Quasi-symmetries of Determinantal Point Process - A.i. Bufetov

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015 QUASI-SYMMETRIES OF DETERMINANTAL POINT

PROCESSES.

ALEXANDER I. BUFETOV

ABSTRACT. The main result of this paper is that determinantal pointprocesses onR corresponding to projection operators with integrablekernels are quasi-invariant, in the continuous case, underthe group ofdiffeomorphisms with compact support (Theorem 1.5); in thediscretecase, under the group of all finite permutations of the phase space (The-orem 1.7). The Radon-Nikodym derivative is computed explicitly and isgiven by a regularized multiplicative functional. Theorem1.5 applies, inparticular, to the sine-process, as well as to determinantal point processeswith the Bessel and the Airy kernels; Theorem 1.7 to the discrete sine-process and the Gamma kernel process. The paper answers a question ofGrigori Olshanski.

1. INTRODUCTION

G. Olshanski [18] established the quasi-invariance of the determinantalmeasure corresponding to the Gamma kernel under the group offinite per-mutations ofZ and expressed the Radon-Nikodym derivative as a multi-plicative functional. S. Ghosh and Y. Peres [9], [10] showed, for the Ginibreensemble and the Gaussian zero process, that the conditional distribution ofparticles in a bounded domain, with the configuration fixed inthe exterior,is equivalent to the Lebesgue measure.

This paper is mainly concerned with determinantal point processes onRgoverned by kernels admitting anintegrablerepresentation

A(x)B(y)−B(x)A(y)

x− y.

Under some additional assumptions it is proved that the measure class ofthe corresponding determinantal measures is preserved, inthe continuouscase, under the group of diffeomorphisms with compact support (Theorem1.5); in the discrete case, under the group of finite permutations of the phasespace (Theorem 1.7). The key step in the proof is the equivalence of reducedPalm measures corresponding to differentl-tuples of points(p1, . . . , pl),

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http://arxiv.org/abs/1409.2068v2

2 ALEXANDER I. BUFETOV

(q1, . . . , ql) in the phase space; the corresponding Radon-Nikodym deriva-tive is the regularized multiplicative functional corresponding to the func-tion

(1)(x− p1)...(x− pl)

(x− q1)...(x− ql).

The Radon-Nikodym derivative thus has the same form for all the processeswith integrable kernels; the normalizing constants do, of course, depend onthe specific process.

Olshanski [18] proves the quasi-invariance of the Gamma-kernel processby a limit transition from finite-dimensional approximations. The argumentin this paper is direct: first, it is shown that the Palm subspaces correspond-ing to conditioning at pointsp andq are taken one to the other by multipli-cation by the function(x− p)/(x− q); after which, the proof is completedusing a general result of [6] (see also [5]) that multiplyingthe range of theprojection operatorΠ inducing a determinantal measurePΠ by a functiong,corresponds, under certain additional assumptions, to taking the product ofthe determinantal measurePΠ by the multiplicative functionalΨg inducedby the functiong.

1.1. Quasi-symmetries of the sine-process.For example, let

S(x, y) =sin π(x− y)

π(x− y), x, y ∈ R

be the standard sine-kernel onR, and letPS be the sine-process, the deter-minantal measure induced by the kernelS on the spaceConf(R) of config-urations onR.

Proposition 1.1. The measurePS is quasi-invariant under the group of dif-feomorphisms ofR with compact support.

To write down the Radon-Nikodym derivative, we need more notation.Given a Borel measureµ on a Borel spaceX and a Borel automorphismT of X , denote byµ ◦ T the measure defined byµ ◦ T (Z ) = µ(T (Z ))for all Borel subsetsZ ⊂ X . SinceT is invertible, the measureµ ◦ T iswell-defined, and, for anyµ-integrable Borel functionf onX , satisfies

∫

X

f ◦ Tdµ ◦ T =

∫

X

fdµ

as well as(fµ) ◦ T = (f ◦ T )(µ ◦ T ).A diffeomorphismF of R acts onConf(R) by sending a configuration

X to the configurationF (X) = {F (x), x ∈ X}; slightly abusing notation,we keep the same symbolF for this induced action.

By definition, the measurePS ◦ F is determinantal with kernel

QUASI-SYMMETRIES OF DETERMINANTAL POINT PROCESSES 3

F ∗S(x, y) =

√F ′(x)F ′(y)S(F (x), F (y)).

If LS is the range inL2(R) of the projection operatorΠS with kernelS, thenthe kernelF ∗S induces the operatorΠF ∗S of orthogonal projection onto thesubspace

LF ∗S = F∗LS = {√F ′ · ϕ ◦ F, ϕ ∈ L}.

By definition of the sine-kernel, the spacesLS andLF ∗S only consist ofcontinuous functions; givenl distinct pointsq1, . . . , ql ∈ R, we set

LS(q1, . . . , ql) = {ϕ ∈ L : ϕ(q1) = . . . = ϕ(ql) = 0};and we denoteΠq1,...,ql

Sthe operator of orthogonal projection onto the sub-

spaceLS(q1, . . . , ql). The subspaceLF ∗S(q1, . . . , ql) and the operatorΠq1,...,qlF ∗S

are defined in the same way. By the Macchı-Soshnikov theorem, the oper-atorΠq1,...,ql

S, a finite-rank perturbation ofΠS, induces on the space of con-

figurations onR a determinantal processPΠq1,...,qlS

. Take distinct points

p1, . . . , pl, q1, . . . , ql ∈ R,

and, for a configurationX onR write

ΨN(p1, . . . , pl; q1, . . . , ql;X) = CN

∏

x∈X,|x|≤N

l∏

i=1

(x− pix− qi

)2

,

where the constantCN is chosen in such a way that∫

Conf(R)

ΨN(p1, . . . , pl; q1, . . . , ql;X)dPΠq1,...,qlS

= 1.

Proposition 1.2. The limit

Ψ(p1, . . . , pl; q1, . . . , ql;X) = limN→∞

ΨN (p1, . . . , pl; q1, . . . , ql;X)

exists bothPS-almost surely andL1(Conf(R),PΠq1,...,qlS

).

Proposition 1.3. LetF : R → R be a diffeomorphism acting as the identitybeyond a bounded open setV ⊂ R. For PS-almost every configurationX ∈ Conf(R) the following holds. IfX

⋂V = {q1, . . . , ql}, then

(2)dPS ◦ FdPS

(X) = Ψ(F (q1), . . . , F (ql); q1, . . . , ql;X)×

× det(S(F (qi), F (qj))i,j=1,...,l

det(S(qi, qj))i,j=1,...,l×

× F ′(q1) . . . F′(ql).


We now proceed to the formulation of the main result of this paper infull generailty. We start by describing the assumptions on the kernels of ourprocesses.

1.2. Integrable kernels. Let µ be aσ-finite Borel measure onR; for ex-ample,µ can be the Lebesgue measure onR or onR+ or else the countingmeasure onZ. The inner product inL2(R, µ) will be denoted〈, 〉. LetL ⊂ L2(R, µ) be a closed subspace, and letΠ be the corresponding opera-tor of orthogonal projection.

We assume that the operatorΠ is locally of trace class and admits a ker-nel, for which, slightly abusing notation, we keep the same symbolΠ.

All kernels considered in this paper will always be supposedto satisfythe following

Assumption 1. There exists a setU ⊂ R, satisfyingµ(R \ U) = 0, suchthat

(1) For anyq ∈ U the functionvq(x) = Π(x, q) lies inL2(R, µ) and foranyf ∈ L2(R, µ) we have

Πf(q) = 〈f, vq〉.In particular, all functions inL are defined everywhere onU .

(2) The diagonal valuesΠ(q, q) of the kernelΠ are defined for allq ∈U . We have〈vq, vq〉 = Π(q, q), and, for any bounded Borel subsetB ⊂ R, we have

tr(χBΠχB) =

∫

B

Π(q, q)dµ(q).

(3) For anyq ∈ U and anyϕ ∈ L satisfyingϕ(q) = 0, we have

ϕ(x)

x− q∈ L2(R, µ).

The last condition is automatically satisfied once the kernel is sufficientlysmooth: indeed, letϕ ∈ L have norm1 and be such thatϕ(q) = 0, let

Kq(x, y) = K(x, y)− K(x, q)K(q, y)

K(q, q)

be the kernel of the orthogonal projection onto the spaceL(q), the orthog-onal complement ofvq in L. Finally, letK be the kernel of the orthogonalprojection onto the orthogonal complement ofϕ in L(q). For anyx ∈ U ,by definition, we have

Kqvx = 〈vx, ϕ〉vx + Kvx,


whence, taking the inner product withvx, we obtain

Kq(x, x) = |ϕ(x)|2 + 〈Kvx, vx〉.In a small neighbourhood ofq, we now haveKq(x, x) = O(|x − q|2),whence also|ϕ(x)| = O(|x− q|), and the desired result follows.

We next assume that our kernelΠ has integrable form : there exists anopen setU ⊂ R satisfyingµ(R\U) = 0 and smooth functionsA,B definedonU such that

(3) Π(x, y) =A(x)B(y)− A(y)B(x)

x− y, x 6= y.

We assume that the functionsA,B never simultaneously take value0 onU .Forp ∈ U we have

vp(x) =A(p)B(x)− B(p)A(x)

p− x;

We havevp ∈ L2(R, µ) for anyp ∈ U and for anyϕ ∈ L2(R, µ) we have

Πϕ(p) = 〈ϕ, vp〉.We consider two cases:

(1) the continuous case: for anyp ∈ R, µ({p}) = 0;(2) the discrete case:µ is the counting measure on a countable subset

E ⊂ R without accumulation points.

In the continuous case we make the additional requirement

(4) Π(x, x) = A′(x)B(x)− A(x)B′(x).

on diagonal values of the kernelΠ; in the discrete case, when the measureµis the counting measure on a countable subsetE ⊂ R without accumulationpoints, the integrability assumption only concerns off-diagonal entries ofthe kernelΠ(x, y), and the smoothness assumption is not needed:A, B arejust arbitrary functions defined onE. Note also that the third requirementof Assumption 1 is only needed in the continuous case.

As before, givenl distinct pointsq1, . . . , ql ∈ R, we set

L(q1, . . . , ql) = {ϕ ∈ L : ϕ(q1) = . . . = ϕ(ql) = 0};and we denoteΠq1,...,ql the operator of orthogonal projection onto the sub-spaceL(q1, . . . , ql).

Remark. The functionsA,B in the definition of integrability are notunique: for example, if one makes a linear unimodular changeof variable

(5) (A,B) → (α11A+ α12B, α21A+ α22B), α11α22 − α12α22 = 1,

then the formula (3) remains valid.


1.3. The main result in the continuous case.In this subsection we as-sume that the measureµ is continuous (for anyp ∈ R, µ({p}) = 0) andadditionally that

(6)∫

R

Π(x, x)

1 + x2dµ(x) < +∞.

Let p1, . . . , pl, q1, . . . , ql ∈ R be distinct. ForR > 0, ε > 0 and a configu-rationX onR write

ΨR,ε(p1, . . . , pl; q1, . . . , ql;X) = C(R, ε)×∏

x∈X,|x|≤R,min |x−qi|≥ε

l∏

i=1

(x− pix− qi

)2

,

where the constantC(R, ε) is chosen in such a way that

(7)∫

Conf(R)

ΨR,ε(p1, . . . , pl; q1, . . . , ql;X)dPq1,...,qlΠ = 1.

Proposition 1.4. If the kernelΠ of an orthogonal projection operator, forwhich Assumption1 holds, is integrable and satisfies (6), then the limit

Ψ(p1, . . . , pl; q1, . . . , ql;X) = limR→∞,ε→0

ΨR,ε(p1, . . . , pl; q1, . . . , ql;X)

exists bothPΠq1,...,ql -almost surely and inL1(Conf(R),PΠq1,...,ql ).

Theorem 1.5.Letµ be a continuous measure onR. LetΠ be an integrablekernel satisfying Assumption1 as well as (6) and inducing a locally trace-class operator of orthogonal projection inL2(R, µ). Let F : R → R bea Borel automorphism preserving the measure class ofµ and acting as theidentity beyond a bounded open setV ⊂ R. For PΠ-almost every configu-rationX ∈ Conf(R) the following holds. IfX

⋂V = {q1, . . . , ql}, then

(8)dPΠ ◦ FdPΠ

(X) = Ψ(F (q1), . . . , F (ql); q1, . . . , ql;X)×

× det(Π(F (qi), F (qj))i,j=1,...,l

det(Π(qi, qj))i,j=1,...,l×

× dµ ◦ Fdµ

(q1) . . .dµ ◦ Fdµ

(ql).

Remark. The open setV can be chosen in many ways; the resultingvalue of the Radon-Nikodym derivative is of course the same.

For example, Theorem 1.5 applies to the sine-process as wellas to theAiry and Bessel point processes of Tracy and Widom [26], [27].


1.4. The main result in the discrete case.The main result is similar inthe discrete case except that we also need to consider measures conditionalon the absence of particles and that, in order to ensure quasi-invarianceof our measures under the infinite symmetric group, we imposethe extrarestriction that our subspaceL not contain functions with finite support.

LetE ⊂ R be a countable subset without accumulation points, endowedwith the counting measure. In what follows, we will need the assumption

(9)∑

n∈E

1

1 + n2< +∞.

LetΠ be an integrable kernel inducing an operator of orthogonal projectiononto a subspaceL ⊂ L2(E), and letPΠ be the corresponding determi-nantal measure on the spaceConf(E) of configurations onE. The infinitesymmetric group naturally acts onE by finite permutations and inducesthe corresponding natural action onConf(E). Given l ∈ N, m < l andan l-tuple(p1, . . . , pl) of distinct points inE such that there does not exista nonzero function inL supported on the set{p1, . . . , pl}, we introduce aclosed subspaceL(p1, . . . , pm, pm+1, . . . , pl) by the formula

(10) L(p1, . . . , pm, pm+1, . . . , pl) =

= {χE\{pm+1,...,pl}ϕ : ϕ ∈ L, ϕ(p1) = · · · = ϕ(pl) = 0}.

Let Πp1,...,pm,pm+1,...,pl be the corresponding orthogonal projection operator,PΠp1,...,pm,pm+1,...,pl the corresponding determinantal measure.

TakeR > 0,m ≤ l, a permutationσ of the pointsp1, . . . , pl, and define

ΨR(p1, . . . , pl, m, σ;X) = CR

∏

x∈X:|x|≤R

m∏

i=1

(x− σ(pi)

x− pi

)2

χE\{p1,...,pl}(x),

where the positive constantCR is chosen in such a way that∫

Conf(E)

ΨR(p1, . . . , pl, m, σ)dPΠp1,...,pm,pm+1,...,pl = 1.

Remark. Our multiplicative functional is positive with positive proba-bility precisely because the subspaceL does not admit functions supportedon{p1, . . . , pl}.

Proposition 1.6. LetE be a countable subset ofR without accumulationpoints satisfying (9). LetΠ be an integrable kernel inducing an operatorof orthogonal projection onL2(E). Let p1, . . . , pl ∈ E be distinct pointssuch that there does not exist a nonzero function inL supported on the set{p1, . . . , pl}. Then, for anym ≤ l and any permutationσ of p1, . . . , pl, the


limitΨ(p1, . . . , pl, m, σ) = lim

R→∞ΨR(p1, . . . , pl, m, σ)

existsPΠp1,...,pm,pm+1,...,pl -almost surely and inL1(Conf(E),PΠp1,...,pm,pm+1,...,pl ).

LetC(p1, . . . , pm, pm+1, . . . , pl)

be the set of configurations onE containing exactly one particle in each ofthe positionsp1, . . . , pm and no particles in the positionspm+1, . . . , pl.

We are now ready to formulate the main result in the discrete case, thequasi-invariance of determinantal measures with integrable kernels underthe natural action of the infinite symmetric group onConf(E). Givena permutationσ of points p1, . . . , pl of the setE, slightly absuing nota-tion, we use the same symbol both for the bijection ofE that acts asσ on{p1, . . . , pl} and as the identity onE \ {p1, . . . , pl} and the automorphisminduced by this bijection on the spaceConf(E) of configurations onE.

Theorem 1.7. Let E be a countable subset ofR without accumulationpoints satisfying (9). LetΠ be an integrable kernel inducing an operatorof orthogonal projection onto a closed subspaceL ⊂ L2(E). Letp1, . . . , plbe distinct points inE such that there does not exist a nonzero function inLsupported on the set{p1, . . . , pl}. Then for anym ≤ l, any permutationσ ofthe pointsp1, . . . , pl andPΠ-almost everyX ∈ C(p1, . . . , pm, pm+1, . . . , pl),we have(11)dPΠ ◦ σdPΠ

(X) = Ψ(p1, . . . , pl, m, σ;X)×det (Π(σ(pi), σ(pj)))i,j=1,...,m

det (Π(pi, pj))i,j=1,...,m

.

In particular, if the subspaceL does not contain functions supported onfinite sets, then the measurePΠ is quasi-invariant under the natural actionof the infinite symmetric group onConf(E).

For example, the discrete sine-process of Borodin, Okounkov and Ol-shanski [2] as well as the Gamma kernel process of Borodin andOlshanski[3] satisfy all the assumptions of Theorem 1.7.

Remark. By the Theorem of Ghosh [8], the sine-process, discrete orcontinuous, isrigid: if, for a bounded subsetB and a configurationX, welet #B(X) stand for the number of particles ofX lying in B, and, for anyBorel subsetC we letFC be theσ-algebra generated by all random variablesof the form#B, B ⊂ C, then, for any boundedB, the random variable#B is measurable with respect to the completion, under the sine-process,of the sigma-algebraFBc, whereBc stands for the complement ofB. AsR. Lyons, developing the method of [1], showed in Theorem 7.15 of [15],the tail sigma-algebra of the discrete sine-process is trivial. It follows now


that thesymmetricsigma-algebra of the sine-process is trivial as well: inother words, the discrete sine-process is ergodic with respect to the actionof the infinite symmetric group. This argument holds, of course, for anyrigid point process.

To further illustrate Theorem 1.7, we now write the Radon-Nikodym de-rivative for a transposition of two pointsp, q ∈ E. Set

L(p, q) = {χZ\{p,q}ϕ, ϕ ∈ L, ϕ(p) = 0}.and letPp,q

Π be the determinantal measure corresponding to the operatoroforthogonal projection onto the subspaceL(p, q). The subspaceL(q, p) andthe measurePq,p

Π are defined in the same way. Write

ΨN(p, q;X) = Cp,q ×∏

x∈X,|x|≤N

(x− p

x− q

)2

,

where the constantCp,q is chosen in such a way that∫

Conf(E)

ΨN (p, q;X) dPp,qΠ (X) = 1.

By definition,Pp,qΠ -almost all configurationsX on E contain no particles

either atp or at q, so the functionΨN is well-defined; by definition it isbounded.

Proposition 1.8.The limitΨ(p, q;X) = limN→∞

ΨN(p, q;X) exists bothPp,qΠ -

almost surely and inL1(Conf(E),Pp,qΠ ).

The regularized multiplicative functionalΨ(q, p;X) is defined in the sameway.

The Radon-Nikodym derivative ofPΠ under the action of the permutationσpq is now given by the following

Proposition 1.9. For PΠ-almost allX ∈ Conf(E) the following holds.If p /∈ X, q ∈ X, then

dPΠ ◦ σpqdPΠ

(X) = Ψ(p, q;X) · Π(p, p)Π(q, q)

.

If p ∈ X, q /∈ X, then

dPΠ ◦ σpqdPΠ

(X) = Ψ(q, p;X) · Π(q, q)Π(p, p)

.

If p, q ∈ X or p, q /∈ X, then

dPΠ ◦ σpqdPΠ

(X) = 1.


Remark. If E is a countable set,P a Gibbs measure onConf(E) corre-sponding to the HamiltonianH of pairwise interaction of particles (cf. e.g.Sinai [24]),p, q are points inE andσpq the transposition ofp andq, then,for almost every configurationX, conditioned to contain a particle atq butnot atp, by definition, we have

dP ◦ σp.qdP

(X) =∏

x∈X:x 6=q

exp(H(p, x)−H(q, x)).

The quasi-invariance property established in this paper isthus, informallyspeaking, the analogue of the Gibbs property, with HamiltonianH(x, y) =2 log |x− y|, for determinantal point processes.

1.5. Examples of regularized multiplicative functionals. Regularizationof a multiplicative functional can take different form depending on the spe-cific process. We illustrate this by two examples.

The Sine-Process.The argument below is valid for the continuous sine-process as well as the discrete sine-process. The sine-process is stationary,therefore, for almost every configurationX the series

(12)∑

x∈X:x 6=0

1

x

divergesabsolutely since so does the harmonic series. Nonetheless,theseries (12) convergesconditionallyin principal value: the limit

limN→∞

∑

x∈X:x 6=0,|x|≤N

1

x

is almost surely finite and, as we shall check below, has finitevariance. Sim-ilarly, for distinct pointsp1, . . . , pl, q1, . . . , ql, taken inR in the continuouscase and inZ in the discrete case,

(13) limN→∞

∏

x∈X,|x|≤N,x 6=q1,...,ql

l∏

i=1

(x− pix− qi

)2

,

the limit exists and has finite expectation. The normalized mutliplicativefunctional is in this case precisely the limit (13) normalized to have expec-tation1.

The Determinantal Point Process with the Gamma-Kernel.The determi-nantal point process with the Gamma-kernel, introduced by Borodin andOlshanski in [3] and for which the quasi-invariance under the action of theinfinite symmetric group is due to Olshanski [18], is a point process onthe phase spaceZ′ = 1/2 + Z of half-integers such that for almost every


configurationX we have

(14)∑

x∈X:x>0

1

x< +∞,

∑

y/∈X:y<0

1

|y| < +∞.

Furthermore, each sum in (14), considered as a random variable on thespace of configurations onZ′, has finite variance with respect to the de-terminantal point process with the Gamma-kernel.

For p, q ∈ Z′, the normalized multiplicative functional correspondingtothe functiongp,q(x) = ((x− p)/(x− q))2 will therefore have the form

C ·∏

x∈X,x>0

g(x) ·∏

y/∈X:y<0

g−1(y),

where the constantC is chosen in such a way that the resulting expressionhave expectation1.

1.6. Outline of the argument. We start with the discrete case and illus-trate the argument in the specific case of a transposition of two distinctpointsp, q ∈ E. A theorem due to Lyons [15], Shirai-Takahashi [22] statesthat the measurePp,q

Π is the conditional measure ofPΠ on the subset of con-figurations containing a particle atp and not containing a particle atq.

Step 1. The Relation Between Palm Subspaces.The key point in the proofof Proposition 1.9 is the equality

(15) L(p, q) =x− p

x− qL(q, p),

which it is more convenient to rewrite in the form

(16) L(p, q) = χE\{p,q}x− p

x− qL(q, p).

The equality (16) directly follows from theintegrability of the discretesine-kernel. The remainder of the argument shows that the relation (16)implies the relation

(17) Pp,qΠ = Ψ(p, q)Pq,p

Π ,

which, in turn, is a reformulation of Proposition 1.9.Step 2. Multiplicative functionals of determinantal pointprocesses.Given

a functiong onZ, the multiplicative functionalΨg is defined onConf(E)by the formula

Ψg(X) =∏

x∈X

g(x).

provided that the infinite product in the right-hand side converges abso-lutely.

At the centre of the argument lies the result of [6] that can informallybe summarized as follows: a determinantal measure times a multiplicative


functional is, after normalization, again a determinantalmeasure. More pre-cisely, letg be a positive function onE bounded away from 0 and∞, andlet Π be an operator of orthogonal projection inL2(E) onto a closed sub-spaceL. LetΠg be the operator of orthogonal projection onto the subspace√gL. Then, under certain additional assumptions we have

(18) PΠg =ΨgPΠ∫

Conf(E)

Ψg dPΠ

The relation (18) together with the relation (16) suggests that the measuresPp,qΠ andPq,p

Π are equivalent, and the Radon-Nikodym derivative is given bythe normalized multiplicative functional corresponding to the function

gp,q(x) =x− p

x− qχE\{p,q}.

Step 3. Regularization of multiplicative functionals.A technical diffi-culty arises that in many examples the multiplicative functional correspond-ing to the functiongp,q might fail to converge absolutely with respect to themeasurePp,q

Π ; indeed, in many examples (in particular, for stationary deter-minantal processes onZ), we have

∑

x∈E

|gp,q(x)− 1| = +∞

and, consequently, also∑

x∈E

|gp,q(x)− 1| · Πp,q(x, x) = +∞.

In order to resolve this difficulty, we go back to the formula (18). Formultiplicative functionalΨg integrable with respect to a determinantal mea-surePΠ set

(19) Ψg =Ψg∫ΨgdPΠ

.

The functionalΨg will be called the normalized multiplicative functionalcorresponding toΨg andPΠ. To keep notation lighter, we do not explicitlyindicate dependence onΠ; in what follows, the precise measure, with re-spect to which normalization is taken, will be clear from thecontext. Wenow rewrite (18) in the form

(20) PΠg = Ψg · PΠ.


The key observation for the remainder of the argument is thatthe definitionof the normalized multiplicative functionalΨg can be extended in such away that (20) continues to hold for a wider class of functionsg, for whichthe multiplicative functional itself diverges almost surely.

We first explain the idea of this extension for additive functionals. Givena measurable functionf on E, the corresponding additive functional onConf(E) is defined by the formula

Sf(X) =∑

x∈X

f(x)

provided the series in the right hand side converges absolutely. The expec-tation of the additive functional with respect toPΠ is given by the formula

(21) EPΠSf =

∑

x∈E

f(x)Π(x, x),

provided, again, that the series in the right hand side converges absolutely.For the variance of the additive functional we have

(22) VarPΠSf =

∑

x,y∈Z

(f(x)− f(y))2(Π(x, y))2.

LetSf = Sf − EPΠ

Sf

be the normalized additive functional corresponding to thefunctionf . It iseasy to give examples of functionsf for which the sum in the right handside of (21) diverges while the sum in the right hand side of (22) converges.For such functions, convergence of the sum in the right hand side of (22)allows one to define the normalized additive functionalSf by continuity,even though the additive functionalSf itself is not defined. In a similarway, for a functiong bounded away from 0 and∞ and satisfying

∑

x∈E

|g(x)− 1|2Π(x, x) < +∞,

one can define the normalized multiplicative functionalΨg, even though themultiplicative functionalΨg itself need not be defined; the relation (20) stillholds.

We thus check that the normalized multiplicative functional Ψgp,q can bedefined with respect to the measureP

q,pΠ ; the relation (16) now implies the

desired equality (17). This completes the outline of the proof of Theorem1.7.

The proof in continuous case follows a similar scheme. The rˆole ofconditional measures is played by reduced Palm measures. The reduced


Palm measurePq1,...,qlΠ of the measurePΠ with respect tol distinct points

q1, ..., ql ∈ R is the determinantal measure corresponding to the operatorΠq1,...,ql of the orthogonal projection onto the subspace

L(q1, ..., ql) = {ϕ ∈ LS : ϕ(q1) = ... = ϕ(ql) = 0}.

The continuous analogue of the equality (16) is the relation

(23) L(p1, ..., pl) =(x− p1)...(x− pl)

(x− q1)...(x− ql)L(q1, ..., ql)

valid for µ-almost any twol-tuples of distinct points(p1, ..., pl), (q1, ..., ql)in R.

The next step is to regularize the multiplicative functional correspondingto the function

(24)(x− p1)...(x− pl)

(x− q1)...(x− ql);

while the overall scheme of regularization is the same as in the discrete case,additional estimates are needed here because the function (24) is boundedaway neither from zero nor from infinity.

The resulting normalized multiplicative functionalΨ(p1, ..., pl, q1, ..., ql)is then seen to be the Radon-Nikodym derivative of the reduced Palm mea-suresPp1,...,pl

Π andPq1,...,qlΠ , which, in turn, implies Theorem 1.5.

1.7. Organization of the paper. The paper is organized as follows. InSection 2, we collect necessary facts about determinantal point processes,their multiplicative functionals and their Palm measures.We recall the re-sults of [6] (see also [5]) showing that the product of a determinantal mea-sure with a multiplicative functional is, after normalization, again a deter-minantal measure, whose kernel is found explicitly. We alsocheck thatequivalence of reduced Palm measures corresponding to distinct l-tuples ofpoints implies the quasi-invariance of the point process under Borel auto-morphisms preserving the class of its correlation measuresand acting by theidentity beyond a bounded set. In Section 3, we start by showing that re-duced Palm measures of determinantal point processes givenby projectionoperators with integrable kernels are themselves determinantal point pro-cesses given by projection operators with integrable kernels and proceed toverify the key relations (50) and (54) showing that the ranges of projectionoperators corresponding to reduced Palm measures at distinct points differby multiplication by a function.

Proposition 4.6 in Section 4 describes the regularization of multiplica-tive functionals. Relations (50) and (54) are then seen to imply that the


reduced Palm measures themselves are equivalent, and that the correspond-ing Radon-Nikodym derivative is a regularized multiplicative functional.This completes the proof of Theorems 1.5 and 1.7.

Acknowledgements. Grigori Olshanski posed the problem to me andsuggested that the Radon-Nikodym derivative be given by a multiplicativefunctional; I am greatly indebted to him. I am deeply grateful to AlexeiKlimenko and Cosme Louart for useful discussions.

This work has been carried out thanks to the support of the A*MIDEXproject (no. ANR-11-IDEX-0001-02) funded by the programme“Investisse-ments d’Avenir ” of the Government of the French Republic, managed bythe French National Research Agency (ANR). It was also supported in partby the Grant MD-2859.2014.1 of the President of the Russian Federation,by the Programme “Dynamical systems and mathematical control theory”of the Presidium of the Russian Academy of Sciences, by the ANR underthe project “VALET” (ANR-13-JS01-0010) of the Programme JCJC SIMI1, and by the RFBR grant 13-01-12449.

Part of this work was done while I was visiting Institut HenriPoincare inParis and the Max Planck Institute in Bonn; I am deeply grateful to theseinstitutions for their warm hospitality.

Remark. After this paper was completed, I became aware of the preprintAbsolute continuity and singularity of Palm measures of theGinibre pointprocess, arXiv:1406.3913, by Hirofumi Osada and Tomoyuki Shirai, inwhich, for the Ginibre ensemble, using its finite-dimensional approxima-tions by orthogonal polynomial ensembles, the authors establish the equiv-alence of reduced Palm measures, conditioned at distinctl-tuples of pointsin C, and represent the Radon-Nikodym derivative as a regularized multi-plicative functional.

2. POINT PROCESSES ANDPALM DISTRIBUTIONS.

2.1. Spaces of configurations.LetE be a locally compact complete met-ric space. AconfigurationonE is a collection of points inE, calledpar-ticles, considered without regard to order and subject to the additional re-quirement that every bounded set contain only finitely many particles ofa configuration. LetConf(E) be the space of configurations onE. To aconfigurationX ∈ Conf(E) assign a Radon measure

∑

x∈X

δx

on the spaceE; this correspondence identifies the spaceConf(E) with thespace of integer-valued Radon measures onE. The spaceConf(E) is thusendowed with a natural structure of a complete separable metric space. The

http://arxiv.org/abs/1406.3913


Borel structure on the spaceConf(E) can equivaletly be defined withoutintroducing a topology: namely, for a bounded Borel setB ⊂ E, let

#B : Conf(E) → N ∪ {0}be the function that to a configuration assigns the number of its particlesbelonging toB. The random variables#B over all bounded Borel setsB ⊂ E determine the Borel sigma-algebra onConf(E).

2.2. Multiplicative functionals. We next recall the definition ofmulti-plicative functionalson spaces of configurations. Letg be a non-negativemeasurable function onE, and introduce themultiplicative functionalΨg :Conf(E) → R by the formula

(25) Ψg(X) =∏

x∈X

g(x).

If the infinite product∏x∈X

g(x) absolutely converges to0 or to∞, then we

set, respectively,Ψg(X) = 0 or Ψg(X) = ∞. If the product in the right-hand side fails to converge absolutely, then the multiplicative functional isnot defined.

2.3. Point processes.A Borel probability measureP onConf(E) is calleda point processwith phase spaceE.

We recall that the processP is said to admit correlation functions of orderl if for any continuous compactly supported functionf onEl the functional

∑

x1,...,xl∈X

f(x1, . . . , xl)

isP-integrable; here the sum is taken over alll-tuples of distinct particles inX. Thel-th correlation measureρl of the point processP is then defined bythe formula

EP

(∑

x1,...,xl∈X

f(x1, . . . , xl)

)=

∫

El

f(q1, . . . , ql)dρl(q1, . . . , ql).

By definition, a point processP is uniquely determined by prescribingjoint distributions, with respect toP, of random variables#B1 , . . . ,#Bl

over all finite collections of disjoint bounded Borel subsetsB1, . . . , Bl ⊂E. Since, for arbitrary nonzero complex numbersz1, . . . , zl inside the unitcircle, the function

(26)l∏

k=1

z#Bl

i


is a well-defined multiplicative functional onConf(E), that, moreover, takesvalues inside the unit circle, a point processP onConf(E) is also uniquelydetermined by prescribing the values of expectations of multiplicative func-tionals of the form (26).

2.4. Campbell Measures.Following Kallenberg [12] and Daley–Vere-Jones[7], we now recall the definition of Campbell measures of point processes.

Take a Borel probability measureP onConf(E) of finite local intensity,that is, admitting the first correlation measureρ1, or, equivalently, such thatfor any bounded Borel setB, the function#B is integrable with respect toP. For any bounded Borel setB ⊂ E, by definition we then have

ρ1(B) =

∫

Conf(E)

#B(X)dP(X).

TheCampbell measureCP of a Borel probability measureP of finite localintensity onConf(E) is a sigma-finite measure onE × Conf(E) such thatfor any Borel subsetsB ⊂ E, Z ⊂ Conf(E) we have

CP(B × Z ) =

∫

Z

#B(X)dP(X).

For a point process admitting correlation functions of order l one canalso define thel-th iterated Campbell measureC(l) of the point processP,that is, by definition, a measure onEl ×Conf(E) such that for any disjointbounded setsB1, ..., Bl ⊂ E and any measurable subsetZ ⊂ Conf(E) wehave

(27) C(l)(B1 × ...× Bl × Z ) =

∫

Z

#B1(X)× ...×#Bl(X)dP(Z )

2.5. Palm Distributions. Following Kallenberg [12] and Daley–Vere-Jones[7], we now recall the construction of Palm distributions from Campbellmeasures. For a fixed BorelZ ⊂ Conf(E) the Campbell measureCP in-duces a sigma-finite measureCZ

PonE by the formula

CZ

P (B) = CP(B × Z ).

By definition, for any Borel subsetZ ⊂ Conf(E) the measureCZ

Pis

absolutely continuous with respect toρ1. For ρ1-almost everyq ∈ E, onecan therefore define a probability measurePq onConf(E) by the formula

Pq(Z ) =

dCZP

dρ1(q) .

The measurePq is calledthe Palm measureof P at the pointq. Equivalently,the Palm measurePq is the canonical conditional measure, in the sense of


Rohlin [20], of the Campbell measureCP with respect to the measurablepartition of the spaceE×Conf(E) into subsets of the form{q}×Conf(E),q ∈ E.

Similarly, using iterated Campbell measures one defines iterated Palmmeasures: for a fixed BorelZ ⊂ Conf(E) thel-th iterated Campbell mea-sureCl

Pinduces a sigma-finite measureCl,Z

PonE by the formula

Cl,ZP

(B) = CP(B × Z ).

By definition, for any Borel subsetZ ⊂ Conf(E) the measureCl,ZP

isabsolutely continuous with respect toρl. Forρl-almost all(q1, . . . , ql) ∈ El,one can therefore define a probability measurePq1,...,ql onConf(E) by theformula

Pq1,...,ql(Z ) =

dCZP

dρ1(q1, . . . , ql) .

The measurePq1,...,ql is called thel-th iterated Palm measure of the pointprocessP. The iterated Palm measurePq is the canonical conditional mea-sure, in the sense of Rohlin [20], of the Campbell measureCl

Pwith respect

to the measurable partition of the spaceEl × Conf(E) into subsets of theform {q1, . . . , ql} × Conf(E), with q1, . . . , ql ∈ E distinct.

For distinct pointsq1, . . . , ql, the l-th iterated Palm measure of coursesatisfies

Pq1,...,ql =

(. . .(Pq1)q2

. . .)ql

.

2.6. Reduced Palm measures.By definition, the Palm measurePq1,...,ql

is supported on the subset of configurations containing a particle at eachpositionq1, . . . , ql. It is often convenient to remove these particles and todefine thereducedPalm measurePq1,...,ql as the push-forward of the PalmmeasurePq1,...,ql under the erasing mapX → X \ {q1, . . . , ql}. ReducedPalm measures allow one to give a convenient representationfor measuresof cylinder sets. TakeX0 ∈ Conf(E) andq(0)1 , . . . , q

(0)l ∈ X0. Take disjoint

bounded open setsB(1), ..., B(l) ⊂ E, setB = ∪B(i) and take an opensetU ⊂ E disjoint from allB(i) in such a way thatq(0)i ∈ B(i) for alli = 1, . . . , l andX0 \ {q(0)1 , . . . , q

(0)l } ⊂ U . Let W be a neighbourhood of

X0 \ {q(0)1 , . . . , q(0)l } in Conf(E) satisfying

W ⊂ {X ∈ Conf(E) : X ⊂ U}.Introduce a neighbourhoodZ of X0 by setting(28)Z = {X ∈ Conf(E) : #B(1)(X) = · · · = #B(l)(X) = 1, X|E\B ⊂ W }.


Proposition 2.1. We have

P(Z ) =

∫

B(1)×...×B(l)

Pq1,...,ql(Z )dρl(q1, ..., ql).

Proof. By definition of iterated Palm measures, we have

C(l)(B(1) × ...×B(l) × Z ) =

∫

B(1)×...×B(l)

Pq1,...,ql(Z )dρl(q1, ..., ql).

By construction and definition of reduced Palm measures, we have

Pq1,...,ql(Z ) = P

q1,...,ql(W ).

By definition, see (27),(28), we have

C(l)(B(1)×...×B(l)×Z ) =

∫

Z

#B(1)(X)×...×#B(l) (X)dP(Z ) = P(Z ).

Consequently,

P(Z ) =

∫

B(1)×...×B(l)

Pq1,...,ql(W )dρl(q1, ..., ql),

as desired.

2.7. Locally trace class operators and their kernels.Let µ be a sigma-finite Borel measure onE. The inner product inL2(E, µ) is always denotedby the symbol〈, 〉.

LetI1(E, µ) be the ideal of trace class operatorsK : L2(E, µ) → L2(E, µ)

(see volume 1 of [19] for the precise definition); the symbol||K||I1 willstand for theI1-norm of the operatorK. Let I2(E, µ) be the ideal ofHilbert-Schmidt operatorsK : L2(E, µ) → L2(E, µ); the symbol||K||I2

will stand for theI2-norm of the operatorK.LetI1,loc(E, µ) be the space of operatorsK : L2(E, µ) → L2(E, µ) such

that for any bounded Borel subsetB ⊂ E we have

χBKχB ∈ I1(E, µ).

Again, we endow the spaceI1,loc(E, µ) with a countable family of semi-norms

(29) ||χBKχB||I1

where, as before,B runs through an exhausting familyBn of bounded sets.A locally trace class operatorK admits akernel, for which, slightly abusingnotation, we use the same symbolK.


2.8. Determinantal Point Processes.A Borel probability measureP onConf(E) is calleddeterminantalif there exists an operatorK ∈ I1,loc(E, µ)such that for any bounded measurable functiong, for which g − 1 is sup-ported in a bounded setB, we have

(30) EPΨg = det

(1 + (g − 1)KχB

).

The Fredholm determinant in (30) is well-defined sinceK ∈ I1,loc(E, µ).The equation (30) determines the measureP uniquely. For any pairwisedisjoint bounded Borel setsB1, . . . , Bl ⊂ E and anyz1, . . . , zl ∈ C from

(30) we haveEPz#B11 · · · z#Bl

l = det

(1 +

l∑j=1

(zj − 1)χBjKχ⊔iBi

).

For further results and background on determinantal point processes, seee.g. [8], [11], [15], [16], [21], [22], [25].

If K belongs toI1,loc(E, µ), then, throughout the paper, we denote thecorresponding determinantal measure byPK . Note thatPK is uniquely de-fined byK, but different operators may yield the same measure. By theMacchı—Soshnikov theorem [17], [25], any Hermitian positive contractionthat belongs to the classI1,loc(E, µ) defines a determinantal point process.For the purposes of this paper, we will only be interested in determinantalpoint processes given by operators of orthogonal projection; there is a stan-dard procedure of doubling the phase space (see e.g. the Appendix in [2])that reduces the case of contractions to the case of projections.

2.9. The product of a determinantal measure and a multiplicativefunc-tional. We start by recalling the results of [6] (see also [5]) showing thatthe product of a determinantal measure with a multiplicative functional is,after normalization, again a determinantal measure, whosekernel is foundexplicitly.

Let g be a non-negative measurable function onE. If the operator1 +(g − 1)K is invertible, then we set

B(g,K) = gK(1+(g − 1)K)−1, B(g,K) =√gK(1+(g − 1)K)−1√g.

By definition,B(g,K), B(g,K) ∈ I1,loc(E, µ) sinceK ∈ I1,loc(E, µ),and, ifK is self-adjoint, then so isB(g,K).

We now quote Proposition 2.1 in [5].

Proposition 2.2. LetK ∈ I1,loc(E, µ) be a self-adjoint positive contrac-tion, and letPK be the corresponding determinantal measure onConf(E).Let g be a nonnegative bounded measurable function onE such that

(31)√g − 1K

√g − 1 ∈ I1(E, µ),


and that the operator1+(g − 1)K is invertible. Then the operatorsB(g,K), B(g,K)induce onConf(E) a determinantal measurePB(g,K) = P

B(g,K) satisfying

(32) PB(g,K) =ΨgPK∫

Conf(E)

Ψg dPK

.

Remark. Of course, from (31) and the invertibility of the operator1 +(g − 1)K we haveΨg ∈ L1(Conf(E),PK) and

∫Ψg dPK = det

(1 +

√g − 1K

√g − 1

)> 0,

so the right-hand side of (32) is well-defined.For the reader’s convenience, we recall the proof of Proposition 2.2 in the

case when the assumption (31) is replaced (cf. [6]) by a simpler assumption

(g − 1)K ∈ I1(E, µ);

for the proof in the general case, see [5]. Take a bounded measurable func-tion f onE such that(f − 1)K ∈ I1(E, µ); for example, one may takefthat is different from1 on a bounded set. We have(fg − 1)K ∈ I1(E, µ)since(f − 1)K ∈ I1(E, µ), (g− 1)K ∈ I1(E, µ). By definition, we have

(33) EPKΨfΨg = det(1 + (fg − 1)K) =

= det(1 + (f − 1)gK(1 + (g − 1)K)−1) det(1 + (g − 1)K).

We rewrite (33) in the form

EPKΨfΨg

EPKΨg

= det(1 + (f − 1)B(g,K)) = det(1 + (f − 1)B(g,K)).

Since a probability measure on the space of configurations isuniquely de-termined by the values of multiplicative functionals corresponding to allbounded functionsf that are different from1 on a bounded set, formula(33) implies Proposition 2.2.

2.10. Projections and subspaces.LetL ⊂ L2(E, µ) be a closed subspace,let Π be the corresponding projection operator, assumed to be locally oftrace class, and letPΠ the corresponding determinantal measure. Out aimis to determine how the measurePΠ changes if the subspaceL is multipliedby a function.

We start with the following clear

Proposition 2.3.Letα(x) be a measurable function such that|α(x)| = 1 µ-almost surely. Then the operator of orthogonal projection onto the subspaceα(x)L induces the same determinantal measurePΠ.


Proof. Indeed, ifΠ(x, y) is the kernel of the operatorΠ, then the kernelof the new operator has the form

α(x)Π(x, y)

α(y),

and such gauge transformations do not change the determinantal measure.

Proposition 2.4. Let g be a non-negative bounded function onE such thatthe operator1 + (g − 1)Π is invertible. Then the operator

(34) Πg =√gΠ(1 + (g − 1)Π)−1√g

is the operator of orthogonal projection onto the closure ofthe subspace√gL.

Proof. First, letϕ ∈ √gL, that is,ϕ =

√gϕ, ϕ ∈ L. Sinceϕ ∈ L, we

have(1 + (g − 1)Π)ϕ = gϕ,

whence(1 + (g − 1)Π)−1√gϕ = ϕ,

and finallyΠgϕ = ϕ

as desired.Now takeϕ to be orthogonal to the subspace

√gL. Sinceg is real-valued,

we have√gϕ ∈ L⊥, whence(1+(g−1)Π)−1√gϕ = ϕ, whenceΠgϕ = 0.

The proposition is proved completely.

2.11. Normalized multiplicative functionals. If the multiplicative func-tionalΨg is PΠ-integrable, then we introduce thenormalizedmultiplicativefunctionalΨg by the formula

(35) Ψg =Ψg∫

Conf(E)

ΨgdPΠ

.

We reformulate Proposition 2.1 in [5] in our new notation (34), (35):

Proposition 2.5. If g is a bounded Borel function onE such that√g − 1Π

√g − 1 ∈ I1(E, µ)

and the operator1+ (g− 1)Π is invertible, then the normalized multiplica-tive functionalΨg is well-defined, the subspace

√gL is closed, and we have

(36) ΨgPΠ = PΠg .


Note that closedness of the subspace√gL is immediate from the invert-

ibility of the operator1+ (g−1)Π: indeed, the operator1+ (g−1)Π takesthe subspaceL to the subspacegL, which is consequently closed; since thefunctiong is bounded from above, the subspace

√gL is, a fortiori, closed

as well.A key point in the argument of this paper is that the normalized multi-

plicative functional (35) can be defined, in such a way that the formula (36)still holds, even when the multiplicative functionalΨg itself is not defined,see Proposition 4.6 below.

2.12. Palm Measures of Determinantal Point Processes.Palm measuresof determinantal point processes admit the following characterization. Asabove, letΠ ∈ I1,loc(E, µ) be the operator of orthogonal projection onto aclosed subspaceL ⊂ L2(E, µ). Forq ∈ E satisfyingΠ(q, q) 6= 0, introducea kernelΠq by the formula

(37) Πq(x, y) = Π(x, y)− Π(x, q)Π(q, y)

Π(q, q).

If Π(q, q) = 0, then we also haveΠ(x, q) = Π(q, y) = 0 almost surely withrespect toµ, and we setΠq = Π.

The operatorΠq defines an orthogonal projection onto the subspace

L(q) = {ϕ ∈ L : ϕ(q) = 0}of functions inL that assume the value zero at the pointq; the spaceL(q)is well-defined by Assumption 1; in other words,L(q) is the orthogonalcomplement ofvq in L. Iterating, letq1, . . . , ql ∈ E be distinct and set

L(q1, . . . , ql) = {ϕ ∈ L : ϕ(q1) = . . . = ϕ(ql) = 0},and letΠq1,...,ql be the operator of orthogonal projection onto the subspaceL(q1, . . . , ql). Shirai and Takahashi [21] have proved

Proposition 2.6 (Shirai and Takahashi [21]). For µ-almost everyq ∈ E,the reduced Palm measure(PΠ)

q of the determinantal point processPΠ atthe pointq is given by the formula

(38) PqΠ = PΠq .

Furthermore, for anyl ∈ N and forρl-almost everyl-tupleq1, . . . , ql ofdistinct points inE, the iterated reduced Palm measurePq1,...,ql

Π is given bythe formula

(39) Pq1,...,qlΠ = PΠq1,...,ql .

Remark. Shirai and Takahashi [21] have in fact established the formula(38) for arbitrary positive self-adjoint locally trace-class contractions; the


formula (37) for the kernel stays the same. Note that the formula for con-tractions is a corollary of the formula for projection operators.

2.13. Conditional measures in the discrete case.In this subsection, weconsider the discrete case, in which the spaceE is a countable set endowedwith the discrete topology, and the measureµ is the counting measure. Inthis case, the reduced Palm measurePq of a point processP on Conf(E)can be described as follows: one takes the conditional measure ofP on thesubset of configurations containing a particle at positionq, and then oneremoves the particle atq; more formally,Pq is the push-forward of the saidconditonal measure under the operation that to a configurationX containingthe particle atq assigns the configurationX \ {q}.

In the discrete case we also have a dual construction: letPq be the con-ditional measure ofP with respect to the event that there is no particle atpositionq. More formally, set

Conf(E;E \ {q}) = {X ∈ Conf(E) : q /∈ X},and write

Pq =

P|Conf(E;E\{q})

P(Conf(E;E \ {q}))be the normalized restriction ofP onto the subsetConf(E;E \ {q}).

We have a dual to Proposition 2.6.

Proposition 2.7. Let q ∈ E be such thatµ({q}) > 0. Then the operator oforthogonal projection onto the subspaceχE\qL has the kernelΠq given bythe formula

(40) Πq(x, y) = Π(x, y) +Π(x, q)Π(q, y)

1−Π(q, q), x 6= q, y 6= q;

Πq(x, q) = Πq(q, y) = 0, x, y ∈ E.

Proof. This is a particular case of Corollary 6.4 in Lyons [15]; see alsoShirai-Takahashi [21], [22].

Given l ∈ N, m < l and anl-tuple (p1, . . . , pl), of distinct points inE,recall that we have introduced a subspaceL(p1, . . . , pm, pm+1, . . . , pl) bythe formula(41)L(p1, . . . , pm, pm+1, . . . , pl) = {χE\{pm+1,...,pl}ϕ : ϕ ∈ L, ϕ(p1) = · · · = ϕ(pl) = 0}.

Let Πp1,...,pm,pm+1,...,pl be the operator of orthogonal projection onto thesubspaceL(p1, . . . , pm, pm+1, . . . , pl). The corresponding determinantal mea-surePΠp1,...,pm,pm+1,...,pl admits the following characterization. Recall that

C(p1, . . . , pm, pm+1, . . . , pl)


is the set of configurations onE containing exactly one particle in eachof the positionsp1, . . . , pm and no particles in the positionspm+1, . . . , pl.There is a naturalerasing bijectionbetweenC(p1, . . . , pm, pm+1, . . . , pl)andC(p1, . . . , pm, pm+1, . . . , pl) obtained by erasing the particles in posi-tionsp1, . . . , pm.

Proposition 2.8. Consider the normalized restriction ofPΠ onto the setC(p1, . . . , pm, pm+1, . . . , pl). The push-forward of this normalized restric-tion onto the setC(p1, . . . , pm, pm+1, . . . , pl) under the erasing bijection isthe measurePΠp1,...,pm,pm+1,...,pl .

Proof. Again, this is a reformulation of Corollary 6.4 in Lyons [15]; seealso Shirai-Takahashi [21], [22].

2.14. Action of Borel automorphisms on point processes.Let T : E →E be an invertible measurable map such that for any bounded setB ⊂ Ethe setT (B) is also bounded. The mapT naturally acts on the space ofconfigurationsConf(E): namely, givenX ∈ Conf(E) we set

T (X) = {Tx, x ∈ X}Note that, by our assumptions,T (X) is a well-defined configuration onE;slightly abusing notation, we thus keep the same symbolT for the inducedaction on the space of configurations.

Let P be a probability measure onConf(E). We assume thatP admitscorrelation measures of all orders, and, forl ∈ N, we let ρl be thel-thcorrelation function of the point processP. Thel-th Cartesian power ofTnaturally acts on the measureρl, and, slightly abusing notation, we denotethe resulting measure byρl ◦ T . The measureρl ◦ T is, of course, thel-th correlation measure of the point processP ◦ T , the push-forward of themeasureP under the induced action of the automorphismT on the space ofconfigurations.

We now prove a simple general statement: if for a point process P andall l ∈ N, the reduced Palm measures corresponding to differentl- tuplesof points are equivalent, then for any Borel automorphismT acting by theidentity beyond a bounded set, the measuresP andP ◦ T are also equiva-lent, and the Radon-Nikodym derivative is found explicitlyin terms of theRadon-Nikodym derivatives of the reduced Palm measures. More precisely,we have the following

Proposition 2.9. Let T : E → E be a Borel automorphism admitting abounded subsetB ⊂ E such thatT (x) = x for all x ∈ E\B. Assume that

(1) for anyl ∈ N, the correlation measuresρl andρl ◦T are equivalent;(2) for any two collections{q1, ..., ql} and{q′1, ..., q′l} of distinct points

ofE, the measuresPq1,...,ql andP q′1,...,q′l are equivalent.


Then the measuresP andP◦T onConf(E) are equivalent, and forP-almostevery configurationX ∈ Conf(E) such thatX ∩ B = {q1, ..., ql} we have

dP ◦ TdP

(X) =dPTq1,...,T ql

dPq1,...,ql(X \ {q1, . . . , ql})×

dρl ◦ Tdρl

(q1, ..., ql).

Proposition 2.9 is a particular case of the following general propositionon absolute continuity of point processes. Letl ∈ N, letP, P be probabilitymeasures onConf(E) admitting correlation functions of orderl. Let ρl, ρlbe the corresponding correlation measures,Pq1,...,ql, Pq1,...,ql are correspond-ing reduced Palm measures. The symbol≪ denotes absolute continuity ofmeasures.

Proposition 2.10. If ρl ≪ ρl and Pq1,...,ql ≪ Pq1,...,ql for ρl-almost any dis-tinct q1, . . . , ql ∈ E, then alsoP ≪ P and forP-almost anyX ∈ Conf(E)and anyl particlesq1, . . . , ql ∈ X we have

dP

dP=dPq1,...,ql

dPq1,...,ql(X \ {q1, . . . , ql})×

dρldρl

(q1, ..., ql).

Proposition 2.10, in turn, is immediate from Proposition 2.1: indeed,takeX0 ∈ Conf(E) andq(0)1 , . . . , q

(0)l ∈ X0; take disjoint bounded open

setsB(1), ..., B(l) ⊂ E, setB = ∪B(i) and take an open setU ⊂ E disjointfrom all B(i) in such a way thatqi(0) ∈ B(i) for all i = 1, . . . , l andX0 \{q(0)1 , . . . , q

(0)l } ⊂ U . Let W be a neighbourhood ofX0 \ {q(0)1 , . . . , q

(0)l }

in Conf(E) satisfyingW ⊂ {X ∈ Conf(E) : X ⊂ U}. Introduce aneighbourhoodZ of X0 by setting

(42)Z = {X ∈ Conf(E) : #B(1)(X) = · · · = #B(l)(X) = 1, X|E\B ⊂ W }.Sets given by (42) form a basis of neighbourhoods ofX0.By definition of Palm measures and Proposition 2.1, we have

P(Z ) =

∫

B(1)×...×B(l)

Pq1,...,ql(Z )dρl(q1, ..., ql).

A similar formula holds forP, and Proposition 2.10 is proved.We now derive Proposition 2.9 from Proposition 2.10. As before, let

Conf(E;E \B) be the subset of those configurations onE all whose parti-cles lie inE \ B. since the automorphismT acts by the identity onE \B,all configurations in the setConf(E;E \B) are fixed byT , and we have

Pq1,...ql|Conf(E;E\B) ◦ T = P

Tq1,...T ql|Conf(E;E\B).


Since, in the notation of Proposition 2.9, we have

X \ {q1, . . . , ql} ∈ Conf(E;E \B),

we also havedPq1,...ql ◦ TdPq1,...,ql

(X \ {q1, . . . , ql}) =dPTq1,...,T ql

dPq1,...,ql(X \ {q1, . . . , ql}),

and Proposition 2.9 follows now from Proposition 2.10. Proposition 2.9 isproved completely.

3. INTEGRABILITY AND CONDITIONING

3.1. Integrability of the Palm kernel.

Lemma 3.1. Let q ∈ U be such thatΠ(q, q) 6= 0. Then the kernel of theoperatorΠq has the integrable form

(43) Πq(x, y) =Aq(x)Bq(y)−Aq(y)Bq(x)

x− y

where

(44) Aq(x) =A(x)B(q)−A(q)B(x)√

(A(q))2 + (B(q))2;

Bq(x) =A(x)A(q) +B(x)B(q)√

(A(q))2 + (B(q))2−√

(A(q))2 + (B(q))2(A(x)B(q)−A(q)B(x))

Π(q, q)(x− q).

Proof. We first consider the case

A(q) = 0, B(q) 6= 0.

Then

Π(x, q) =A(x)B(q)

x− qand

Πq(x, y) = Π(x, y)− B(q)2A(x)A(y)

Π(q, q)(x− q)(y − q)=Aq(x)Bq(y)−Aq(y)Bq(x)

x− y

with

Aq(x) = A(x), Bq(x) = B(x)− B(q)2A(x)

Π(q, q)(x− q),

as desired. The general case is reduced to this particular one by a rotation

A(x) → A(x)B(q)− A(q)B(x)√(A(q))2 + (B(q))2

;

B(x) → A(x)A(q) +B(x)B(q)√(A(q))2 + (B(q))2

.

The proposition is proved.


3.2. The relation between Palm subspaces.

3.2.1. The subspaceL′.

Proposition 3.2. If ϕ ∈ L2(R, µ) is such thatxϕ ∈ L2(R, µ), then theintegrals ∫

R

ϕ(x)A(x)dµ(x),

∫

R

ϕ(x)B(x)dµ(x)

are well-defined.

The proof is immediate from the clear relation

〈vp(x), (x−p)ϕ(x)〉 = A(p)

∫

R

ϕ(x)B(x)dµ(x)−B(p)

∫

R

ϕ(x)A(x)dµ(x).

Let(45)

L′ = {ψ ∈ L : xψ ∈ L2(R, µ),

∫

R

ψ(x)A(x)dµ(x) =

∫

R

ψ(x)B(x)dµ(x) = 0}.

Proposition 3.3. Let p ∈ U andϕ ∈ L satisfyϕ(p) = 0. Then there existsψ ∈ L′ such that

(46) ϕ(x) = (x− p)ψ(x).

Proof. It suffices to consider the casep = 0, A(0) = 0, B(0) 6= 0:the general case is reduced to this one by a translation ofR and a linearunimodular change of variable (5).

Let ψ′ be such that (46) holds (in the continuous case, such a functionψ′ is unique; in the discrete case, however, there are many suchfunctions,differing by the their value atp = 0). Applying the commutatorxΠ − Πxto the functionψ′, we obtain

(47) xΠψ′(x)− ϕ(x) =

= A(x)

∫

R

B(y)ψ′(y)dµ(y)−B(x)

∫

R

A(y)ψ′(y)dµ(y).

Sinceϕ ∈ L, ϕ(0) = 0, A(0) = 0, B(0) 6= 0, substitutingx = 0 into(47) we obtain

(48)∫

R

A(y)ψ′(y)dµ(y) = 0.

Recall that by definition we have

v0(x) =A(x)

x∈ L.


Dividing (47) byx and keeping (48) in mind, we obtain that there existsα ∈ C such that

(49) Πψ′(x)− ψ(x)− αδ0 = v0(x)

∫

R

B(y)ψ′(y)dµ(y).

The extra termαδ0 is only necessary in the case whenµ(0) > 0. It followsthat we haveψ = ψ′+αδ0 ∈ L and, consequently, applying the commutatorxΠ− Πx to the functionψ, that we also have

∫

R

A(y)ψ(y)dµ(y) =

∫

R

B(y)ψ(y)dµ(y) = 0.

The proposition is proved completely.We proceed with the proofs of Theorems 1.5, 1.7. We must now sepa-

rately consider the case of continuous and the case of purelyatomic mea-suresµ.

3.2.2. The case of continuous measures.Assume that the measureµ satis-fiesµ({p}) = 0 for anyp ∈ E and, as before, letΠ be a locally trace classoperator with an integrable kernel defined on an open subsetU ⊂ R whosecomplement has measure0.

3.2.3. The relation between Palm subspaces.

Proposition 3.4. For any distinct pointsp1, . . . , pl, q1, . . . , ql ∈ U we have

(50) L(p1, . . . , pl) =(x− p1) . . . (x− pl)

(x− q1) . . . (x− ql)L(q1, . . . , ql).

Remark. The coincidence of subspaces is understood as coincidenceof subspaces inL2; the functions from the right-hand side subspace are ofcourse not defined at the pointsq1, . . . , ql; they are nonetheless well-definedas elements ofL2 since the measureµ is continuous. For discrete measuresthe fomulation will be modified.

Proof. The proof proceeds by induction onl. We start withl = 1. ByProposition 3.3, we have

x− p1x− q1

L(q1) ⊂ L(q1) + L′ ⊂ L.

By definition, now, any function belonging to the subspacex− p1x− q1

L(q1)

assumes value0 at the pointp1, whence the inclusion

x− p1x− q1

L(q1) ⊂ L(p1).


Interchanging the pointsp1 andq1, we obtain the converse inclusion (usingagain continuity of the measureµ), and the proposition is proved forl = 1.

By Lemma 3.1, Palm measures of determinantal point processes given byprojection operators with integrable kernels are themselves given by projec-tion operators with integrable kernels. Applying, step by step, Proposition3.4 for l = 1, we obtain

(51) L(p1, . . . , pl) =(x− p1)

(x− q1)L(q1, p2, . . . , pl) =

(x− p1)(x− p2)

(x− q1)(x− q2)L(q1, q2, . . . , pl) = . . .

· · · = (x− p1) . . . (x− pl)

(x− q1) . . . (x− ql)L(q1, . . . , ql),

and Proposition 3.4 is proved completely.

3.2.4. The case of discrete measures.We now proceed to the case of atomicmeasures; without losing generality, we assume that the setE is countableand the measureµ is the counting measure. Since we are only interestedin determinantal measures, we need only consider configurations withoutmultiple points: in other words, the spaceConf(E) can be identified withthe space of infinite binary sequences.

In the discrete case, we have a dual to Proposition 3.1.

Proposition 3.5. LetΠ be a projection operator with an integrable kernel.Let q ∈ U be such thatΠ(q, q) 6= 1. Then the kernel of the operatorΠq hasthe integrable form

(52) Πq(x, y) =Aq(x)B q(y)−Aq(y)B q(x)

x− y

whereAq(q) = B q(q) = 0 and forx 6= q, y 6= q we have

(53) Aq(x) =A(x)B(q)−A(q)B(x)√

(A(q))2 + (B(q))2;

B q(x) =A(x)A(q) +B(x)B(q)√

(A(q))2 + (B(q))2+

√(A(q))2 + (B(q))2(A(x)B(q)− A(q)B(x))

(1− Π(q, q))(x− q).

Proof. This is a straightforward verification using Proposition 2.7 andproceeding in the same way as the proof of Proposition 3.1. Asin Proposi-tion 3.1, the integrable representation for the kernel is, of course, far fromunique.


Proposition 3.6. Let the kernelΠ be integrable. Letp1, . . . , pl ∈ E bedistinct, and letπ be a permutation of{1, . . . , l}. Then we have

(54) L(pπ(1), . . . , pπ(m), pπ(m+1), . . . , pπ(l)) =

= χE\{p1,...,pl}

(x)(x− pπ(1)) . . . (x− pπ(m))

(x− p1) . . . (x− pm)L(p1, . . . , pm, pm+1, . . . , pl).

Proof. As in the continuous case, we proceed by induction andstart withthe casel = 2, m = 1: we need to show, for any distinctp, q ∈ E, theequality

(55) L(p, q) = χE\{p,q}x− p

x− qL(q, p).

Now, by Proposition 3.3, we have

χE\qL(q)

x− q⊂ χE\qL.

Sincex− p

x− q= 1 +

q − p

x− q,

we also havex− p

x− qχE\qL(q) ⊂ χE\qL = L(q)

Now, multiplying any function byχE\qx− p

x− qyields a function that assumes

value 0 at the pointp; we thus conclude

(56) χE\{p,q}x− p

x− qL(p, q) ⊂ L(p, q).

Interchanging the variablesp, q, we obtain the inverse inclusion, and (55) isproved.

We proceed with the induction argument. Our kernelΠ is integrable.We know from Proposition 3.1 that for anyq ∈ E the kernelΠq is alsointegrable, while Proposition 3.5 implies that the kernelΠq is integrable aswell. The conclusion of Proposition 3.6 in the particular casel = 2, m = 1is thus valid for any kernel of the formΠq1,...,qr,qr+1,...,qs with r ≤ s andq1, . . . , qs arbitrary distinct points inE. Representing the permutationπ asa product of transpositions and applying (56) repeatedly, we conclude theproof of Proposition 3.6.

4. MULTIPLICATIVE FUNCTIONALS AND REGULARIZATION

The main result of this Section is Proposition 4.6 below; before formu-lating it, we make some preliminary remarks.


4.1. An estimate of diagonal values of the kernelΠg. As the followingproposition shows, diagonal values of the kernel ofΠg can be estimatedfrom above by the diagonal values of the kernelΠ.

Proposition 4.1. Let the kernelΠ satisfy Assumption1, and letg be a non-negative bounded function onE such that the operator1 + (g − 1)Π isinvertible. Then for anyq ∈ U we have

Πg(q, q) ≤ g(q)||(1 + (g − 1)Π)−1||Π(q, q).Proof.As before, we let〈, 〉 be the standard inner product inL2(E, µ) and

we writevq(x) = Π(x, q) so thatΠ(q, q) = 〈vq, vq〉. By definition then

(57) Πg(q, q) = g(q)〈Π(1 + (g − 1)Π)−1vq, vq〉 ≤≤ g(q)||(1 + (g − 1)Π)−1||〈vq, vq〉,

and the proposition is proved.

4.2. Multiplicative functionals corresponding to a function satisfyingg ≤ 1. Let g be a nonnegative function satisfyingg ≤ 1. If

(58) ||χ{x∈E:g(x)<1}Π|| < 1,

then the space√gL is closed and the operator1 + (g − 1)Π is invertible.

Indeed, the assumption (58) together with the inequality0 ≤ g ≤ 1 imme-diately implies the inequality||(g − 1)Π|| < 1. Invertibility of the operator1 + (g − 1)Π is established, and since this operator takes the subspaceLto the subspacegL, it follows that the subspacegL is closed, whence, afortiori, since0 ≤ g ≤ 1, the subspace

√gL is also closed. Summing up,

we obtain

Proposition 4.2. Let g be a bounded measurable function onE satisfyingg ≤ 1 and (58). If

(59) tr(χ{x∈E:g(x)<1}Πχ{x∈E:g(x)<1}

)< +∞,

then all the conclusions of Proposition2.5hold for the functiong.

Remark. The condition (58) can be equivalently reformulated as fol-lows. LetE1 = {x ∈ E : g(x) = 1}, E2 = {x ∈ E : g(x) < 1}. Therequirement (58) is equivalent to the existence of a positive constantC suchthat for anyϕ ∈ L we have

||χE2ϕ|| ≤ C||χE1ϕ||.Consequently, if the condition (58) holds for the subspaceL, then it alsoholds for any subspace of the formhL, whereh is a positive functionbounded above and bounded away from0.


4.3. Multiplicative functionals corresponding to a function g satisfyingg ≥ 1.

4.3.1. The case when the functiong is bounded.Proposition 2.5 takes asimpler form when our bounded functiong satisfiesg ≥ 1. First, in thiscase the subspace

√gL is automatically closed. Second, if

√g − 1Π

√g − 1

belong to the trace class, then the operator1 + (g − 1)Π is automaticallyinvertible. To verify this, observe first that in this case the operator

√g − 1Π

is Hilbert- Schmidt, consequently, so is(g− 1)Π. To check the invertibilityof the operator1 + (g − 1)Π, it thus suffices to prove that a functionϕsatisfying

(60) ϕ+ (g − 1)Πϕ = 0

must be the zero function. Setψ = −√g − 1Πϕ so thatϕ =

√g − 1ψ.

Note that bothϕ andψ are by definition zero on the set{x ∈ E : g(x) = 1}.From (60) we now have

ψ +√g − 1Π

√g − 1ψ = 0,

whence〈ψ, ψ〉+ 〈Πϕ, ϕ〉 = 0,

whence finallyϕ = ψ = 0.We can now reformulate Proposition 2.5 in the following simpler form

Proposition 4.3. Let g be a bounded measurable function onE satisfyingg ≥ 1 and such that the operator

√g − 1Π

√g − 1 belongs to the trace

class. Then all the conclusions of Proposition2.5hold for the functiong.

4.3.2. The case of unboundedg. The function(x−p)/(x−q) is unboundedonR, and we prepare, for future use, a proposition on multiplicative func-tionals corresponding to unbounded functions.

Proposition 4.4. Let g be a measurable function onE satisfyingg ≥ 1.Assume that

(1) the space√gL is a closed subspace ofL2(E, µ);

(2) the operator√g − 1Π is Hilbert-Schmidt;

(3) the operatorΠg of orthogonal projection onto the space√gL is

locally of trace class, and there existsR > 0 such that

(61) tr(χ{x∈E:g(x)>R}Π

gχ{x∈E:g(x)>R}

)< +∞.

ThenΨg ∈ L1(Conf(E),PΠ), and the corresponding normalized multi-plicative functionalΨg satisfies

(62) ΨgPΠ = PΠg .


Remark. Of course, the subspace√gL, provided it lies inL2, is auto-

matically closed.Proof. The Hilbert-Schmidt norm of the operator

√g − 1Π is given by

the expression∫

E

(g(x)− 1)Π(x, x)dµ(x).

Convergence of this integral implies, in particular, for any R > 1, the esti-mate

tr(χ{x∈E:g(x)>R}Πχ{x∈E:g(x)>R}

)< +∞,

whence it follows that the functiong is bounded onPΠ-almost every con-figuration.

For R > 0 setgR(x) = g(x) if g(x) < R andgR(x) = 1 otherwise.Since the operator

√g − 1Π is Hilbert-Schmidt, the operator

√gR − 1Π is

a fortiori Hilbert-Schmidt, and we have∫

Conf(E)

ΨgRdPΠ = det(1 +√gR − 1Π

√gR − 1).

By definition and since the functiong is bounded onPΠ-almost every con-figuration, we have thePΠ-almost sure convergence

Ψg = limR→∞

ΨgR.

The determinant in the right-hand side is bounded above by a constant de-pending only on the Hilbert-Schmidt norm of

√g − 1Π, and integrability

of Ψg is thus established.It remains to check the equality (62). LetR be big enough in such a way

that


gχ{x∈E:g(x)>R}

)< 1.

Setg = gR/g. We clearly haveΨg ∈ L1(Conf(E),PΠg), and since√g(√gL) =√

gRL, Proposition 4.2 implies the relation

ΨgPΠg = PΠgR .

By definition,ΨgΨg = ΨgR. Since we already know that

ΨgRPΠ = PΠgR ,

the equality (62) is proved, and Proposition 4.4 is proved completely.


4.4. Regularization of additive functionals. Let f : E → C be a Borelfunction. We setSf to be the corresponding additive functional, and, ifSf ∈ L1(Conf(E),PΠ), then we set

(64) Sf = Sf − ESf .

The random variableSf will be called thenormalizedadditive functionalcorresponding tof . We shall now see that the normalized additive func-tional can be defined even when the additive functional itself is not well-defined. Set

Var(Π, f) =1

2

∫

E

∫

E

|f(x)− f(y)|2|Π(x, y)|2dµ(x)dµ(y).

Note that the valueVar(Π, f) does not change if the functionf is changedby an additive constant. IfSf ∈ L2(Conf(E),PΠ), thenVar(Π, f) < +∞and

(65) Var(Sf) = E|Sf |2 = Var(Π, f).

Note also the clear inequality

(66) Var(Π, f) ≤ 2

∫

E

|f(x)|2Π(x, x)dµ(x)

which is obtained by summing the inequality|f(x)− f(y)|2 ≤ 2(|f(x)|2+|f(y)|2) over allx, y and using the Pythagoras theorem

Π(x, x) =

∫

E

|Π(x, y)|2dµ(y).

These formulae show that the integral defining the variance of an additivefunctional may converge even when the integral defining its expectationdoes not. The normalized additive functional can thus by continuity bedefined inL2 even when the additive functional itself diverges almost surely.

We therefore introduce the Hilbert spaceV(Π) in the following way: theelements ofV(Π) are functionsf onE satisfyingVar(Π, f) < +∞; func-tions that differ by a constant are identified, but, slightlyabusing terminol-ogy we still refer to elements ofV(Π) as functions. The square of the normof an elementf ∈ V(Π) is preciselyVar(Π, f). By definition, boundedfunctions that are identically zero in the complement of a bounded set forma dense subset ofV(Π). The correspondencef → Sf is thus an isometricembedding of a dense subset ofV(Π) into L2(Conf(E),PΠ); it thereforeadmits a unique isometric extension onto the whole spaceV(Π), and weobtain the following


Proposition 4.5. There exists a unique linear isometric embedding

S : V(Π) → L2(Conf(E),PΠ), S : f → Sf

such that

(1) ESf = 0 for all f ∈ V(Π);(2) if Sf ∈ L1(Conf(E),PΠ), thenSf is given by (64).

4.5. Regularization of multiplicative functionals. Given a functiong suchthatVar(Π, log g) < +∞, set

Ψg = exp(S log g).

By definition, we haveΨg1g2 = Ψg1Ψg2.

SinceES log g = 0, by Jensen’s inequality, for any positive functiong wehave

EΨg ≥ 1.

The expectationEΨg may however be infinite, and our next aim is to giveconditions for its finiteness.

It will be convenient for us to allow zero values for the function g: letthereforeg be nonnegative, setE0 = {x ∈ E : g(x) = 0}, assume that thesubsetConf(E;E\E0) of those configurations all whose particles lie inE\E0 has positive probability, consider the restriction of our measureP ontothe subspaceConf(E;E \ E0), introduce the corresponding functionalΨg

and extend it to the whole ofE by settingΨg(X) = 0 for all configurationscontaining a particle atE0. Assume thattrχE0ΠχE0 < +∞. Then we havePΠ(Conf(E;E \E0)) = det(1−χE0ΠχE0). In particular,PΠ(Conf(E;E \E0)) > 0 provided that the following holds: ifϕ ∈ L satisfiesϕ(x) = 0 forall x ∈ E \E0, thenϕ = 0 identically.

If Ψg ∈ L1(Conf(E),PΠ), then, as before, we write

Ψg =Ψg

EΨg

.

The main result of this section is

Proposition 4.6. Let E0 ⊂ E be a Borel subset satisfyingtrχE0ΠχE0 <+∞ and such that ifϕ ∈ L satisfiesϕ(x) = 0 for all x ∈ E \ E0, thenϕ = 0 identically .

Let g be a nonnegative function, positive onE \ E0 and admittingε > 0such that the setEε = {x ∈ E : |g(x)− 1| > ε} is bounded and

(67)∫

Eε

|g(x)− 1|Π(x, x)dµ(x) +∫

R\Eε

|g(x)− 1|2Π(x, x)dµ(x) < +∞.


ThenΨg ∈ L1(Conf(E),PΠ). If the subspace√gL is closed and the cor-

responding operator of orthogonal projectionΠg satisfies, for a sufficientlylargeR, the condition


gχ{x∈E:g(x)>R}

)< +∞,

then we also have

(69) PΠg = ΨgPΠ.

As we shall see below, Propositions 1.4, 1.6 are immediate Corollaries ofProposition 4.6.

Proof. First, our assumptions implyPΠ(Conf(E;E \ E0)) > 0 and so,without losing generality, restricting ourselves, if necessary, to the subsetConf(E;E \ E0), we can assume that the functiong is positive (observehere that, by Proposition 4.1, applied to the functionχE\E0

, the condition(67) continues to hold for the restricted kernel).

We will need a simple auxiliary

Lemma 4.7. If f is a bounded function onE, then the Hilbert-Schmidtnorm||fΠf ||2 of the operatorfΠf satisfies

(70) ||fΠf ||2 ≤∫

E

(f(x))2Π(x, x)dµ(x).

Proof. Indeed, the Cauchy-Bunyakovsky-Schwarz inequality implies

(Π(x, y))2 ≤ Π(x, x)Π(y, y),

whence

||fΠf ||22 =∫

E

∫

E

f(x)2f(y)2Π(x, y)2dµ(x)dµ(y) ≤

∫

E

f(x)2Π(x, x)dµ(x)

2

.

The proof is complete. We proceed to the proof of Proposition4.6 and firstconsider the case of bounded functions. LetA2(Π) be the set of positiveBorel functionsg onE satisfying

(1) ∞ > supEg ≥ inf

Eg > 0;

(2) ∫

E

|g(x)− 1|2Π(x, x)dµ(x) < +∞.

By definition, the setA2(Π) is a semigroup under multiplication.


Endow the setA2(Π) with a metric by setting the distance between twofunctionsg1 andg2 to be

√√√√∫

E

|g1(x)− g2(x)|2Π(x, x)dµ(x).

Using the second condition in the definition ofA2(Π) and the estimate(66), for anyg ∈ A2(Π) we have

Var(Π, g − 1) < +∞.

Since on any interval of the positive half-line, bounded away from zeroand infinity, the quantity| log t−t+1|/t2 is bounded both above and below,for any functiong ∈ A2(Π), we also have

Var(Π, log g) < +∞.

In particular, for any functiong ∈ A2(Π) the functionalΨg is well-defined.Our next aim is to establish its integrability.

Proposition 4.8.For any functiong ∈ A2(Π)we haveΨg ∈ L1(Conf(E),PΠ).The correspondences

g → Ψg, g → Ψg

are continuous mappings fromA2(Π) toL1(Conf(E),PΠ).

First we give an upper bound for theL2-norm ofΨg.

Proposition 4.9. For anyε > 0,M > 0 there exists a constantCε,M > 0such that ifg ∈ A2(Π) satisfies

(71) M ≥ supEg ≥ inf

Eg ≥ ε

then

(72) logE|Ψg|2 ≤ Cε,M

∫

E

|g(x)− 1|2Π(x, x)dµ(x);

Proof. It suffices to prove the estimate

(73) logEΨg ≤ Cε,M

∫

E

|g(x)− 1|2Π(x, x)dµ(x),

and (72) follows by multiplicativity (perhaps with a different constant). Itsuffices to establish (73) in the case when the set{x ∈ E : g(x) 6= 1} is


bounded, as the general case follows by Fatou’s lemma. Now there exists aconstantC2 > 0 such that

(74) logEΨg ≤ tr(√g − 1Π

√g − 1) + C2||

√g − 1Π

√g − 1||22 ≤

≤∫

E

(g(x)− 1)Π(x, x)dµ(x) + C2

∫

E

|g(x)− 1|2Π(x, x)dµ(x).

Note that we have assumed boundedness of the set{x ∈ E : g(x) 6= 1} inorder that the integral

∫

E

(g(x)− 1)Π(x, x)dµ(x)

be well-defined; this integral will, however, disappear from the final result.Indeed, from (71), again using the fact that the quantity| log t − t + 1|/t2is bounded both above and below by constants only depending on ε andM ,we obtain

(75)

∣∣∣∣∣∣

∫

E

(g(x)− 1)Π(x, x)dµ(x)−∫

E

log g(x)Π(x, x)dµ(x)

∣∣∣∣∣∣≤

≤ CM,ε

∫

E

|g(x)− 1|2Π(x, x)dµ(x),

whence

logEΨg = logEΨg − ESlog g ≤ C ′M,ε

∫

E

|g(x)− 1|2Π(x, x)dµ(x),

and the proposition is proved.

Proposition 4.10. For anyε > 0,M > 0 there exists a constantCε,M > 0such that ifg1, g2 ∈ A2(Π) satisfy

M ≥ supEg1 ≥ inf

Eg1 ≥ ε,M ≥ sup

Eg2 ≥ inf

Eg2 ≥ ε,

then

E|Ψg1−Ψg2 | ≤ E|Ψg1|2

exp

Cε,M

∫

E

|g1(x)− g2(x)|2Π(x, x)dµ(x)

− 1

.

Proof. SinceEΨg ≥ 1, we have

E|Ψg − 1|2 ≤ EΨg2 − 1.


From the estimate (72) we have

(76) E|Ψg − 1|2 ≤ exp

C

∫

E

|g(x)− 1|2Π(x, x)dµ(x)

− 1.

Applying (76) tog = g1/g2, recalling the boundedness of bothg1 andg2and using multiplicativity, we obtain the proposition.

Proposition 4.10 implies Proposition 4.8.We next check that the regularized multiplicative functional correspond-

ing to a functiong is indeed the Radon-Nikodym derivative of the measurePΠg with respect to the measurePΠ.

First note that ifg ∈ A2(Π), then the subspace√gL is automatically

closed; as always, we setΠg to be the corresponding operator of orthogonalprojection.

Corollary 4.11. Let g ∈ A2(Π) satisfy

supx∈E

|g(x)− 1| < 1.

Then the operatorΠg is locally of trace class, and we have

(77) PΠg = ΨgPΠ.

Proof. By our assumptions, we have||(g − 1)Π|| < 1. The operator

Πg =√gΠ(1 + (g − 1)Π)−1√g

is locally of trace class since so isΠ. Let E(n) be a sequence of boundedsets exhaustingE, and setgn = 1 + (g − 1)χE(n). We have

supn

||(gn − 1)Π|| < 1.

The operatorsΠgn =

√gnΠ(1 + (gn − 1)Π)−1√gn

are by definition locally of trace class since so isΠ. As n → ∞, we havethe strong operator convergence

(1 + (gn − 1)Π)−1 → (1 + (g − 1)Π)−1

and, consequently, also the convergence

Πgn → Πg

in the space of locally trace class operators. It follows that, asn → ∞,the sequence of measuresPΠgn weakly converges toPΠg in the space ofprobability measures onConf(E).

Furthermore,Ψgn → Ψg in L1(Conf(E),PΠ), whence also

ΨgnPΠ → ΨgPΠ


weakly in the space of probability measures onConf(E). We finally obtainthe desired equality (77).

4.6. Conclusion of the proof of Proposition 4.6.Chooseε ∈ (0, 1) insuch a way that we have

(78) ||χ{x∈E:g(x)<1−ε}Π|| < 1.

Set

(79) g1 = (g − 1)χ{x∈E:|g(x)−1|≤ε} + 1.

(80) g2 = (g − 1)χ{x∈E:g(x)<1−ε} + 1.

(81) g3 = (g − 1)χ{x∈E:g(x)>1+ε} + 1.

By definition,g = g1g2g3.By definition, the subspace

√g1L is closed, and, by Corollary 4.11, we

havePΠg1 = Ψg1PΠ.

Proposition 4.1 implies the existence of a positive constant C such that

Πg1(x, x) ≤ CΠ(x, x)

for µ-amost allx ∈ E.Next, the inequality (78) implies that all the assumptions of Proposition

4.2 are verified for the functiong2 and the operatorΠg1 (cf. Remark follow-ing the proof of Proposition 4.2); applying Proposition 4.2to the functiong2 and the operatorΠg1, we arrive at the formula

PΠg1g2 = Ψg2PΠg1 = Ψg1g2PΠ.

Again, Proposition 4.1 implies the existence of a positive constantC suchthat

(82) Πg1g2(x, x) ≤ CΠ(x, x)

for µ-amost allx ∈ E.In the third step, we apply Proposition 4.4 to the functiong3 and the op-

eratorΠg1g2 . In order to be able to do so, we first verify, one by one, theassumptions of Proposition 4.4 for the functiong3 and the operatorΠg1g2.First, the subspace

√g3(

√g1g2)L =

√gL is closed. Second, the assump-

tion (67) of Proposition 4.6, together with the estimate (82), implies theestimate ∫

E

|g3(x)− 1|Πg1g2(x, x)dµ(x) < +∞,


and, consequently, that the operator√g3 − 1Πg1g2 is Hilbert-Schmidt. Fi-

nally, from (68), keeping in mind that(Πg1g2)g3 = Πg, we immediatelyhave, for sufficiently largeR, the desired estimate

tr(χ{x∈E:g3(x)>R} (Π

g1g2)g3 χ{x∈E:g3(x)>R}

)< +∞.

Applying Proposition 4.4 to the functiong3 and the operatorΠg1g2 , we have

PΠg1g2g3 = Ψg3PΠg1g2 .

Observe that we only used regularized multiplicative functionals at the veryfirst step of our argument. In other words, there exist constantsC1, C2, C3

such that we havePΠg1 = C1Ψg1PΠ,

PΠg1g2 = C2Ψg2PΠg1,

PΠg = PΠg1g2g3 = C3Ψg3PΠg1g2 .

By definition, we have

Ψg = Ψg1Ψg2Ψg3

and, consequently, for a suitable positive constantC4, also

Ψg = C4Ψg1Ψg2Ψg3.

Summing up, we finally obtain

PΠg = ΨgPΠ.

Proposition 4.6 is proved completely.

4.7. Proof of Proposition 1.4. We check that forl ∈ N and any distinctpointsp1, . . . , pl, q1, . . . , ql ∈ U , the function

g(x) =

((x− p1) . . . (x− pl)

(x− q1) . . . (x− ql)

)2

satisfies

(1) Var(log g,Πq1,...,ql) < +∞;(2) For any bounded intervalI ⊂ R we have

∫

I

g(x)dΠq1,...,ql(x, x)dµ(x) < +∞;

(3) For anyε > 0, we have

(83)∫

{x∈R: mini=1,...,l

|x−qi|>ε}

|g(x)− 1|2Πq1,...,ql(x, x)dµ(x) < +∞.


Remark. Of course, convergence (83) for someε > 0 is equivalent toconvergence (83) for anyε > 0.

Indeed, since the kernelΠ is smooth, for anyi = 1, . . . , l, asx ranges ina sufficiently small neighbourhood of pointqi, we have

|Πq1,...,ql(x, x)| < C|x− qi|2.

Consequently, the integrandg(x)Πq1,...,ql(x, x) is bounded onI, and thesecond condition follows. The third condition is an immediate corollaryof (6): indeed, on the set{x ∈ R : min

i=1,...,l|x− qi| > ε} we have

|g(x)− 1|2 ≤ const× 1

1 + x2,

with the constant depending only onq1, . . . , ql andε. It follows that∫

{x∈R: mini=1,...,l

|x−qi|>ε}

|g(x)− 1|2Π(x, x)dµ(x) < +∞,

and, since the operatorΠq1,...,ql is a finite-rank perturbation of the operatorΠ, the desired inequality follows as well.

The second and the third conclusion together imply

(84)∫

R

| log g(x)|2dΠq1,...,ql(x, x)dµ(x) < +∞

(indeed, for small values ofg integrability in (84) follows from the secondcondition, for large values ofg from the third.) The first conclusion followsfrom (84). In particular, all assumptions of Proposition 4.6 are satisfiedfor the functiong and the kernelΠq1,...,ql and the normalized multiplica-tive functionalΨg is well-defined with respect to the measurePΠq1,...,ql .Theproposition is proved completely.

Proposition 4.6 together with Proposition 3.4 imply the following imme-diate

Corollary 4.12. Under the assumptions of Theorem1.5, for any distinctpointsp1, . . . , pl, q1, . . . , ql ∈ U , for the corresponding reduced Palm mea-sures are equivalent, and we have

dPΠp1,...,pl

dPΠq1,...,ql

= Ψ∣∣∣ (x−p1)...(x−pl)

(x−q1)...(x−ql)

∣∣∣2.

Together with Proposition 2.9, Corollary 4.12 implies Theorem 1.5. The-orem 1.5 is proved completely.


4.8. Proof of Proposition 1.6. Denoteqi = σ(pi), i = 1, . . . , l; of course,we have{p1, . . . , pl} = {q1, . . . , ql}. Apply Proposition 4.6 to the function

(85) g(x) =

m∏

i=1

(x− qix− pi

)2

χE\{p1,...,pl}(x).

The functiong is bounded, and the condition (9) implies∑

E

|g(x)− 1|2 < +∞,

which, in turn, immediately implies (67). Since the subspace L does notcontain functions supported on finite sets, all other assumptions of Proposi-tion 4.6 are verified for the kernelΠp1,...,pm,pm+1,...,pl and the functiong givenby (85). Proposition 1.6 is proved completely.

In a similar way to the continuous case, Proposition 4.6 and Proposition3.6 together imply that, under the assumptions of Proposition 1.6, we have

(86) PΠq1,...,qm,qm+1,...,ql = Ψ(p1, . . . , pl, m, σ)PΠp1,...,pm,pm+1,...,pl .

The relation (86) together with Proposition 2.9 implies Theorem 1.7.Theorem 1.7 is proved completely.

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A IX -MARSEILLE UNIVERSITE, CNRS, CENTRALE MARSEILLE, I2M, UMR 7373

THE STEKLOV INSTITUTE OFMATHEMATICS, MOSCOW

THE INSTITUTE FORINFORMATION TRANSMISSION PROBLEMS, MOSCOW

NATIONAL RESEARCHUNIVERSITY HIGHER SCHOOL OFECONOMICS, MOSCOW

E-mail address: [email protected]

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