This means that we can extend Q to a complex analytic function of two variables in some neighbourhood in 2 of the anti-diagonal {(z, z) : z } 2. We will use the same letter Q for this extension, so Q(z) = Q(z, z). We have and Q(z, w) = Q( w, z) 1 Q(z, z) = Q(z), 1 2 Q(z, z) = Q(z), 2 1 Q(z, z) = 2 Q(z), etc.

13 RANDOM NORMAL MATRIE AND WARD IDENTITIE 13 With the help of this extension, one can show that the leading contribution to the kernel K n is of the form (3.1) K # n(z, w) = 2 π ( 1 2 Q)(z, w) ne n[2q(z, w) Q(z) Q(w)]. In particular, we have K # n(w, w) = n Q(w), (w ). 2π We shall use the following estimate in the interior. Theorem 3.2. If z, δ(z) > 2δ n, and if z w < δ n, then K n (z, w) K = # n (z, w) + O(1), where the constant in O(1) depend on Q and h but not on n. imilar types of expansions are discussed e.g. in [5], [1], [2]. As there is no convenient reference for this particular result, and to make the paper selfcontained, we include a proof in the appendix. We now turn to the proof that the error terms ε 1 n[v] and ε 2 n[v] are negligible. ee (2.10) and (2.11). Our proof uses only the estimates of the kernels K n mentioned above. ince the form of these estimates is the same for all perturbation functions h, we can without loss of generality set h = 0, which will simplify our notation no need to put tildes on numerous letters First error term. We start with the observation that if w and δ(w) > 2δ n then at short distances the so called Berezin kernel rooted at w is close to the heat kernel B w n (z) = K n(z, w) 2 K n (w, w) H w n (z) = 1 π cne cn z w 2, c := 2 Q(w). Both kernels determine probability measures indexed by w. Most of the heat kernel measure is concentrated in the disc D(w, δ n ), H n w (z) da(z) 1 \D(w,δ n ) n, N where N denotes an arbitrary (large) positive number. Lemma 3.3. uppose that w, δ(w) > 2δ n and z w < δ n. Then Proof. By Theorem 3.2 we have B w n (z) H n w (z) nδ n. B w n (z) = K# n(z, w) 2 K # n(w, w) + O(1).

14 14 YAIN AMEUR, HAAKAN HEDENMALM, AND NIKOLAI MAKAROV Next, we fix w and apply Taylor s formula to the function z K # (z, w) at z = w. Using the explicit formula (3.1) for this function, and that we get Q(z, w) + Q(z, w) Q(z, z) Q(w, w) = [Q(z, w) Q(w, w)] + [Q(w, z) Q(z, z)] = Q(w)(z w) Q(w)(z w) 2 + Q(z)(w z) Q(z)(z w) 2 + = [ Q(w) Q(z)](z w) + 2 Q(w)(z w) 2 + = Q(w) z w and the assertion follows. K # n(z, w) 2 = 1 K # n(w, w) π [(c + O( z w )] +O(n z w 3 ) ne cn z w 2 = H n w (z) + O(n z w ), orollary 3.4. If w and δ(w) > 2δ n, then B w n (z) da(z) nδ 3 n = o(1). \D(w,δ n ) Proof. We write D n = D(w, δ n ) and notice that B n w = 1 B n w = 1 H n w + (H n w \D n D n D n D n The statement now follows from Lemma 3.3. B w n ) = H n w + (H n w B w n ). \D n D n Proposition 3.5. If v is uniformly Lipschitz continuous on, then ɛ 1 n[v] 0 as n. Proof. We represent the error term as follows: ɛ 1 n[v] = u n (w)f n (w) ; F n (w) = (w) (z) [ ] v(z) v(w) v(w) z w By the assumption that v is globally Lipschitzian, we have that u n (w)f n (w) u {δ(w)<2δ n } n (w) = o(1). {δ(w)<2δ n } If δ(w) > 2δ n, then F n (w) z D(w,δ n ) [ ] v(z) v(w) v(w) z w B w n (z) + const. z D(w,δ n ) B w n (z). B w n (z), where the last term is o(1) by orollary 3.4. Meanwhile, the integral over D(w, δ n ) is bounded by [ ] v(z) v(w) v(w) H w n (z) z w + const. B w n (z) H n w (z), z D(w,δ n ) where we can neglect the second term (see Lemma 3.3). Finally, v(z) v(w) z w v(w) = D(w,δ n ) v(w) z w z w + o(1), (this is where we use the assumption v 1 ()), so the bound of the first term is o(1) by the radial symmetry of the heat kernel.

15 RANDOM NORMAL MATRIE AND WARD IDENTITIE econd error term. We shall prove the following proposition. Proposition 3.6. If v is uniformly Lipschitzian, then ɛ 2 n[v] := 1 v D2 n 0, as n. 2 n The proof will involve certain estimates of the function K n (ζ) K # D n (z) = n(ζ) d 2 ζ. z ζ (Here K # (ζ) = nu 1.) It will be convenient to split the integral into two parts: ( ) Kn (ζ) K # D n (z) = n (z) + R n (z) := n(ζ) d 2 ζ, z ζ where B n + \B n B n = {z : δ(z) < 2δ n }. By Theorem 3.2 and Lemma 3.1 we have K n K # n 1 in \ B n, and therefore R n 1. Hence we only need to estimate n, the auchy transform of a real measure supported in B n a narrow "ring" around. We start with a simple uniform bound. Lemma 3.7. The following estimate holds, D n L n log 3 n. Proof. This follows from the trivial bound K n K # n n and the following estimate of the integral d 2 ζ z ζ. B n B n Without loosing generality, we can assume that z = 0 and replace B n by the rectangle x < 1, y < δ n. We have d 2 ζ 1 δn dy = ζ x2 + y = I + II, 2 dx 1 δ n where I is the integral over { x < δ n } and II is the remaining term. Passing to polar coordinates we get and I δn 0 II δ n 1 δ n rdr r = δ n, dx x δ n log δ n. Lemma 3.7 gives us the following estimate of the second error term, (3.2) ε 2 n[v] log 6 n v Lip,

16 16 YAIN AMEUR, HAAKAN HEDENMALM, AND NIKOLAI MAKAROV which comes rather close but is still weaker than what we want. Our strategy will be to use (3.2) and iterate the argument with Ward s identity. This will give a better estimate in the interior of the droplet. Lemma 3.8. We have that n (z) log6 n δ(z) 3, z. Proof. Let ψ be a function of Lipschitz norm less than 1 supported inside the droplet. Then we have ε 1 n[ψ] 1, ε 2 n[ψ] log 6 n, where the constants don t depend on ψ. (The first estimate follows from Proposition 3.5, and the second one is just (3.2)). This means that the error ε n := ε 1 n + ε 2 n in the identity (2.12) is bounded by log 6 n for all such ψ, i.e. (since Q = ˇQ a.e. on ), ψd n Q = O(1) + εn [ψ] log 6 n, and therefore, ψ n Q log6 n For z with δ(z) > 4δ n, we now set 2δ = δ(z) and consider the function { } δ ζ z ψ(ζ) = max, 0. Q(ζ) Then ψ has Lipschitz norm 1, and by analyticity of n we have the mean value identity δ ψ n Q = 2π n (z) (δ r)rdr = π n (z)δ 3 /3. We conclude that n (z) δ 3 log 6 n. 0 Finally, we need an estimate of n in the exterior of the droplet. This will be done in the next subsection by reflecting the previous interior estimate in the curve Γ :=. Let us fix some sufficiently small positive number, e.g. ε = 1 10 will do, and define Denote γ n = n ε. Γ n = {ζ + γ n ν(ζ) : ζ Γ}, where ν(ζ) is the unit normal vector to Γ at ζ Γ pointing outside from. We will write intγ n and extγ n for the respective components of \ Γ n. In the following, the notation a b will mean inequality up to a multiplicative constant factor times some power of log n (thus e.g. 1 log 2 n). Let L 2 (Γ n ) be the usual L 2 space of functions on Γ n with respect to arclength measure. We will use the following lemma. Lemma 3.9. We have that n 2 L 2 (Γ n ) nγ n. Given this estimate, we can complete the proof of Proposition 3.6 as follows.

17 RANDOM NORMAL MATRIE AND WARD IDENTITIE 17 Proof of Proposition 3.6. Applying Green s formula to the expression for ε 2 n[v] (see (2.11)), using that D n is, to negligible terms, analytic in extγ n, we find that ε 2 n[v] 1 n D n 2 L 2 (intγ n ) + 1 n D n 2 L 2 (Γ n ) + o(1). The second term is taken care of by Lemma 3.9. To estimate the first term denote A n = {δ(z) < γ n }. The area of A n is γ n, and in \ A n we have D n (z) γ 3 n (Lemma 3.8). We now apply the uniform bound D n n in A n (Lemma 3.7). It follows that D n 2 L 2 (intγ n ) A = + n A n + γ 6 n, n whence This finishes the proof of the proposition. \A n D n 2 L 2 (intγ n ) = o(n) Proof of Lemma 3.9. Let us first establish the following fact: (3.3) Im [ ν(ζ) n (ζ + γ n ν(ζ)) ] nγn, ζ Γ. Proof. Without loss of generality, assume that ζ = 0 and ν(ζ) = i. The tangent to Γ at 0 is horizontal, so Γ is the graph of y = y(x) where y(x) = O(x 2 ) as x 0. We will show that (3.4) Re[ n (iγ n ) n ( iγ n )] nγn. This implies the desired estimate (3.3), because by Lemma 3.8 n ( iγ n ) γ 3 n nγ n. To prove (3.4) we notice that [ ] 1 1 I := Re[ n (iγ n ) n ( iγ n )] = Re ρ n (z)da(z), B n z iγ n z + iγ n where we have put ρ n = K n K # n, viz. ρ n n. We next subdivide the belt B n = {δ(z) < 2δ n } into two parts: learly, (3.5) I n B n = B n { x γ n }, B n = B n \ B n. B n 1 1 z iγ n z iγ n B + n 1 1 z iγ n n z + iγ n. The integral over B n in the right hand side of (3.5) is estimated by y B n δ x 2 + γ 2 n γn n because if z = x + iy B n then y x 2 + δ n x 2 + γ 2 n. B n B n

18 18 YAIN AMEUR, HAAKAN HEDENMALM, AND NIKOLAI MAKAROV We estimate the integral over B n in (3.5) by It follows that B n B n γ n x 2 δ nγ n 1 γn I nδ n γn nγ n. This establishes (3.4), and, as a consequence, (3.3). dx x δ 2 n γn. To finish the proof of Lemma 3.9 we denote by ν n ( ) the outer unit normal of Γ n. Using (3.3) and Lemma 3.7 we deduce that Im ν n n nγ n on Γ n. Next let D be the exterior of the closed unit disk and consider the conformal map We put φ n : ext(γ n ) D,. F n = φ n n φ. n Then F n is analytic in ext(γ n ) including infinity, and we have (3.6) Im F n 2 L 2 (Γ n ) nγ n. To see this note that Im F n = Im[ν n n ] φ, n and recall that we have assumed that Γ is regular (A3), which means that φ n is bounded below by a positive constant. Now note that φ n (z)/φ n(z) = r n z + O(1) as z, where the r n are uniformly bounded. This gives F n ( ) = r n (K n K B # n) = r n (K # n K n ) = O(1), n \B n where we have used Lemma 3.1 and Theorem 3.2 to bound the integrand. Therefore, by (3.6), since the harmonic conjugation operator is bounded on L 2 (Γ n ), This completes the proof of Lemma 3.9. Re F n 2 L 2 (Γ n ) Im F n 2 L 2 (Γ n ) + O(1) nγ n. 4. Proof of the main formula In this section we will use the limit form of Ward s identity (orollary 2.5) to derive our main formula (1.8): for every test function f the limit ν( f ) := lim n ν n ( f ) exists and equals (4.1) ν( f ) = 1 [ ] f + f L + f N(L )ds + 1 f h. 8π 2π

19 RANDOM NORMAL MATRIE AND WARD IDENTITIE Decomposition of f. The following statement uses our assumption that is a (real analytic) Jordan curve. Lemma 4.1. Let f (). Then f has the following representation: 0 f = f + + f + f 0, where (i) all three functions are smooth on ; (ii) f + = 0 and f = 0 in \ ; (iii) f ± = O(1) at ; (iv) f 0 = 0 on. Proof. onsider the inverse conformal maps φ : D \, ψ : \ D,, where D = { z > 1}. On the unit circle T, we have F := f φ = a n ζ n (T). The functions F + (z) = 0 a n z n, F (z) = 1 a n z n, (z D ) are up to the boundary so we can extend them to some smooth functions F ± in. conformal map ψ also extends to a smooth function ψ :. It follows that and f ± satisfy (ii) (iii). Finally, we set f ± := F ± ψ 0 (), f 0 = f f + f. The onclusion. It is enough to prove the main formula (4.1) only for functions of the form f = f + + f + f 0 as in the last lemma with an additional assumption that f 0 is supported inside any given neighborhood of the droplet. Indeed, either side of the formula (4.1) will not change if we "kill" f 0 outside the neighborhood. The justification is immediate by Lemma 3.1. In what follows we will choose a neighborhood O of such that the potential Q is real analytic, strictly subharmonic in O, and and will assume supp( f 0 ) O. Q ˇQ in O \,

20 20 YAIN AMEUR, HAAKAN HEDENMALM, AND NIKOLAI MAKAROV 4.2. The choice of the vector field in Ward s identity. We will now compute the limit (and prove its existence) in the case where ν( f ) := lim ν n ( f ) f = f + + f 0. To apply the limit Ward identity 2 [ (4.2) v Q + v( Q ˇQ) ] D n 1 σ( v) 2σ(v h), (n ), π 2 (see orollary 2.5), we set where and This gives But in \ we have v 0 = v = so comparing with (2.9), we find that (4.3) 2 ν n ( f ) = 2 π v = v + + v 0, f 0 Q 1 f 0 + Q ˇQ 1 \, v + = f + Q 1. f Q 1 f 0 + Q ˇQ 1 \. v Q + v (Q ˇQ) = f, [ v Q + v( Q ˇQ) ] D n. However, to justify that (4.2) holds, we must check that v satisfies the conditions (i)-(iii) of orollary 2.5. Lemma 4.2. The vector field v defined above is Lip() and the restrictions of v to and to := (\) are. Proof. We need to check the following items: (i) v is smooth, and (i ) v is smooth; (ii) v 0 is continuous on, and (ii ) same for v +. The items (i ) and (ii ) are of course trivial. (E.g., f + 1 = f +.) Proof of (i). We have v = f 0 /g in \ where g = Q ˇQ. ince the statement is local, we consider a conformal map φ that takes a neighbourhood of a boundary point in onto a neighbourhood of a point in R and takes (parts of) to R. If we denote F = f 0 φ and G = g φ, then F = 0 and G = 0 on R. Moreover, G is real analytic with non-vanishing derivative G y. Thus it is enough to check that F(x, y) H(x, y) = y

21 RANDOM NORMAL MATRIE AND WARD IDENTITIE 21 has bounded derivatives of all orders. We will go through the details for H, H y, H yy,.... Applying the same argument to H x, we get the boundedness of the derivatives H x, H xy, H yy,..., etc. Let us show, e.g., that H := H y is bounded. We have H = yf F = y(f 0 + O(y)) (yf 0 + O(y2 )) = O(1), y 2 y 2 where F 0 := F (, 0) and all big O s are uniform in x. (They come from the bounds for the derivatives of F.) imilarly, H = y2 F 2yF + 2F y 3. The numerator is y 2 (F 0 + O(y)) 2y(F 0 + yf 0 + O(y2 )) + 2(yF y2 F 0 + O(y3 )) = O(y 3 ), etc. (We can actually stop here because we only need 2 smoothness to apply Theorem 2.3.) Proof of (ii). Let n = n(ζ) be the exterior unit normal with respect to. We have f 0 (ζ + δn) δ n f 0 (ζ) = 2δ ( f 0 )(ζ) n(ζ), as δ 0. imilarly, if g := Q ˇQ, so g = 0 on and g = Q in \, then g(ζ + δn) δ n g(ζ) = 2δ ( g)(ζ) n(ζ), as δ 0, where g(ζ) denotes the -derivative in the exterior sense. It follows that f 0 (ζ + δn) g(ζ + δn) f0 (ζ), (δ 0), Q(ζ) which proves the continuity of v 0. We have established that v = v 0 + v + satisfies conditions (i)-(iii) of orollary 2.5. Thus the convergence in (4.2) holds, and by (4.3) we conclude the following result. orollary 4.3. If f = f 0 + f +, then ν( f ) = 1 σ( v) + σ(v h). 4

22 22 YAIN AMEUR, HAAKAN HEDENMALM, AND NIKOLAI MAKAROV 4.3. onclusion of the proof. (a). Let us now consider the general case By the last corollary we have f = f + + f 0 + f. ν( f + ) = 1 4 σ( v +) + σ(v + h), v + := Using complex conjugation we get a similar expression for ν( f ): f + Q 1. Indeed, ν( f ) = 1 4 σ( v ) + σ(v h), v := f Q 1. (Recall that h is real-valued.) umming up we get (4.4) ν( f ) = 1 4 and ν( f ) = ν( f ) = 1 4 σ( v ) + σ( v h) = 1 4 σ( v ) + σ(v h). [ σ( v0 ) + σ( v + ) + σ( v ) ] (4.5) ν( f ) ν( f ) = σ(v + h) + σ(v 0 h) + σ(v h). (b) omputation of ν( f ). Recall that dσ(z) = 1 2π Q(z)1 (z)d 2 z, L = log Q. Using (4.4) we compute ν( f ) = 1 ( f0 + ) f + 2π Q Q + 1 ( ) f 2π Q Q = 1 ( ) f0 Q 1 f 0 2π Q 2π Q ( Q) + 1 ( ) f+ 2π Q Q + 1 ( ) f 2π Q Q = 1 2π f 1 f 0 L 1 f + L 1 f L 2π 2π 2π At this point, let us modify L outside some neighborhood of to get a smooth function with compact support. We will still use the notation L for the modified function. The last expression clearly does not change as a result of this modification. We can now transform the integrals involving L as follows: f 0 L f + L f L = = f L + f L, and we conclude that \ ν( f ) = 1 [ ] f + f L + f L 8π \ f 0 L + ( f + + f ) L

23 RANDOM NORMAL MATRIE AND WARD IDENTITIE 23 Note. The formula for ν( f ) was stated in this form in [3]. Let us finally express the last integral in terms of Neumann s jump. We have ( f L = f L L f ) In conclusion, \ (4.6) ν( f ) = 1 8π = = = [ \ f + ( f n L n f L ) ds ( f n L f n L ) ds f N(L ) ds ] f L + f N(L ). (c) omputation of [ ν( f ) ν( f )]. Using the identity (4.5), we can deduce that ν( f ) ν( f ) = 2 [ ] f + h + f h + f 0 h π = 1 f h. 2π This is because and similarly On the other hand, Therefore, and this is equal to Applying (4.6) we find that ν( f ) = 1 8π f + h = f + h = f h = 4 f 0 h = 1 4 f + h = 1 4 f h. f 0 h = 1 4 f 0 h. f + h, ν( f ) ν( f ) = 1 [ ] f h + f h, 2π \ [ 1 f h = 1 f h. 2π 2π f + ] f L + f N(L ) + 1 f h, 2π and the main formula (4.1) has been completely established. q.e.d.

24 24 YAIN AMEUR, HAAKAN HEDENMALM, AND NIKOLAI MAKAROV Appendix: Bulk asymptotics for the correlation kernel Polynomial Bergman spaces. For a suitable (extended) real valued function φ, we denote by L 2 φ the space normed by f 2 φ = f 2 e 2φ. We denote by A 2 φ the subspace of L2 consisting of a.e. φ entire functions; P n (e 2φ ) denotes the subspace consisting of analytic polynomials of degree at most n 1. Now consider a potential Q, real analytic and strictly subharmonic in some neighborhood of the droplet, and subject to the usual growth condition. We put Q Q n = Q 1 n h, where h is a smooth bounded real function. We denote by K the reproducing kernel for the space P n (e 2nQ ), and write K w (z) = K(z, w). The corresponding orthogonal projection is denoted by P n : L 2 nq P n(e 2nQ ) : f ( f, K w )nq. The map P n : L 2 n Q P n(e 2n Q ) is defined similarly, using the reproducing kernel K for the space P n (e 2n Q ). We define approximate kernels and Bergman projection as follows. In the case h = 0, the wellknown first order approximation inside the droplet is given by the expression K # w(z) = 2 π ( 1 2 Q)(z, w) ne 2nQ(z, w), where Q(, ) is the complex analytic function of two variables satisfying Q(w, w) = Q(w). If the perturbation h 0 is a real-analytic function, we can just replace Q by Q in this expression. Note that in this case, the analytic extension h(, ) satisfies h(z, w) = h(w) + (z w) h(w) +..., (z w). This motivates the definition of the approximate Bergman kernel in the case where h is only a smooth function: we set K # w(z) = K # w(z) e 2h w(z), where h w (z) := h(w) + (z w) h(w). The approximate Bergman projection is defined accordingly: P # n f (w) = ( f, K # w) n Q. The kernels K # n(z, w) do not have the Hermitian property. The important fact is that they are analytic in z.

25 RANDOM NORMAL MATRIE AND WARD IDENTITIE 25 Proof of Theorem 3.2. We shall prove the following estimate. Lemma A.1. If z, δ(z) > 2δ n, and if z w < δ n, then K w (z) K w(z) # e nq(z) e nq(w). Before we prove the lemma, we use it to conclude the proof of Theorem 3.2. Recall that K n (z, w) = K w (z) e n Q(z) e n Q(w). If we define then by Lemma A.1, On the other hand, we have K # n(z, w) = K # w(z) e n Q(z) e n Q(w), K n (z, w) = K # n(z, w) + O(1). K # n(z, w) = K # n(z, w) e h(z)+h(w) 2h w(z), so It follows that as claimed in Theorem 3.2. It remains to prove Lemma A.1. K # n(z, w) = K # n(z, w) (1 + O( w z 2 ) = K # n(z, w) + O(1). K n (z, w) = K # n(z, w) + O(1), Lemma A.2. If f is analytic and bounded in D(z; 2δ n ) then f (z) P # n(χ z f )(z) 1 n e nq(z) f nq. Here χ = χ z is a cut-off function with χ = 1 in D(z; 3δ n /2) and χ = 0 outside D(z; 2δ n ) satisfying χ 2 1. Proof. Wlog, z = 0, so P # n(χ f )(0) is the integral I # = 1 χ(ζ) f (ζ) 2( 1 2 Q)(0, ζ) e 2[h(ζ) h(0) ζ h(0)] ne 2n[Q(ζ, ζ) Q(0, ζ)]. π ince ζ [ e 2n[Q(ζ, ζ) Q(0, ζ)] ] = 2[ 2 Q(ζ, ζ) 2 Q(0, ζ)] ne 2n[Q(ζ, ζ) Q(0, ζ)], we can rewrite the expression as follows: I # = 1 1 π ζ f (ζ)χ(ζ)a(ζ)b(ζ) [ e 2n[Q(ζ, ζ) Q(0, ζ)] ], where and A(ζ) = A trivial but important observation is that ζ ( 1 2 Q)(0, ζ) 2 Q(ζ, ζ) 2 Q(0, ζ), B(ζ) = e 2[h(ζ) h(0) ζ h(0)]. A, B = O(1), A = O( ζ ), B = O( ζ ),

26 26 YAIN AMEUR, HAAKAN HEDENMALM, AND NIKOLAI MAKAROV where the O-constants have uniform bounds throughout. Integrating by parts we get where Using that ɛ 1 = ɛ 1 1 δ n f ( χ) AB ζ e 2n[Q(ζ, ζ) Q(0, ζ)] f χ e 2n[Q(ζ) Re Q(0, ζ)] and noting that Taylor s formula gives I # = f (0) + ɛ 1 + ɛ 2,, ɛ 2 = f χ (AB) ζ e 2n[Q(ζ) Q(0, ζ)]., ɛ 2 χ f e 2n[Q(ζ) Re Q(0, ζ)], e n[q(ζ) 2 Re Q(0, ζ)] e nq(0) cn ζ 2, (c Q(0) > 0, ζ 2δ n ) we find, by the auchy chwarz inequality, (since ζ δ n when χ(ζ) 0) and ɛ 1 e nq(0) e cnδ2 n δ n f nq χ L 2 1 n f nq ɛ 2 e nq(0) f nq ( e nc ζ 2 ) 1/2 1 n f nq. The proof is finished. uppose now that dist(z, \ ) 2δ n and w z δ n. From Lemma A.2, we conclude that (4.7) K [ w (z) P n χz K w] # (z) e nq(z) e nq(w). This is because P # n [ ] χz K z (w) = (χz K z, K w) # n Q = (χ z K w, # K z ) n Q = [ ] P n χ K w # (z), so K w (z) P n [ χz K # w] (z) = and because (cf. [2], ection 3) K z = K [ ] z (w) P # n χz K z (w) K z (z) ne nq(z). On the other hand, we will prove that (4.8) K [ z(w) # P n χz K z] # (w) e nq(z) e nq(w), which combined with (4.7) proves Lemma A.1. The verification of the last inequality is the same as in [5] or [1], depending on the observation that L 2 nq = L2 with equivalence of norms. We give n Q a detailed argument, for completeness. For given smooth f, consider u, the L 2 -minimal solution to the problem nq (4.9) u = f and u f Pn 1. ince u n Q u nq, the L 2 n Q -minimal solution ũ to the problem (4.9) satisfies ũ n Q u nq. We next observe that P n f is related to the L 2 nq -minimal solution u to the problem (4.9) by u = f P n f.

27 RANDOM NORMAL MATRIE AND WARD IDENTITIE 27 We write [ u(ζ) = χ z (ζ) K w(ζ) # P n χz K w] # (ζ), i.e., u is the L 2 nq -minimal solution to (4.9) for f = χ z K w. # Let us verify that (4.10) u nq 1 ( χz K # nq w). n To prove this, we put 2φ(ζ) = 2 ˇQ(ζ) + n 1 log ( 1 + ζ 2), and consider the function v 0, the L 2 nφ -minimal solution to the problem v = ( χ z K w) #. Notice that φ is strictly subharmonic on. By Hörmander s estimate (e.g. [12], p. 250) v 0 2 nφ ( χ z k ) w # 2 e 2nφ n φ. ince χ z is supported in, we hence have v 0 nφ 1 n ( χz K # w) nq. We next observe that by the growth assumption on Q near infinity, we have an estimate nφ nq + const. on, which gives v 0 nq v 0 nφ. It yields that v 0 nq 1 n ( χz K # w) nq. But v 0 χ z K # w belongs to the weighted Bergman space A 2 nφ. ince 2nφ(ζ) = (n + 1) log ζ 2 + O(1) as ζ, the latter space coincides with P n 1 as sets. This shows that v 0 solves the problem (4.9). ince u nq v 0 nq, we then obtain (4.10). By norms equivalence, (4.10) implies that (4.11) ũ n Q 1 n ( χz K # w) n Q, where ũ = χ z K # w P n [ χz K # w is the L 2 n Q -minimal solution to (4.9) with f = χ z K # w. We now set out to prove the pointwise estimate (4.12) ũ(z) ne cnδ2 n e n(q(z)+q(w)). To prove this, we first observe that ũ(ζ) = ( χ z K # w) (ζ) = χz (ζ) K # w(ζ), whence, by the form of K w # and Taylor s formula, 2 ũ(ζ) e 2nQ(ζ) n 2 χz (ζ) 2 e 2n(Q(w) c ζ w 2 ) with a positive constant c Q(z). ince ζ w δ n /2 when χ(ζ) 0, it yields ( χ z K w) # 2 e 2nQ(ζ) n 2 χz (ζ) 2 e 2nQ(w) cnδ 2 n. ]

28 28 YAIN AMEUR, HAAKAN HEDENMALM, AND NIKOLAI MAKAROV We have shown that ( χ z K w) # nq ne ncδ2 n e nq(w). In view of the estimate (4.11), we then have ũ nq ne ncδ2 n e nq(w). ince ũ is analytic in D(z; 1/ n) we can now invoke the simple estimate (e.g. [2], Lemma 3.2) to get ũ(z) 2 e 2nQ(z) n ũ 2 nq ũ(z) ne ncδ2 n e n(q(z)+q(w)). This gives (4.8), and finishes the proof of Lemma A.1. Remark 4.4. The corresponding estimate in [1], though correct, contains an unnecessary factor " n. References [1] Ameur, Y., Near boundary asymptotics for correlation kernels, J. Geom. Anal. (2011), available online at DOI /s [2] Ameur, Y., Hedenmalm, H., Makarov, N., Berezin transform in polynomial Bergman spaces, omm. Pure Appl. Math. 63 (2010), [3] Ameur, Y., Hedenmalm, H., Makarov, N., Fluctuations of eigenvalues of random normal matrices, Duke Math. J. 159 (2011), [4] Ameur, Y., Kang, N.-G., Makarov, N., In preparation. [5] Berman, R., Bergman kernels and weighted equilibrium measures of n. Indiana Univ. J. Math. 5 (2009). [6] Berman, R., Determinantal point processes and fermions on complex manifolds: bulk universality, Preprint in 2008 at arxiv.org/abs/math.v/ [7] Borodin, A., Determinantal point processes, Preprint in 2009 at ArXiv.org/ [8] Borodin, A., inclair,. D., The Ginibre ensemble of real random matrices and its scaling limits, ommun. Math. Phys. 291 (2009), [9] Elbau, P., Felder, G., Density of eigenvalues of random normal matrices, ommun. Math. Phys. 259 (2005), [10] Hedenmalm, H., Makarov, N., oulomb gas ensembles and Laplacian growth, Preprint in 2011 at arxiv.org/abs/math.pr/ [11] Hedenmalm, H., himorin,., Hele-haw flow on hyperbolic surfaces, J. Math. Pures. Appl. 81 (2002), [12] Hörmander, L., Notions of convexity, Birkhäuser [13] Johansson, K., On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J. 91 (1998), [14] Mehta, M. L., Random matrices, Academic Press [15] Rider, B., Virág, B., The noise in the circular law and the Gaussian free field, Internat. Math. Research notices 2007, no. 2. [16] aff, E. B., Totik, V., Logarithmic potentials with external fields, pringer [17] akai, M., Regularity of a boundary having a chwarz function, Acta Math. 166 (1991), [18] oshnikov, A., Determinantal random point fields, Russ. Math. urv. 55 (2000), [19] Wiegmann, P., Zabrodin, A., Large N expansion for the 2D Dyson gas, J. Phys. A.: Math. Gen. 39 (2006), [20] Wiegmann, P., Zabrodin, A., Large N expansion for the normal and complex matrix ensembles, Frontiers in number theory, physics, and geometry I (2006), Part I, [21] Wiegmann, P., Zabrodin, A., Large scale correlations in normal non-hermitian matrix ensembles, J. Phys. A.: Math. Gen. 36 (2003), [22] Zabrodin, A., Matrix models and growth processes: from viscous flows to the quantum Hall effect, Preprint in 2004 at arxiv.org/abs/hep-th/

29 RANDOM NORMAL MATRIE AND WARD IDENTITIE 29 Yacin Ameur, entre for Mathematical ciences, Lund University, weden address: Makarov: Mathematics, alifornia Institute of Technology, Pasadena, A 91125, UA address: makarov@caltech.edu