Determinantal point processes associated with Hilbert spaces of holomorphic functions

spaces of holomorphic functions

Alexander I. Bufetov, Yanqi Qiu

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arXiv:1411.4951v2 [math.PR] 12 Sep 2015

Hilbert spaces of holomorphic functions

Alexander I. Bufetov, Yanqi Qiu

Abstract

We study determinantal point processes on C induced by the reproducing ker-nels of generalized Fock spaces as well as those on the unit disc D induced by the reproducing kernels of generalized Bergman spaces. In the first case, we show that all reduced Palm measures of the same order are equivalent. The Radon-Nikodym derivatives are computed explicitly using regularized multiplicative functionals. We also show that these determinantal point processes are rigid in the sense of Ghosh and Peres, hence reduced Palm measures of different orders are singular. In the sec-ond case, we show that all reduced Palm measures, of all orders, are equivalent. The Radon-Nikodym derivatives are computed using regularized multiplicative function-als associated with certain Blaschke products. The quasi-invariance of these deter-minantal point processes under the group of diffeomorphisms with compact supports follows as a corollary.

Keywords. Determinantal point processes, Palm measures, generalized Fock spaces,

generalized Bergman spaces, regularized multiplicative functionals, rigidity.

Contents

1 Introduction 2

1.1 Main results . . . 2

1.1.1 The case of C . . . 2

1.1.2 The case of D . . . 4

Alexander I. Bufetov: Aix-Marseille Universit´e, Centrale Marseille, CNRS, I2M, UMR7373, 39 Rue F. Joliot Curie 13453, Marseille Cedex 13; Steklov Institute of Mathematics, Moscow; Institute for In-formation Transmission Problems, Moscow; National Research University Higher School of Economics, Moscow; e-mail: [email protected]

Yanqi Qiu: Aix-Marseille Universit´e, Centrale Marseille, CNRS, I2M, UMR7373, 39 Rue F. Joliot Curie 13453, Marseille Cedex 13; e-mail: [email protected]

Mathematics Subject Classification (2010): Primary 60G55; Secondary 30H20, 30B20

1.2 Quasi-invariance . . . 6

1.3 Unified approach for obtaining Radon-Nikodym derivatives . . . 7

1.3.1 Relations between Palm subspaces . . . 7

1.3.2 Radon-Nikodym derivatives as regularized multiplicative func-tionals . . . 8

1.4 Organization of the paper . . . 8

2 Spaces of configurations and determinantal point processes 9 2.1 Additive functionals and multiplicative functionals . . . 9

2.2 Locally trace class operators and their kernels . . . 10

2.3 Definition of determinantal point processes . . . 11

2.4 Palm measures and Palm subspaces . . . 11

2.5 Rigidity . . . 12

3 Generalized Fock spaces and Bergman spaces 13 4 Regularized multiplicative functionals 14 5 Case of C 16 5.1 Examples . . . 16

5.2 Proof of Theorem1.1 . . . 18

5.3 Proof of Proposition1.2. . . 21

6 Case of D 23 6.1 Analysis of the conditions on the weightω . . . 23

6.2 Proof of Theorem1.4and Proposition1.5 . . . 24

7 Proof of Theorem4.1 25

8 Appendix 38

1

Introduction

1.1

Main results

1.1.1 The case of C

Let_{ψ : C → R be a C}2-smooth function and equip the complex plane C with the

mea-sure e−2ψ(z)_{dλ(z), where dλ is the Lebesgue measure. Assume that there exist positive}

constantsm, M > 0 so that

where∆ is the Euclidean Laplacian differential operator.

Denote by Fψthe generalized Fock space with respect to the weighte−2ψ(z)and letBψ

be the reproducing kernel of Fψ, whose definition is recalled in Definition3.1. The

con-dition (1) implies in particular the useful Christ’s pointwise estimate for the reproducing kernelBψ, see Theorem3.1below.

By the Macch`ı-Soshnikov theorem, the kernelBψ induces a determinantal point

pro-cess on C, which will be denoted by PBψ. For more background on determinantal point processes, see, e.g. [11], [14], [21], [15] and_{§2 below.}

Letp _{∈ C}ℓ andq_{∈ C}k be two tuples of distinct points in C. Denote by Pp_B

ψ and P

q

Bψ

the reduced Palm measures of PBψ conditioned atp and q respectively. For the definition, see, e.g. [12], here, we follow the notation and conventions of [1].

Our first main result is that, under the assumption (1), Palm measures Pp_B

ψ and P

q

Bψ of

the same order are equivalent.

Theorem 1.1 (Palm measures of the same order). Letψ satisfy (1) and let p_{, q ∈ C}ℓ be any two tuples of distinct points in C. Then

1) The limit Σp,q(Z) := lim R→∞  X z∈Z:|z|≤R log    (z − p_{(z − q}1₁) . . . (z − p_{) . . . (z − qℓ}ℓ₎)     − EPq Bψ X z∈Z:|z|≤R log    (z − p1) . . . (z − pℓ ) (z − q1) . . . (z − qℓ)      exists for Pq_B

ψ-almost every configuration Z and the function Z → e

2Σp,q(Z)_is inte-grable with respect to Pq_B

ψ. 2) The Palm measures Pp_B

ψ and P

q

Bψ are equivalent. Moreover, for P

q Bψ-almost every configuration Z, we have dPp_Bψ dPq_Bψ(Z) = e2Σp,q(Z) E_Pq Bψ(e 2Σp,q). (2)

Definition 1.1 (Ghosh [8], Ghosh-Peres[9]). A point process P on C is said to be rigid if for any bounded open set_{D ⊂ C with Lebesgue-negligible boundary ∂D, there exists a} functionFDdefined on the set of configurations, measurable with respect to theσ-algebra

generated by the family of random variables _{#A : A ⊂ C \ D bounded and Borel},

where#Ais defined by

#A(Z) = the cardinality of the finite set Z ∩ A,

such that

Proposition 1.2 (Rigidity). Under the assumption (1), the determinantal point process

P_B

ψ is rigid in the sense of Ghosh and Peres. Proposition8.1in the Appendix now implies

Corollary 1.3 (Palm measures of different orders). Under the assumption (1), if_{ℓ 6= k,} then the reduced Palm measures Pp_Bψ and Pq_Bψ are mutually singular.

Remark 1.1. In the particular case ψ(z) = 1

2|z|

2 _{(Ginibre point process), the results of}

Theorem1.1and Corollary1.3were obtained in [17] with a different approach, where the authors used finite dimensional approximation by orthogonal polynomial ensembles. The rigidity in the caseψ(z) = 1

2|z|

2 _{is due to Ghosh and Peres [9], their original approach}

will be followed in our proof of Proposition1.2.

1.1.2 The case of D

In the case of Bergman spaces on the unit disc D, the situation becomes quite different and the corresponding determinantal point processes in this case are not rigid.

Consider a weight function_{ω : D → R}+ and equip D with the measure ω(z)dλ(z).

Assume thatω satisfies that Z

D

(1 − |z|)2Bω(z, z)ω(z)dλ(z) < ∞. (3)

We will denote by Bωthe generalized Bergman space on D with respect to the weightω,

and byBω its reproducing kernel, the definition is recalled in Definition3.2.

Again, by the Macch`ı-Soshnikov theorem, the reproducing kernelBω induces a

deter-minantal point process on D, which we denote by PBω.

Letp _{∈ D}ℓ be anℓ-tuple of distinct points in D and denote by Pp_Bω the reduced Palm measures of PBω atp.

Under the assumption (3), we show, for anyp_{∈ D}ℓof distinct points in D, the reduced Palm measure Pp_Bω is equivalent to PBω. In particular, any two reduced Palm measures are equivalent. For the weight_{ω ≡ 1, this result is due to Holroyd and Soo [}10].

We now proceed to the statement of our main result in the case of D. For anℓ-tuple p= (p1, · · · , pℓ) of distinct points in D, set

bp(z) = ℓ Y j=1 z − pj 1 − ¯pjz . (4)

Theorem 1.4. Letω be a weight such that (3) holds. Letp _{∈ D}ℓ be anℓ-tuple of distinct

1) The limit Sp(Z) := lim r→1−   X z∈Z:|z|≤r log |bp(z)| − EP_Bω X z∈Z:|z|≤r log |bp(z)|   (5)

exists for PBω-almost every configuration Z and the function Z → e

2Sp(Z) is inte-grable with respect to PBω.

2) The Radon-Nikodym derivativedPp_Bω/dPBω is given by the formula:

dPp_Bω dPBω (Z) = e 2Sp(Z) E_P Bω(e

2Sp), for PBω-almost every configuration Z. (6) Theorem1.4will be obtained from

Proposition 1.5. Let ω be a weight such that (3) holds. Letp _{∈ D}ℓ andq _{∈ D}k be two tuples of distinct points in D. Then the Radon-Nikodym derivativedPp_Bω/dPq_Bω is given by

dPp_Bω dPq_Bω(Z) = e2Sp,q(Z) E_Pq Bω(e 2Sp,q), for P q

Bω-almost every configuration Z, (7)

whereSp,q(Z) is defined for Pq_Bω-almost every configuration Z, given by

Sp,q(Z) := lim r→1−   X z∈Z:|z|≤r log |bp(z)bq(z)−1| − EPq Bω X z∈Z:|z|≤r log |bp(z)bq(z)−1|   . (8)

Remark 1.2. Ifψ (resp. ω) is a radial function, then the monomials (zn₎

n≥0 are

orthogo-nal in the corresponding Hilbert space, hence the determinantal point process PBψ (resp.

P_B

ω) can be naturally approximated by orthogonal polynomial ensembles. In particular, ifψ(z) = 1_2|z|2 _{for all z ∈ C, then P}

Bψ is the Ginibre point process, see chapter 15 of Mehta’s book [16]; if_{ω(z) ≡ 1 for all z ∈ D, then P}Bω is the determinantal point pro-cess describing the zero set of a Gaussian analytic function on the hyperbolic disc D, see [18]. Our study, however, goes beyond the radial setting and our methods work for more general phase spaces as well.

Remark 1.3. The regularized multiplicative functionals are necessary in Theorem 1.1, Theorem1.4and Proposition1.5: indeed, when _{ω ≡ 1, for P}Bω-almost every configura-tion Z on D, the points in the configuraconfigura-tion Z violate the Blaschke condiconfigura-tion:

X

z∈Z

(1 − |z|) = ∞, (9)

whence for anyp_{∈ D}ℓ, we have,

Y

z∈Z

so the simple multiplicative functional is identically0. To see (9), we use the Kolmogorov three-series theorem and the fact (Peres and Vir´ag [18]) that, for PBω-distributed random configurations Z, the set of moduli_{{|z| : z ∈ Z} has same law as the set of random} vari-ables _{Uk1/(2k)_{}, where U}1, U2, . . . are independent identically distributed random

vari-ables such thatU1has a uniform distribution in[0, 1]. A direct computation shows that

E_P Bω X z∈Z (1 − |z|) =X k (1 − E(Uk1/(2k))) = ∞.

The determinantal point process PBω in the case ω ≡ 1 describes the zero set of a Gaussian analytic function on D:

FD(z) = ∞

X

n=0

gnzn,

where (gn)n≥0 is a sequence of independent identically distributed standard complex

Gaussian random variables. Direct computation shows that

E_kFDk2

H2 = ∞ and EkFDk2_Bω = ∞,

hence the random holomorphic function almost surely belongs neither to the Hardy space

H2_{nor to the Bergman space, thus it is not surprising that the zero set ofF}

Dalmost surely

violates Blaschke condition.

1.2

Quasi-invariance

Let_{U = C or D. Let F : U → U be a diffeomorphism. Its support, denoted by supp(F ),} is defined as the relative closure in_{U of the subset {z ∈ U : F (z) 6= z}. The totality of} diffeomorphisms with compact supports is a group denoted byDiffc(U), i.e.,

Diffc(U) :=

n

F : U → UF is a diffeomorphism and supp(F ) is compacto.

The groupDiffc(U) naturally acts on the set of configurations on U: given any

diffeomor-phism_{F ∈ Diff}c(U) and any configuration Z on U,

(F, Z) 7→ F (Z) := {F (z) : z ∈ Z}.

Recall that the JacobianJF of the functionF : U → U is defined by

JF(z) = | det DF (z)|.

Corollary 1.6. Let PKbe a determinantal point process onU, which is either the

Assumption (1) in the case of C or, in the case of D Assumption (3), PK is quasi-invariant

under the induced action of the groupDiffc(U).

More precisely, let _{F ∈ Diff}c(U) and let V ⊂ U be any precompact subset

con-tainingsupp(F ). For PK-almost every configuration Z the following holds: if ZTV =

{q1, . . . , qℓ}, then dPK◦ F dPK (Z) =det[K(F (qi), F (qj))] ℓ i,j=1 det[K(qi, qj)]ℓi,j=1 ·dP p K dPq_K(Z) · ℓ Y i=1 JF(qi), whereq= (q1, . . . , qℓ) ∈ Uℓandp= (F (q1), . . . , F (qℓ)) ∈ Uℓ

Proof. This is an immediate consequence of Theorem1.1, Proposition1.5 and Proposi-tion 2.9 of [1].

1.3

Unified approach for obtaining Radon-Nikodym derivatives

In this section, let us describe briefly the main idea of our unified approach for obtaining the Radon-Nikodym derivatives in Theorem1.1, Theorem1.4and Proposition1.5.

1.3.1 Relations between Palm subspaces

Ifp_{∈ C}ℓ is anℓ-tuple of distinct points of C, we define the Palm subspace:

F_{ψ(p) := {ϕ ∈ Fψ : ϕ(p1) = · · · = ϕ(pℓ) = 0} . (11)}

LetBp_ψdenote the reproducing kernel of Fψ(p).

Similarly, ifp_{∈ D}ℓ is anℓ-tuple of distinct points of D, we define the Palm subspace B_{ω(p) = {ϕ ∈ Bω : ϕ(p1) = · · · = ϕ(pℓ) = 0} , (12)}

and denote its reproducing kernel byBp

ω.

By Shirai-Takahashi’s theorem, which motivates our terminology, see Theorem 2.1

below, these Palm subspaces are related to the reduced Palm measures:B_ψp (resp.Bp

ω) is

the correlation kernel of Pp_B

ψ (resp. P p Bω), i.e., we have Pp Bψ = PBψp (resp. P p Bω = PBωp).

Proposition 1.7. For any pair of_{ℓ-tuples p, q ∈ C}ℓof distinct points in C, we have

F_{ψ(p) =} (z − p1) · · · (z − pℓ) (z − q1) · · · (z − qℓ) · Fψ

Proposition 1.8. Let _{k, ℓ ∈ N ∪ {0} and let p ∈ D}ℓ_{, q ∈ D}k be two tuples of distinct points in D, then B_{ω(p) =} ℓ Y j=1 z − pj 1 − ¯pjz k Y j=1 z − qj 1 − ¯qjz !−1 · Bω(q). (14) In particular, we have B_{ω(p) =} ℓ Y j=1 z − pj 1 − ¯pjz · B ω.

Comments. _{• The proofs of Propositions}1.7and 1.8are immediate from the defini-tions (11) and (12) and basic properties of holomorphic functions.

• Notice the analogy of the above Propositions1.7 and 1.8 with Proposition 3.4 in [1].

• A common feature, which is crucially used later, of Propositions1.7and1.8, is the following relations lim |z|→∞    (z − p1) · · · (z − pℓ ) (z − q1) · · · (z − qℓ)     = 1 and lim|z|→1−      ℓ Y j=1 z − pj 1 − ¯pjz     = 1. (15)

The rate of convergence in (15) also plays an important rˆole for defining the regu-larized multiplicative functionals, see_§5.2and_§6.2.

1.3.2 Radon-Nikodym derivatives as regularized multiplicative functionals

For obtaining the Radon-Nikodym derivatives in question, we will first develop in The-orem4.1, the most technical result of this paper, a general method on regularized mul-tiplicative functionals. This result, an extension of Proposition 4.6 of [1], is, we hope, interesting in its own right; the stronger statement is also necessary for our argument in the case of C, in which Proposition 4.6 of [1] is not applicable.

By Theorem4.1, under the assumption (1) onψ, we can show that the regularized

mul-tiplicative functional, i.e., the formula (7), is well-defined. This regularized multiplicative functional is then shown to be exactly the Radon-Nikodym derivative between the desired reduced Palm measures of the same order for the determinantal point process PBψ.

The regularized multiplicative functionals in the case of D are technically simpler and the full force of Theorem4.1is not needed.

1.4

Organization of the paper

The paper is organized as follows. In the introduction section_§1, we give necessary defi-nitions and notation and state our main results. The basic materials in the theory of deter-minantal point processes are recalled in_§2. The definitioins concerning generalized Fock

spaces and generalized Bergman spaces are given in_§3. In _§4, our main ingredient, reg-ularized multiplicative functionals, is defined. We also state the most technical Theorem

4.1 in_§4. We then apply Theorem4.1 to prove our main results for determinantal point processes associated with generalized Fock spaces in_§5and to prove the main results in the case of generalized Bergman spaces in_§6. The section_§7 is devoted to the proof of Theorem4.1. In the Appendix _§8, we give details for the fact that rigid point processes have singular Palm measures with different orders.

Remark 1.4. Part of our main results in this paper were announced in [3].

2

Spaces of configurations and determinantal point

pro-cesses

For the reader’s convenience, we recall the basic definitions and notation on determinantal point processes.

LetE be a locally compact complete separable metric space equipped with a

sigma-finite Borel measureµ. The space E will be later referred to as phase space. The measure µ is referred to as reference measure or background measure. By a configuration X on the

phase space_{E, we mean a locally finite subset of X ⊂ E. By identifying any configuration}

X_{∈ Conf(E) with the Radon measure} mX:=

X

x∈X

δx,

whereδx is the Dirac mass on the pointx, the space of configurations Conf(E) is

iden-tified with a subset of the space M(E) of Radon measures on E and becomes itself a

complete separable metric space. We equipConf(E) with its Borel sigma algebra.

Points in a configuration will also be called particles. In this paper, the italicized letters as X, Y, Z always denote configurations.

2.1

Additive functionals and multiplicative functionals

We recall the definitions of additive and multiplicative functionals on the space of config-urations.

If_{ϕ : E → C is a measurable function on E, then the additive functional (which is} also called linear statistic)Sϕ : Conf(E) → C corresponding to ϕ is defined by

Sϕ(X) =

X

x∈X

ϕ(x)

provided the sumP_x∈Xϕ(x) converges absolutely. If the sum P_x∈Xϕ(x) fails to

Similarly, the multiplicative functional Ψg : Conf(E) → [0, ∞] associated with a

non-negative measurable function_{g : E → R}+, is defined as the function

Ψg(X) :=

Y

x∈X

g(x),

provided the product Q

x∈Xg(x) absolutely converges to a value in [0, ∞]. If the product

Q

x∈X

g(x) fails to converge absolutely, then the multiplicative functional is not defined at

the configuration X.

2.2

Locally trace class operators and their kernels

LetL2_{(E, µ) denote the complex Hilbert space of C-valued square integrable functions}

onE. Let S1(E, µ) be the space of trace class operators on L2(E, µ) equipped with the

trace class norm_{k · k}S₁. Let S_1,loc(E, µ) be the space of locally trace class operators, that

is, the space of bounded operatorsK : L2_{(E, µ) → L}2_{(E, µ) such that for any bounded}

subset_{B ⊂ E, we have}

χBKχB ∈ S1(E, µ).

A locally trace class operatorK admits a kernel, for which we use the same symbol K. In this paper, we are especially interested in locally trace class orthogonal projection

operators. Let, therefore,_{Π ∈ S}1,locbe an operator of orthogonal projection onto a closed

subspace_{L ⊂ L}2(E, µ). All kernels considered in this paper are supposed to satisfy the

following

Assumption 1. There exists a subset e_{E ⊂ E, satisfying µ(E \ e}E) = 0 such that

• For any q ∈ eE, the function vq(·) = Π(·, q) lies in L2(E, µ) and for any f ∈

L2_{(E, µ), we have}

(Πf )(q) = hf, vqiL2_(E,µ).

In particular, if_{f is a function in L, then by letting f (q) = hf, v}qiL2_(E,µ), for any

q ∈ eE, the function f is defined everywhere on eE (which is slightly stronger than

almost everywhere defined onE).

• The diagonal values Π(q, q) of the kernel Π are defined for all q ∈ eE and we have Π(q, q) = hvq, vqiL2_(E,µ). Moreover, for any bounded Borel subsetB ⊂ E,

tr(χBΠχB) =

Z

B

2.3

Definition of determinantal point processes

A Borel probability P on Conf(E) will be called a point process on E. Recall that the

point process P is said to admitk-th correlation measure ρkonEkif for any continuous

compactly supported functionϕ : Ek_{→ C, we have}

Z Conf(E) ∗ X x1,...,xk∈X ϕ(x1, . . . , xk)P(dX) = Z Ek ϕ(q1, . . . , qk)dρk(q1, . . . , qk), where ∗ P

denotes the sum over all orderedk-tuples of distinct points (x1, . . . , xk) ∈ Xk.

Given a bounded measurable subset_{A ⊂ E, we define #}A: Conf(E) → N ∪ {0} by

#A(X) = the number of particles in X ∩ A.

Then the point process P is determined by the joint distributions of #A1, . . . , #An, if

A1, . . . , Anrange over the family of bounded measurable subsets ofE.

A Borel probability measure P onConf(E) is called determinantal if there exists an

operator_{K ∈ S}1,loc(E, µ) such that for any bounded measurable function g, for which

g − 1 is supported in a bounded set B, we have

E_{PΨg = det (1 + (g − 1)KχB) . (16)}

The Fredholm determinant is well-defined since_{(g − 1)Kχ}B ∈ S1(E, µ). The equation

(16) determines the measure P uniquely and we will denote it by PK and the kernelK

is said to be a correlation kernel of the determinantal point process PK. Note that PK is

uniquely determined byK, but different kernels may yield the same point process.

By the Macch`ı-Soshnikov theorem [15], [21], any Hermitian positive contraction in

S_{1,loc(E, µ) defines a determinantal point process. In particular, the projection operator}

on a reproducing kernel Hilbert space induces a determinantal point process.

Remark 2.1. If_{α : E → C is a Borel function such that |α(x)| = 1 for µ-almost every}

x ∈ E, and if Π ∈ S1,locis the operator of orthogonal projection onto a closed subspace

L ⊂ L2_{(E, µ), then Π and αΠα define the same determinantal point process, i.e.,}

P_{αΠα= PΠ.}

Note thatαΠα is the orthogonal projection onto the subspace α(x)L.

2.4

Palm measures and Palm subspaces

In this paper, by Palm measures, we always mean reduced Palm measures. We refer to [12], [5] for more details on Palm measures of general point processes.

Let P be a point process onConf(E). Assume that P admits k-th correlation measure ρk onEk. Then for ρk-almost everyq = (q1, . . . , qk) ∈ Ek of distinct points in E, one

can define a point process onE, denoted by Pq_{and is called (reduced) Palm measure of P}

conditioned atq, by the following disintegration formula: for any non-negative Borel test

function_{u : Conf(E) × E}k _{→ R,} Z Conf(E) ∗ X q1,...,qk∈X u(X; q)P(dX) = Z Ek ρk(dq) Z Conf(E) u(X ∪ {q1, . . . , qk}; q)Pq(dX), (17) where ∗ P

denotes the sum over all mutually distinct pointsq1, . . . , qk∈ X.

Informally, Pq is the conditional distribution of X_{\ {q}1, . . . , qk} on Conf(E)

condi-tioned to the event that all particlesq1, . . . , qkare in the configuration X, providing that X

has as distribution P.

Now let PΠ be a determinantal point process on Conf(E) induced by the projection

operatorΠ. Let q = (q1, . . . , qk) ∈ eEkbe ak-tuple of distinct points in eE ⊂ E, where eE

is as in Assumption1. Set

L(q) = {ϕ ∈ L : ϕ(q1) = · · · = ϕ(qk) = 0}. (18)

The space L(q) will be called the Palm subspace of L2_{(E, µ) corresponding to q. Both}

the operator of orthogonal projection fromL2_{(E, µ) onto the subspace L(q) and the}

re-producing kernel ofL(q) will be denoted by Πq_.

Explicit formulae for Πq _{in terms of the kernel Π are known, see Shirai-Takahashi}

[20]. Here we recall that for a single point_{q ∈ e}E, we have Πq_{(x, y) = Π(x, y) −} Π(x, q)Π(q, y)

Π(q, q) . (19)

IfΠ(q, q) = 0, we set Πq _{= Π. In general, we have the iteration}

Πq

= (· · · (Πq1₎q2_{· · · )}qk_.

Note that the order of the pointsq1, q2, · · · qk has no effect in the above iteration.

Theorem 2.1 (Shirai and Takahashi [20]). For any_{k ∈ N and for ρ}k-almost everyk-tuple

q_{∈ E}k of distinct points inE, the Palm measure Pq_Πis induced by the kernelΠq_:

Pq

Π= PΠq.

2.5

Rigidity

Let P be a point process over C. We will use the following result on the rigidity of point processes (see Definition1.1).

Theorem 2.2 (Ghosh [8], Ghosh and Peres [9]). Let P be a point process on C whose first correlation measureρ1is absolutely continuous with respect to the Lebesgue measure and

let D be an open bounded set D with Lebesgue-negligible boundary. Letϕ be a continuous

function on C. Suppose that for any0 < ε < 1, there exists a C2

c-smooth functionΦεsuch

thatΦε= ϕ on D, and VarP(SΦε) < ε. Then the point process P is rigid.

3

Generalized Fock spaces and Bergman spaces

Let_{ψ : C → R be a function satisfying the assumption (}1) and denote

dvψ(z) = e−2ψ(z)dλ(z),

where dλ is the Lebesgue measure on C. Let O(C) denote the space of holomorphic

functions on C.

Definition 3.1. If the linear subspace

F_{ψ := L}2_{(C, dvψ) ∩ O(C)}

is closed in L2_{(C, dv}

ψ), then it will called generalized Fock space with respect to the

measure dvψ. The orthogonal projection P : L2(dvψ) → Fψ is given by integration

against a reproducing kernelBψ(z, w) (analytic in z and anti-analytic in w):

(P f )(z) = Z

C

f (w)Bψ(z, w)e−2ψ(w)dλ(w). (20)

Definition 3.2. Let D _{⊂ C be the open unit disc. A weight function ω : D → R}+ is called a Bergman weight, if it is integrable with respect to the Lebesgue measure and the generalized Bergman space

B_{ω := L}2_{(D, ωdλ) ∩ O(D)}

is closed and the evaluation functionals _{f → f(z) on B}ω are uniformly bounded on

any compact subset of D. In such situation, the space Bω is a reproducing kernel Hilbert

space, its reproducing kernel will be denoted asBω.

We shall need Christ’s pointwise estimate (cf. [4], [6], [19]) of the reproducing kernel

Bψ(z, w). Theorem 3.2 in [19] gives the estimate in the form most convenient for us.

Theorem 3.1 (Christ). Let_{ψ ∈ C}2(C) be a real-valued function satisfying (1). Then there are contants_{δ, C > 0 such that for all z, w ∈ C,}

|Bψ(z, w)|2e−2ψ(z)−2ψ(w) ≤ Ce−δ|z−w|. (21)

In particular, for all_{z ∈ C,}

Remark 3.1. For the Gaussian caseψ(z) = 1_2|z|2_{, we have the following explicit formula}

|Bψ(z, w)|2e−2ψ(z)−2ψ(w) = π−2e−|z−w|

2

.

4

Regularized multiplicative functionals

As (10) shows, simple multiplicative functionals cannot be used in our situation. Follow-ing [1], we use regularized multiplicative functionals whose definition we now recall.

Let_{f : E → C be a Borel function. Set}

Var(Π, f ) = 1 2

ZZ

E2|f(x) − f(y)|

2_{|Π(x, y)|}2_{dµ(x)dµ(y). (23)}

Introduce the Hilbert space V(Π) in the following way: the elements of V(Π) are functions f on E satisfying Var(Π, f ) < ∞; functions that differ by a constant are identified. The

square of the norm of an element_{f ∈ V(Π) is precisely Var(Π, f).}

LetSf : Conf(E) → C to be the corresponding additive functional, such that Sf ∈

L1_{(Conf(E), P}

Π), then we set

Sf = Sf − EP_ΠSf. (24)

If moreover,Sf ∈ L2(Conf(E), PΠ), then it is easy to see that

E_P

Π|Sf|

2 _{= Var}

P_Π(Sf) = Var(Π, f ). (25)

Definition 4.1. Let V0(Π) be the subset of functions f ∈ V(Π), such that there exists an

exhausting sequence of bounded subsets(En)n≥1, depending onf , so that

f χEn

V(Π)

−−−→_n→∞ f.

The identity (25) implies that there exists a unique isometric embedding (as metric spaces)

S : V0(Π) → L2(Conf(E), PΠ)

extending the definition (24), so that we have

Sf = lim n→∞ X x∈X∩En f (x) − EP_Π X x∈X∩En f (x).

Definition 4.2. Given a non-negative function_{g : E → R such that log g ∈ V}0(Π), then

we set

e

Ψg = exp(Slog g).

If moreover, eΨg ∈ L1(Conf(E), PΠ), then we set

Ψg = e Ψg E_P ΠΨeg .

The functionΨg is called the regularized multiplicative functional associated tog and

P_{Π. For specifying the dependence on PΠ, the notationΨ}Π

g will also be used. By definition,

for PΠ-almost every configuration X, the following identity holds:

log ΨΠ_g(X) = lim n→∞ X x∈X∩En log g(x) − EP_Π X x∈X∩En log g(x) ! . (26)

Clearly,ΨΠ_g is a probability density for PΠ, since EP_Π(Ψ Π g) = 1.

Theorem 4.1. LetE0 ⊂ E be a Borel subset satisfying tr(χE0ΠχE0) < ∞ and such that if_{ϕ ∈ L satisfies χ}E\E0ϕ = 0, then ϕ = 0 identically.

Let_{g be a nonnegative Borel function on E satisfying g|}E0 = 0, g|E0c > 0 and such that for anyε > 0 the subset Eε = {x ∈ E : |g(x) − 1| ≥ ε} is bounded. Assume moreover

that there exists an increasing sequence of bounded subsets(En)n≥1exhausting the whole

phase spaceE and

Z En |g(x) − 1|Π(x, x)dµ(x) < ∞; (27) Z Ec n |g(x) − 1|3Π(x, x)dµ(x) < ∞; (28) ZZ Ec n×Ecn

|g(x) − g(y)|2|Π(x, y)|2dµ(x)dµ(y) < ∞. (29)

And lim n→∞tr(χEnΠ|g − 1| 2_χ Ec nΠχEn) = 0. (30) Then e Ψg ∈ L1(Conf(E), PΠ).

If the subspace √gL is closed and the corresponding operator of orthogonal projection Πg _{satisfies, for sufficiently largeR, the condition}

tr(χg>RΠgχg>R) < ∞ (31)

then we also have

P_Πg = Ψ

Π

Remark 4.1. Note that tr(χEnΠ|g − 1| 2_χ Ec nΠχEn) = Z En dµ(y) Z Ec n |g(x) − 1|2|Π(x, y)|2dµ(x).

Remark 4.2. The above theorem is an extension of Proposition 4.6 of [1]: we replace the convergence ofR_Ec

ε|g(x) − 1|

2_{Π(x, x)dµ(x) in Proposition 4.6 of [1] by the convergence}

of R_Ec

ε|g(x) − 1|

3_{Π(x, x)dµ(x). This extension is crucial for treating the case of Fock}

space, since the former condition is already violated in the case of the Ginibre point process.

5

Case of C

5.1

Examples

In this section, we assume that_{ψ : C → R is a measurable function on C, the condition} (1) is not necessarily satisfied. Recall that we denote dvψ(z) = e−2ψ(z)dλ(z) and

de-note Fψ =  f : C → Cf holomorphic,R C |f|2_dv ψ < ∞ 

. If the evaluation functionals evz(f ) := f (z) defined on Fψ are uniformly bounded on compact subsets, then Fψ is a

closed subspace ofL2_{(C, dv}

ψ). In this case, denote by Bψ the reproducing kernel of Fψ,

we have Bψ(z, w) = ∞ X j=1 fj(z)fj(w),

where(fj)∞j=1is any orthonormal basis of Fψ.

Assumption 2. The measuredvψ satisfies

(1) the evaluation functionalsevz defined on Fψ are uniformly bounded on compact

subsets;

(2) the polynomials are dense in Fψ;

(3) R_C1+|z|1 2Bψ(z, z)dvψ(z) < ∞.

Example 5.1 (A radial case). Let α > 0, and set ψα(z) = 1₂|z|α, then the measure

dvψα(z) = e

−|z|α

dλ(z) satisfies Assumption 2 if and only if 0 < α < 2. Indeed, the

first two conditions in Assumption2are satisfied bydvψα by allα > 0. Now one can see that the third condtion is equivalent to

∞ X n=1 kzn−1_k2 L2_(dv ψ) kzn_k2 L2_(dv ψ) < ∞. (32)

A direct computation shows that kznk2L2_(dv ψ) = 2π α Γ  2n + 2 α  and kz n−1_k2 L2_(dv ψ) kzn_k2 L2_(v ψ) ∼ 1 n2/α. (33)

The series (32) converges if and only if0 < α < 2.

Remark 5.2. As shown in Example5.1, the third condition in Assumption2is too strict: indeed, it fails already for the Ginibre point process (corresponding toψ(z) = 1

2|z|

2_).

Let PBψbe the determinantal point process induced by the operatorBψ. For anyℓ-tuple

q= (q1, . . . , qℓ) ∈ Cℓof distinct points, set

F_{ψ(q) :=}n_{f ∈ Fψf(q1) = · · · = f(qℓ) = 0}o_,

and letB_ψq denote the operator of orthogonal projection onto F_ψq. Recall that the Palm distribution Pq_B ψ of PBψ conditioned atq is induced byB q ψ, i.e., Pq Bψ = PBqψ.

Given a positive integer_{ℓ ∈ N, introduce the closed subspace}

F(ℓ) ψ := n f ∈ Fψ   f(0) = f′(0) = · · · = f(ℓ−1)(0) = 0o. (34)

DenoteB(ℓ)_ψ the operator of orthogonal projection onto F_ψ(ℓ). Let P(ℓ)_B

ψ be the determinantal point process induced byB_ψ(ℓ).

Remark 5.3. In general, we do not have F_ψ(ℓ) = zℓ_F

ψ. Indeed, let ψ(z) = 1₂|z|2, we have

zFψ 6⊂ Fψ. This can be seen from the closed graph theorem: otherwise, the operator

Mz : Fψ → Fψ of multiplication by the function z is bounded, which contradicts the

explicit computation (33): kMzkF_ψ→Fψ ≥ sup n kzn+1_k F_ψ kzn_k F_ψ = ∞;

see also the related discussion after Theorem 2 in [7].

Proposition 5.1. Ifψ satisfies Assumption 2, then for any _{ℓ ∈ N and any ℓ-tuple q =}

(q1, . . . , qℓ) ∈ Cℓ of distinct points, we have equivalence of measures:

Pq

Bψ ≃ P

(ℓ)

Bψ.

Moreover, if one sets

gq(z) =    (z − q1) . . . (z − qℓ ) zℓ     2 ,

then the Radon-Nikodym derivative is given by the regularized multiplicative functional

dPq_Bψ dP(ℓ)_Bψ = Ψ

B_ψ(ℓ)

gq .

In particular, given any twoℓ-tuples q and q′ _{of distinct points, the corresponding Palm}

measures Pq_B

ψ and P

q′

Bψ are equivalent.

Proof. First note that, under Assumption2, for any_{ℓ ∈ N and any ℓ-tuple q = (q}1, . . . , qℓ) ∈

Cℓ_{of distinct points,}

F_{ψ(q) =} (z − q1) . . . (z − qℓ)

zℓ F

(ℓ)

ψ .

Next we use Proposition 4.6 of [1]. We now verify the assumption of Proposition 4.6 of [1] for the pair(B_ψ(ℓ), g). Note that B_ψ(ℓ)_{(z, z) = O(|z|}2ℓ_{) for |z| → 0 and |g(z) − 1|}2 ₌

O (1/|z|2_{), for |z| → ∞. Recall that B}(ℓ)

ψ (z, z) ≤ Bψ(z, z). Hence, under Assumption2,

we have Z |z|≤1 |g(z) − 1|Bψ(ℓ)(z, z)dvψ(z) + Z |z|≥1 |g(z) − 1|2B_ψ(ℓ)(z, z)dvψ(z) < ∞.

The pair(B_ψ(ℓ), g) satisfies all assumptions of Proposition 4.6 in [1], and Proposition5.1

follows immediately.

5.2

Proof of Theorem

1.1

We now derive Theorem1.1from Theorem4.1. From now on, the functionψ is assumed

to satisfy the condition (1) until the end of this paper.

Let_{ℓ ≥ 1 and let p = (p}1, . . . , pℓ), q = (q1, . . . , qℓ) ∈ Cℓ be any two fixedℓ-tuples of

distinct points; letg be the function defined by the formula g(z) = |gp,q(z)|2 =    (z − p_{(z − q}1₁) · · · (z − p_{) · · · (z − qℓ}ℓ₎)     2 . (35)

Let0 < ε < 1 be a small fixed number. Choose Rε > max{|pk|, |qk| : k = 1, . . . , ℓ},

large enough, such that outsideEε := {z ∈ C : |z| ≤ Rε}, we have |g(z) − 1| ≤ ε.

Finally, for_{n ∈ N, let E}n = {z ∈ C : |z| ≤ max(Rε, n)}.

We start with a simple but very useful observation that conditions (28), (29), (30) and (31) in Theorem4.1are preserved under taking finite rank pertubation.

Remark 5.4. Assume that the pair(g, Π) satisfies the conditions (28), (29), (30) and (31) in Theorem4.1. If eΠ = Π + Π′_{, where Π}′ _{has finite rank and Ran(Π) ⊥ Ran(Π}′_{), or}

e

Π = Π − Π′_{, whereΠ}′_{has finite rank andRan(Π}′_{) ⊂ Ran(Π), then conditions (28), (29),}

(30) and (31) hold for the new pair(g, eΠ) . If g is unbounded, then the condition (27) for the pair(g, Π) does not imply the condition for the pair (g, eΠ). The condition (27) is on the other hand usually easy to check directly.

Lemma 5.2. Letg be the function defined by the formula (35). We have Z En |g(z) − 1|Bψq(z, z)e −2ψ(z) dλ(z) < ∞ and Z Ec n |g(z) − 1|3B_ψq(z, z)e−2ψ(z)_{dλ(z) < ∞.}

Proof. We first note that for any smallε > 0, there exists Cε > 0, such that if |z −qk| < ε,

then

B_ψq_{(z, z) ≤ C}ε|z − qk|2. (36)

Indeed,B_ψq is the orthogonal projection to the subspace Fψ(q), hence we have

B_ψq(z, w) =

∞

X

k=1

fj(z)fj(w), (37)

where (fj)∞j=1 is any orthornomal basis of Fψ(q). The convergence is uniform on any

compact subset of C. Thus the function_{|g(z) − 1|B}q_ψ(z, z)e−2ψ(z) _{is bounded onE} n, this

implies the first inequality in the lemma.

By Theorem3.1, there exists a constantC > 0, such that B_ψp(z, z)e−2ψ(z) _{≤ B}ψ(z, z)e−2ψ(z) ≤ C.

Since_{|g(z) − 1|}3_{= O(1/|z|}3_{) as |z| → ∞, there exists C}′ > 0, such that Z Ec n |g(z) − 1|3B_ψq(z, z)e−2ψ(z)_{dλ(z) ≤ C}′ Z |z|≥Rε 1 |z|3dλ(z) < ∞.

Lemma 5.3. Letg be the function defined by the formula (35). We have

ZZ

Ec

ε×Eεc

|g(z) − g(w)|2|Bψp(z, w)|2dvψ(z)dvψ(w) < ∞. (38)

Proof. Since B_ψp is a finite rank perturbation of Bψ, and since g is bounded on Eεc, it

suffices to show that

I1 :=

ZZ

|z|≥Rε,|w|≥Rε

|g(z) − g(w)|2|Bψ(z, w)|2dvψ(z)dvψ(w) < ∞. (39)

Christ’s pointwise estimate, (21) in Theorem3.1, implies that there exists_{α ∈ C, such} that g(z) = 1 +α z + ¯ α ¯ z + O 1/|z| 2
_as |z| → ∞.

Thus it suffices to show that I2 := ZZ |z|≥Rε,|w|≥Rε     1 z − 1 w     2 e−δ|z−w|_{dλ(z)dλ(w) < ∞.} (40)

To this end, write

I2 = Z |w|≥Rε dλ(w) Z |ζ+w|≥Rε |ζ|2 |w(w + ζ)|2e −δ|ζ|_dλ(ζ) ≤ Z |w|≥Rε dλ(w) Z |ζ+w|≥Rε χ{|w|≥2|ζ|} |ζ| 2 |w(w + ζ)|2e −δ|ζ|_dλ(ζ) + Z |w|≥Rε dλ(w) Z |ζ+w|≥Rε χ{|w|<2|ζ|} |ζ| 2 |w(w + ζ)|2e −δ|ζ|_dλ(ζ).

The first integral is controlled by

4 Z |w|≥Rε dλ(w) Z C |ζ|2 |w|4e −δ|ζ|_{dλ(ζ) < ∞,}

while the second integral is controlled by

Z |w|≥Rε dλ(w) Z C χ{|w|<2|ζ|} |ζ| 2 |Rεw|2 e−δ|ζ|dλ(ζ) =2π Z 2|ζ|≥Rε log  2|ζ| Rε  |ζ|2 |Rε|2 e−δ|ζ|_{dλ(ζ) < ∞.}

The proof of the lemma is complete.

Lemma 5.4. Letg be the function defined by the formula (35). We have

lim n→∞tr(χEnB p ψ|g − 1|2χEc nB p ψχEn) = 0. (41)

Proof. SinceB_ψp is a finite rank perturbation ofBψ, by Remark 5.4, it suffices to check

the same condition (41) for the new pair (g, Bψ). By applying again Christ’s pointwise

estimate (21), we have I3(n) :=tr(χEnBψ|g − 1| 2_χ Ec nBψχEn) = kχEnBψ|g − 1|χEnck 2 HS = Z |z|≤n Z |w|≥n |g(w) − 1|2|Bψ(z, w)|2e−2ψ(z)−2ψ(w)dλ(z)dλ(w) ≤C Z |z|≤n Z |w|≥n |g(w) − 1|2e−δ|z−w|dλ(z)dλ(w) ≤C′ Z |z|≤n Z |w|≥n 1 |w|2e −δ|z−w|_{dλ(z)dλ(w) = C}′ Z |w|≥n dλ(w) |w|2 Z |w+ζ|≤n e−δ|ζ|dλ(ζ) ≤C′ Z |w|≥n dλ(w) |w|2 Z |w|−n≤|ζ|≤|w|+n e−δ|ζ|dλ(ζ) = 4π2C′ Z s≥n ds s Z s+n s−n re−δrdr.

Now since there existsC′′ _{> 0, such that re}−δr_{≤ C}′′_e−δr/2_{for allr ≥ 0, we have} I3(n) ≤ C′′′ Z s≥n e−δ(s−n)/2 s ds = C ′′′ Z ∞ 1 e−δn(t−1)/2 t dt.

By dominated convergence theorem, we have

lim

n→∞I3(n) = 0.

Proof of Theorem1.1. By Lemma5.2, Lemma5.3and Lemma 5.4, the conditions (27), (28), (29), (30) are satisfied by the pair(g, B_ψq). Moreover, let

α(z) = |gp,q(z)| gp,q(z)

,

then by Proposition1.7, we have

p

g(z)Fψ(q) = α(z)gp,q(z)Fψ(q) = α(z)Fψ(p).

Hence pg(z)Fψ(q) is a closed subspace of L2(dvψ). And (B_ψq)g = αB_ψpα is locally¯

of trace class, this implies the condition (31). Now the formula (2) of Radon-Nikodym derivativedPp_Bψ/dPq_Bψ follows from Theorem4.1and Remark2.1.

Remark 5.5. Under the condition (1), we also have the same result as in Proposition5.1.

5.3

Proof of Proposition

1.2

Lemma 5.5. There exists a constant C > 0, depending only on ψ, such that for any C2_{-smooth compactly supported functionϕ : C → R, we have}

VarP_Bψ(S_ϕ) ≤ C

Z

Ck∇ϕ(w)k 2

2dλ(w). (42)

Proof. Let _{ϕ : C → R be a C}2-smooth compactly supported function. Our convention for the Fourier transform ofϕ will be

b ϕ(ξ) =

Z

C

ϕ(w)e−i2πhw,ξi_{dλ(w), where hz, wi := ℜ(z)ℜ(w) + ℑ(z)ℑ(w).}

By definition, we have VarP_Bψ(Sϕ) = 1 2 ZZ C2|ϕ(z) − ϕ(w)| 2 |Bψ(z, w)|2e−2ψ(z)−2ψ(w)dλ(z)dλ(w).

By Theorem3.1and Plancherel identity for Fourier transform, we obtain VarP Bψ(Sϕ) ≤ C Z Z C2|ϕ(z) − ϕ(w)| 2_e−δ|z−w|_dλ(z)dλ(w) = C ZZ C2|ϕ(ζ + w) − ϕ(w)| 2_e−δ|ζ|_{dλ(w)dλ(ζ)} = C ZZ C2|e i2πhξ,ζi − 1|2| bϕ(ξ)|2e−δ|ζ|dλ(ξ)dλ(ζ).

Now since_|ei2πhξ,ζi_{− 1| = 2| sin(πhξ, ζi)| ≤ 2π|ξ||ζ|, we have}

VarP_Bψ(Sϕ) ≤ C′ ZZ C2|ξ| 2 | bϕ(ξ)|2|ζ|2e−δ|ζ|dλ(ξ)dλ(ζ) ≤ C′′ Z C|ξ| 2 | bϕ(ξ)|2dλ(ξ) = C′′ Z Ck∇ϕ(w)k 2 2dλ(w).

Proof of Proposition1.2. We will follow the argument of Ghosh and Peres [9]. By The-orem2.2, it suffices, for any fixed bounded open set D with Lebesgue-negligible bound-ary and any ε > 0, to construct a function Φε ∈ Cc2(C) such that Φε|D ≡ 1 and

VarP_Bψ(SΦε) < ε.

Letr0 = 2 sup{|z| : z ∈ D}. By Lemma5.5, it suffices to construct a radial function

Φε(z) = φε(|z|),

withφεa function inCc2(R+) such that φε|[0,r0/2] ≡ 1 and

Z ∞

0 |φ

′

ε(r)|2rdr < ε.

To this end, first we take ˜φε(r) = (1 − ε log+(r/r0))+, wherelog+(x) = max(log x, 0).

Note that ˜φε|[r0exp(1/ε),∞) ≡ 0 and ˜φ

′

ε(r) = −ε/r on the interval (r0, r0exp(1/ε)).

Next we smooth the function ˜φε at the points r0 and r0exp(1/ε) to obtain a function

φε ∈ Cc2(R+) such that φε identically equals to 1 on [0, r0/2] and φ′ε is supported on

[r0/2, 2r0exp(1/ε)] such that |φ′ε(r)| ≤ ε/r for all r > 0. Hence we have

Z ∞ 0 |φ ′ ε(r)|2rdr ≤ Z 2r0exp(1/ε) r0/2 ε2 r dr = ε + ε 2_{log 4.}

6

Case of D

6.1

Analysis of the conditions on the weight

ω

Let_{ω : D → R}+be a Bergman weight. We collect some known results from the literature on the sufficient conditions on the Bergman weightω, so that the inequality (3):

Z

D

(1 − |z|)2Bω(z, z)ω(z)dλ(z) < ∞

holds.

Example 6.1 (Classical weights). Assume_{ω(z) = (1 − |z|}2)α_{, α > −1. Then}

Bω(z, w) =

α + 1 π

1 (1 − z ¯w)α+2,

hence_{(1 − |z|)}2Bω(z, z)ω(z) is bounded and the inequality (3) holds.

Example 6.2 (A class of logarithmatically superharmonic weights). Let

ω(z) = e−2ϕ(z).

Assume

1) _{ϕ ∈ C}2(D) and ∆ϕ > 0;

2) the function(∆ϕ(z))−1/2_{is Lipschitz on D;}

3) there existC1, a > 0 and 0 < t < 1, such that

(∆ϕ(z))−1/2 _{≤ C}1(1 − |z|);

(∆ϕ(z))−1/2 _{≤ (∆ϕ(w))}−1/2_{+ t|z − w| for |z − w| > a(∆ϕ(w))}−1/2.

By [13, Lemma 3.5], the weightω is a Bergman weight and sup

z∈D(1 − |z|)

2_B

ω(z, z)ω(z) < ∞.

Hence the inequality (3) holds. Some concrete such examples are

• ω(z) = (1 − |z|2₎α_{exp(h(z)) with α > 0 and h(z) any real harmonic function on}

D;

• ω(z) = (1 − |z|2₎α_{exp(−β(1 − |z|}2₎−γ_{+ h(z)) with α ≥ 0, β > 0, γ > 0 and h(z)}

Proposition 6.1. Letω1, ω2 be two Bergman weights on D such that

Z

D(1 − |z|)

2_B

ω1(z, z)ω2(z)dλ(z) < ∞.

Letω be a Bergman weight on D and assume that there exist c, C > 0 such that cω1(z) ≤ ω(z) ≤ Cω2(z)

thenω satisfies the condition (3).

Proof. Since Bω(z, z) = supkf k_{Bω ≤1}|f(z)|2, we have Bω(z, z) ≤ c2Bω1(z, z). By the assumption, we have Z D(1 − |z|) 2_B ω(z, z)ω(z)dλ(z) ≤ c2C Z D(1 − |z|) 2_B ω1(z, z)ω2(z)dλ(z) < ∞.

Example 6.3. Letω be a Bergman weight. Assume that there exist c, C > 0 and let α, β

be either_{0 ≥ α ≥ β > −1 or α ≥ β > α − 1 ≥ −1, such that}

c(1 − |z|2)α _{≤ ω(z) ≤ C(1 − |z|}2)β

thenω satisfies the condition (3).

6.2

Proof of Theorem

1.4

and Proposition

1.5

Let_{k, ℓ ∈ N ∪ {0}, let p ∈ D}ℓ be an _{ℓ-tuple of distinct points and q ∈ D}k a k-tuple of

distinct points. Set

g(z) = |bp(z)bq(z)−1|2 = ℓ Y j=1    _{1 − ¯p}z − p_jj_z     2 · k Y j=1    1 − ¯qj z z − qj     2 .

By virtue of Proposition1.8, to prove Proposition1.5and hence Theorem 1.4, it suf-fices to prove that the pair(g, Bq

ω) satisfies the assumption of Proposition 4.6 of [1]. This

is done in the following

Lemma 6.2. Takeε > 0 small enough and let Eε = Sk_i=1Uε(qi), where Uε(qi) is a disc

centred at pointqiwith radiusε in D. Then we have

Z Eε |g(z) − 1|Bq ω(z, z)ω(z)dλ(z) + Z Ec ε |g(z) − 1|2Bq ω(z, z)ω(z)dλ(z) < ∞. (43)

Proof. For_{ε > 0 small enough, there exists C > 0 such that for any z ∈ E}ε, we have Bq ω(z, z) ≤ C k Y i=1 |z − qi|2, whence_{|g(z) − 1|B}q

ω(z, z) is bounded on Eε, and the first integral in (43) is bounded.

For the second integral, the identities

   _{1 − ¯p}z − p_jj_z     2 = 1 − (1 − |z| 2_{)(1 − |p} j|2) |1 − ¯pjz|2 ,

together with the same identities forqj : j = 1, . . . , k, imply that there exists C′ > 0 such

that

|g(z) − 1| ≤ C′(1 − |z|) for z ∈ Eεc.

Note also that sinceRan(Bq

ω) ⊂ Ran(Bω), we have Bωq(z, z) ≤ Bω(z, z), hence by our

assumption (3), we have Z Ec ε |g(z) − 1|2Bq ω(z, z)ω(z)dλ(z) ≤ C′ Z Ec ε (1 − |z|)2Bω(z, z)ω(z)dλ(z) < ∞.

7

Proof of Theorem

4.1

Recall that we denote byΠ an orthogonal projection on L2_{(E, µ) which is locally in trace}

class.

In [1], a class of Borel functions onE, denoted there by A2(Π), plays a central role in

the proof of the main result. Recall that, by definition, A2(Π) is the set of positive Borel

functionsg on E satisfying

(1) 0 < inf

E g ≤ sup_E g < ∞;

(2) R_{E|g(x) − 1|}2_{Π(x, x)dµ(x) < ∞.}

If_{g ∈ A}2(Π), then the subspace √gL, where L is the range of the orthogonal projection

Π, is automatically closed; we set Πg _{to be the corresponding operator of orthogonal}

projection. The main property of A2(Π) that will be used later is stated in the following

Proposition 7.1 (Cor. 4.11 of [1]). If_{g ∈ A}2(Π) satisfies

sup

E |g(x) − 1| < 1.

Then the operatorΠg_{is locally of trace class, and we have}

P_Πg = Ψ

Π

Let_{g : E → R be a Borel function, set} L(g) := Z E|g(x) − 1| 3_{Π(x, x)dµ(x) ∈ [0, ∞] (45)} and V (g) := ZZ E2|g(x) − g(y)| 2 |Π(x, y)|2dµ(x)dµ(y) ∈ [0, ∞]. (46)

And then, we introduce a new class of Borel functions onE as follows. Let A3(Π) be the

set of positive Borel functionsg on E satisfying

(1) 0 < inf

E g ≤ sup_E g < ∞;

(2) L(g) =R_{E|g(x) − 1|}3_{Π(x, x)dµ(x) < ∞;}

(3) V (g) =RR_E2|g(x) − g(y)|2|Π(x, y)|2dµ(x)dµ(y) < ∞;

(4) there exists an exhausting sequence (En)n≥1 of bounded subsets of E, possibly

depending ong, such that lim

n→∞tr(χEnΠ|g − 1|

2_χ

Ec

nΠχEn) = 0. (47)

More precisely, Relation (47) can equivalently be rewritten as follows:

lim n→∞ Z Z E2 χEc n(x)χEn(y)|g(x) − 1| 2 |Π(x, y)|2dµ(x)dµ(y) = 0. (48)

Remark 7.1. We have the following useful identity

V (g) = k[g, Π]k2HS, (49)

where_k·kHSstands for the Hilbert-Schmidt norm and[g, Π] = gΠ−Πg is the commutator

of the operator of multiplication byg and the projection operator Π.

Remark 7.2. The sequence(En)n≥1 in the definition of A3(Π) is an analogue of the

se-quence of the subsets_{({z ∈ C : |z| ≤ n})}n≥1in the proof of Lemma5.4.

The most technical result in this section is the following Proposition 7.2. If_{g ∈ A}3(Π) satisfies

sup

E |g(x) − 1| < 1.

(50)

Then the operatorΠg_{is locally of trace class, and we have}

P_Πg = Ψ

Π

Remark 7.3. Note that the condition (47) holds automatically for any_{g ∈ A}2(Π), hence

we have

A_{2(Π) ⊂ A3(Π).}

Proof of Theorem4.1. We now derive Theorem 4.1 from Proposition 7.2. The proof is similar to the proof of Proposition 4.6 of [1]. Proving the statement for A3(Π) instead of

A_{2(Π) requires extra effort, however. For sake of completeness, let us sketch the proof}

here.

Let_{Conf(E; E \ E}0)) stand for the subset of Conf(E) consisting of those

configu-rations whose particles all lie in _{E \ E}0. The assumptions of Theorem 4.1 imply that

P_{Π(Conf(E; E \ E0)) > 0. Replacing, if necessary, g by g|E}c

0 andL by χE0cL, we may assume thatg is positive on E.

By our assumption, we may choose0 < ε1 < ε2 < 1 and a bounded subset E1 ⊂ E,

such that {x ∈ E : |g(x) − 1| ≥ ε2} ⊂ E1 ⊂ {x ∈ E : |g(x) − 1| ≥ ε1}, and kχ{x∈E:|g(x)−1|≤ε2}Πk < 1. DecomposeE1 = E1+⊔ E1−by setting E₁+ _{= {x ∈ E : g(x) > 1} ∩ E}1 and E1− = {x ∈ E : g(x) < 1} ∩ E1. Note that E₁+_{⊂ {x ∈ E : g(x) > 1 + ε}1} and E1− ⊂ {x ∈ E : g(x) < 1 − ε1}.

Then we can decomposeg as g = g1g2g3 with

g1 = (g − 1)χEc 1 + 1,

g2 = (g − 1)χE₁−+ 1,

g3 = (g − 1)χE₁+ + 1.

Claim. We haveg1 ∈ A3(Π).

Indeed, the first two and the last condition in the definition of A3(Π) are immediate

forg1. We now check the third condition. We have

|g1(x) − g1(y)| =        |g(x) − g(y)| (x, y) ∈ Ec 1× E1c |g(x) − 1| (x, y) ∈ Ec 1× E1 |g(y) − 1| (x, y) ∈ E1× E1c 0 _{(x, y) × E}1× E1 ,

whence

V (g1) =

ZZ

E2|g

1(x) − g1(y)|2|Π(x, y)|2dµ(x)dµ(y)

= ZZ

Ec

1×E1c

|g(x) − g(y)|2|Π(x, y)|2dµ(x)dµ(y)

+ 2 Z E1 dµ(y) Z Ec 1 |g(x) − 1|2|Π(x, y)|2dµ(x).

By (29), (30) and Remark4.1, we haveV (g1) < ∞.

By Proposition7.2, we have

P_{Πg1 = Ψ}Π

g1 · PΠ.

The rest of the proof of Theorem4.1 follows the scheme of the proof of Proposition 4.6 of [1]. First, we have

Πg1g2 _{= (Π}g1₎g2 _{and Π}g _{= Π}g1g2g3 _{= (Π}g1g2₎g3_.

Sinceg2is bounded andg2−1 is compactly supported, the usual multiplicative functional

Ψg2(X) =

Y

x∈X

g2(x),

is well defined and

P_{Πg1g2 = C1Ψg}

2PΠg1.

The functiong3− 1, although not necessarily bounded, is compactly supported and

pos-itive. The usual multiplicative functionalΨg3 is also well defined for PΠg1g2-almost every configuration. Indeed, since g1g2 is bounded and by Proposition 4.1 of [1], there exists

C > 0 such that Πg1g2_{(x, x) ≤ CΠ(x, x).} Consequently, we have Z E|g 3(x) − 1|Πg1g2(x, x)dµ(x) ≤ C Z E₁+|g 3(x) − 1|Π(x, x)dµ(x) < ∞. (52)

In the relation (52), we used the fact thatg3− 1 is supported on E1+and our assumption

(27). It follows that

EP

Πg1g2(Ψg3) = det(1 + (g3− 1)Π

g1g2_{) < ∞.} Hence, by Proposition 4.4 in [1], we have

P_Πg = C′Ψ_g 3PΠg1g2 = C ′_CΨ g3Ψg2 · PΠg1 = C ′_CΨ g3Ψg2Ψ Π g1 · PΠ, whence PΠg = Ψ Π

Introduce a topology T on A3(Π) generated by the open sets

U(ε, g) = {g′ ∈ AN(Π) : L(g′/g) < ε, V (g′/g) < ε} ,

whereL, V are defined by formulae (45), (46). With respect to the topology T , a sequence

gntends tog in A3(Π) if and only if

L(gn/g) → 0 and V (gn/g) → 0. (53)

Lemma 7.3. Let _{g ∈ A}3(Π) and let (En)n≥1 be the exhausting sequence of bounded

subsets ofE such that condition (47) holds. Denote

gn= 1 + (g − 1)χEn. Then

gn T

−−−→_n→∞ g.

Proof. Assume that_{g ∈ A}3(Π). First, by definition, we have

|gn/g − 1| = |1/g − 1|χEc n ≤ 1 infEg|g − 1|. It follows thatL(gn/g) → 0. Next, setting Vn(x, y) = |gn(x)/g(x) − gn(y)/g(y)|2|Π(x, y)|2, we have V (gn/g) = ZZ En×Enc Vn+ ZZ Ec n×En Vn+ ZZ Ec n×Enc Vn. (54)

The first and second terms in (54) are equal and

ZZ

En×Ecn

Vn =

ZZ

En×Enc

|1 − 1/g(y)|2|Π(x, y)|2dµ(x)dµ(y)

≤ _inf1

Eg2

ZZ

En×Enc

|g(y) − 1|2|Π(x, y)|2dµ(x)dµ(y)

= 1 infEg2kχEnΠ|g − 1|χE c nk 2 2 → 0.

The third term in (54) converges to 0 since

ZZ Ec n×Enc Vn ≤ 1 infEg2 ZZ Ec n×Enc

|g(x) − g(y)|2|Π(x, y)|2dµ(x)dµ(y),

and the latter integral tends to _{0 as n → ∞. Thus V (g}n/g) → 0, and Lemma 7.3 is

Lemma 7.4. Letgn ∈ A3(Π), n ≥ 1, g ∈ A3(Π), and assume that the sequence (gn) is

uniformly bounded. Ifgn T

−−−→_n→∞ g, then L(gn) → L(g) and V (gn) → V (g).

Proof. By definition, we haveL(gn/g) → 0 and V (gn/g) → 0.

The relationL(gn/g) → 0 together with the inequality

Z |gn(x) − g(x)|3Π(x, x)dµ(x) ≤ sup E g · Z |gn(x)/g(x) − 1|3Π(x, x)dµ(x) implies that lim n→∞k(gn− 1) − (g − 1)kL3(E;Π(x,x)dµ(x))= 0, whence lim n→∞kgn− 1kL3(E;Π(x,x)dµ(x) = kg − 1kL3(E;Π(x,x)dµ(x)). This is equivalent toL(gn) → L(g) as n → ∞.

We turn to the proof of the convergence V (gn) → V (g). It suffices to prove any

convergent subsequence (in _{[0, ∞]) of the sequence (V (g}n))n≥1 converges to V (g). We

have already shown that

Z

E

|gn(x) − g(x)|3Π(x, x)dµ(x) → 0.

Passing perhaps to a subsequence, we may assume thatgn → g almost everywhere with

respect toΠ(x, x)dµ(x). Set

Fn(x, y) = gn(x) − gn(y) and F (x, y) = g(x) − g(y).

The desired relationV (gn) → V (g) is equivalent to the relation

lim

n→∞kFnkL2(E×E; |Π(x,y)|2dµ(x)dµ(y)) = kF kL2(E×E; |Π(x,y)|2dµ(x)dµ(y))

To simplify notation, we denotedM2(x, y) = |Π(x, y)|2dµ(x)dµ(y). It suffices to prove

that

lim

n→∞kFn− F kL2(E×E; dM2) = 0. (55)

A direct computation shows that

Fn(x, y) − F (x, y) g(x) = gn(x) g(x) − gn(y) g(y) +

F (x, y)(gn(y) − g(y))

g(x)g(y) . Hence we have |Fn(x, y) − F (x, y)| ≤ sup E g ·     gn(x) g(x) − gn(y) g(y)     + 1

and kFn− F kL2_{(E×E; dM2)} ≤ sup E g · gn(x) g(x) − gn(y) g(y) _L2_{(E×E; dM2)} + 1

infEg kF (x, y) · |gn(y) − g(y)|kL

2_{(E×E; dM2)}

The limit relationV (gn/g) → 0 implies that

lim n→∞ g_g(x)n(x)− g_g(y)n(y) L2_{(E×E; dM2)} = 0.

By definition,_{F ∈ L}2_{(E × E; dM}2). Since the sequence (gn) is uniformly bounded and

gn → g almost everywhere with respect to Π(x, x)dµ(x), the dominated convergence

theorem yields

lim

n→∞kF (x, y) · |gn(y) − g(y)|kL2(E×E; dM2)= 0. This completes the proof of (55). Lemma7.4is proved completely.

Recall that, in Definition4.1 and Definition 4.2, we introduced the subset V0(Π) ⊂

V_{(Π) and the functional eΨg} for functions_{g such that log g ∈ V}0(Π). Recall also that we

introduced in (23) the notation_{Var(Π, f ) for any Borel function f : E → C.} Lemma 7.5. If_{g ∈ A}3(Π), then

Var(Π, log g) < ∞ and log g ∈ V0(Π).

In particular, for any function_{g ∈ A}3(Π), the functional eΨg is well-defined.

Proof. By the third condition in the definition of A3(Π), if g ∈ A3(Π), then

Var(Π, g − 1) < ∞. Define a function F (t) :=  log(1+t)−t t2 ift 6= 0 −1 2 ift = 0 ,

so that_{F is continuous on (−1, ∞). It follows that for any 0 < ε ≤ 1 and M ≥ 1, there} existsCε,M > 0, such that if t ∈ [−1 + ε, −1 + M], then

|log(1 + t) − t| ≤ Cε,Mt2. (56)

By the first condition in the definition of A3(Π), we can apply the above inequality to

g − 1. A simple computation yields

|log g(x) − log g(y)|2 ≤20M2|g(x) − g(y)|2

whereε = min(1, infEg) and M = max(1, supEg). Inequality (57), combined with the reproducing property Π(x, x) = Z E|Π(x, y)| 2_dµ(y)

and the second and third conditions on g in the definition of A3(Π), yields the desired

result:

Var(Π, log g) < ∞.

We turn to the proof of the relation _{log g ∈ V}0(Π). By definition, there exists a

se-quence (En) of exhausting bounded subsets of E, such that the relation (48) holds. It

suffices to show that

lim

n→∞kχEnlog g − log gkV(Π)= limn→∞kχE

c nlog gkV(Π) = 0. (58) We have kχEc nlog gk 2 V(Π)= 1 2 ZZ Ec n×Ecn

| log g(x) − log g(y)|2|Π(x, y)|2dµ(x)dµ(y)

+1 2 ZZ E2 χEc n(x)χEn(y)| log g(x)| 2 |Π(x, y)|2dµ(x)dµ(y) + 1 2 ZZ E2 χEc

n(y)χEn(x)| log g(y)|

2

|Π(x, y)|2dµ(x)dµ(y).

The fact that first integral in the above identity tends to0 when n tends to infinity follows

from the fact that _{Var(Π, log g) < ∞. The second and the third integrals are equal, and} since_{ε ≤ g ≤ M, we may use | log g(x)| ≤ C}ε,M|g(x) − 1| and we get

ZZ E2 χEc n(x)χEn(y)| log g(x)| 2 |Π(x, y)|2dµ(x)dµ(y) ≤Cε,M2 ZZ E2 χEc n(x)χEn(y)|g(x) − 1| 2 |Π(x, y)|2dµ(x)dµ(y). (59)

The assumption (48) implies that the last integral in (59) tends to0 as n tends to infinity.

This completes the proof of the desired relation (58).

Proposition 7.6. For any_{ε, M : 0 < ε ≤ 1, M ≥ 1, there exists a constant C}ε,M > 0

such that if_{g ∈ A}3(Π) satisfies

ε ≤ inf

E g ≤ sup_E g ≤ M (60)

then

log E|eΨg|2 ≤ Cε,M(L(g) + V (g)). (61)

Definition 7.1. Let A₃ε,M_{(Π) ⊂ A}3(Π) be the subset of functions satisfying the condition

By definition_|eΨg|2 = eΨg2. Ifg ∈ A₃ε,M(Π), then

L(g2_{) ≤ 8M}3L(g) and V (g2_{) ≤ 4M}2V (g). (62) Consequently, in order to establish (61), it suffices to obtain the estimate (63) in Lemma

7.7below.

Lemma 7.7. For any_{ε, M : 0 < ε ≤ 1, M ≥ 1, there exists a constant C}ε,M > 0 such

that if_{g ∈ A3}ε,M(Π), then log EeΨg ≤ C(L(g) + V (g)). (63) Denote g+_{= 1 + χ} g≥1(g − 1) and g− = 1 + χg≤1(g − 1). Then g = g+g−andg+_{≥ 1, g}−_{≤ 1.} (64) Our aim here is to reduce Lemma7.7forg to the same statement for g+_{, g}−_.

Lemma 7.8. Bothg+_andg−_{are in the class A}ε,M

3 (Π), moreover, we have

L(g±_{) ≤ L(g) and V (g}±_{) ≤ V (g).} (65) Proof. Inequalities (65) follow from the elementary inequalities

|g±− 1| ≤ |g − 1| and |g±(x) − g±(y)| ≤ |g(x) − g(y)|. (66) Let(En)n≥1be the exhausting sequence of bounded subsets such that (47) holds. The first

inequality in (66) yields the following inequalities for self-adjoint operators:

χEnΠ|g ± − 1|2χEc nΠχEn ≤ χEnΠ|g − 1| 2_χ Ec nΠχEn. Hence (47) holds forg±with respect to the sequence(En)n≥1.

Denote by A₃ε,M(Π)+_{the subclass of functions in A}ε,M

3 (Π) such that

g ∈ A3(Π) and g ≥ 1.

Similarly, denote by A₃ε,M(Π)−_{the subclass of functions in A}ε,M

3 (Π) such that

g ∈ A3ε,M(Π) and g ≤ 1.

Let

Aε,M

We reduce the statement of Lemma7.7for generalg in A3ε,M(Π) to the particular case

g in A₃ε,M(Π)±_{. Indeed, assume that we have established (63) in the case of A}ε,M

3 (Π)±,

then by multiplicativity, for generalg in A₃ε,M(Π), we have

E_Ψe_{g = E( eΨg}+Ψe_g−) ≤ (EeΨ2_g+ · EeΨ2_g−)1/2 = (EeΨ_(g+₎2 · EeΨ_(g−₎2)1/2

≤ 1₂(EeΨ(g+₎2 + EeΨ_(g−₎2).

Now we may apply (63) for functions (g+₎2 _{∈ A}ε,M

3 (Π)+ and (g−)2 ∈ A

ε,M

3 (Π)−

respectively and use the relations (62) together with Lemma7.8, to obtain that

E_Ψe_{g ≤ C}′h_L((g+₎2_{) + V ((g}+₎2_{) + L((g}−₎2_{) + V ((g}−₎2i ≤ C′′hL(g+) + V (g+) + L(g−) + V (g−))i

≤ C′′′(L(g) + V (g)).

We now proceed to the proof of (63) for functionsg in A₃ε,M(Π)±_{and, consequently,}

Lemma7.7. By definition, if_{g ∈ A3}ε,M(Π)±_{, then the sequences(g}

n)n≥1 defined in the

proof of Lemma7.3all stay in the set A₃ε,M(Π)±_{. Since}

kSlog gn− Slog gk

2

2 = Var(Π, log gn/g),

passing perhaps to a subsequence, we may assume that

e

Ψgn = exp(Slog gn)

a.e.

−−−→_n→∞ Ψeg = exp(Slog g).

By Fatou’s Lemma and Lemma 7.4 , it suffices to establish (63) for a function _{g ∈}

Aε,M

3 (Π)± such that the subset {x ∈ E : g(x) 6= 1} is bounded. We will assume the

boundedness of_{{x ∈ E : g(x) 6= 1} until the end of the proof of Proposition}7.6.

For any_{0 < ε ≤ 1 and any M ≥ 1, there exists C}ε,M > 0 such that if t ∈ [−1 +

ε, −1 + M], then    log(1 + t) − t + 1 2t 2     ≤ Cε,M · |t|3. (67)

Recall that for any bounded linear operator_{A acts on a Hilbert space, we set |A| =}

√

A∗_{A. The inequality (67) applied to the eigenvalues of trace class operator with}

spec-trum contained in_{[−1 + ε, −1 + M] yields the following}

Lemma 7.9. Letε, M, Cε,M be as in the inequality (67). For any self-adjoint trace class

operator_{A whose spectrum σ(A) satisfies σ(A) ⊂ [−1 + ε, −1 + M], we have}

log det(1 + A) ≤ tr(A) −1 2tr(A

2_{) + C}

Proof. The lemma is an immediate consequence of the inequality (67) and the identity log det(1 + A) = ∞ X i=1 log(1 + λi(A)),

where(λi(A))∞i=1is the sequence of the eigenvalues ofA.

In order to simplify notation, for_{g ∈ A3}ε,M(Π)+_{, set}

h = g − 1 ≥ 0 and Tg+ =

√

hΠ√_{h ≥ 0;} (69)

and for_{g ∈ A3}ε,M(Π)−_{, set}

h = g − 1 ≤ 0 and Tg−= ΠhΠ ≤ 0. (70)

By applying the relation (68), for_{g ∈ A3}ε,M(Π)±_{, we have}

log EΨg = log det(1 + (g − 1)Π) = log det(1 + Tg±)

≤ tr(Tg±) − 1 2tr((T ± g )2) + Cε,Mtr(|Tg+|3). (71)

Clearly, the tracestr(T+

g ) and tr(Tg−) are given by the formula:

tr(T_g±) = Z

E

h(x)Π(x, x)dµ(x). (72)

Recall that the inner product on the space of Hilbert-Schmidt operators is defined by the formula

ha, biHS = tr(ab∗).

Lemma 7.10. For any_{g ∈ A3}ε,M(Π)±_{, we have}

tr((T_g±)2) = Z E h(x)2_{Π(x, x)dµ(x) −} 1 2V (g). (73) Proof. If_{g ∈ A3}ε,M(Π)+_{, then} tr((T_g+)2) = tr(√hΠhΠ√_{h) = tr(ΠhΠh) = hΠh, hΠi}HS. (74) Note that kΠhk2HS = khΠk2HS = Z E h(x)2Π(x, x)dµ(x). (75) By (49), we have V (g) = k[g, Π]k2HS = k[h, Π]k2HS = khΠ − Πhk2HS = khΠk2HS + kΠhk2HS− 2hhΠ, Πhi. (76)