Summability of joint cumulants of nonindependent lattice fields

0.25 MB·

by  Jani Lukkarinen

Summability of joint cumulants of nonindependent lattice
fields
Jani Lukkarinen∗, Matteo Marcozzi†, Alessia Nota‡
∗, †, ‡University of Helsinki, Department of Mathematics and Statistics
6
1 P.O. Box 68, FI-00014 Helsingin yliopisto, Finland
0
2 ‡University of Bonn, Institute for Applied Mathematics
n Endenicher Allee 60, D-53115 Bonn, Germany
a
J February 1, 2016

9
2

]
R Abstract
P Weconsidertwononindependentrandomfieldsψ andφdefinedonacountablesetZ. For
h. instance, Z = Zd or Z = Zd ×I, where I denotes a finite set of possible “internal degrees
t of freedom” such as spin. We prove that, if the cumulants of both ψ and φ are ℓ1-clustering
a
m up to order 2n, then all joint cumulants between ψ and φ are ℓ2-summable up to order n,
in the precise sense described in the text. We also provide explicit estimates in terms of
[ the related ℓ1-clustering norms, and derive a weighted ℓ2-summation property of the joint
1 cumulants if the fields are merely ℓ2-clustering. One immediate application of the results is
v givenbyastochasticprocessψt(x)whosestateisℓ1-clusteringatanytimet: thentheabove
3 estimatescanbeappliedwithψ=ψt andφ=ψ0andweobtainuniformintestimatesforthe
6 summability of time-correlations of the field. The above clustering assumption is obviously
1 satisfied by any ℓ1-clustering stationary state of the process, and our original motivation for
8 thecontrol of thesummability of time-correlations comes from a quest for a rigorous control
0 oftheGreen-Kubocorrelation functioninsuchasystem. Akeyroleintheproofisplayedby
1. theproperties of non-Gaussian Wick polynomials and theirconnection to cumulants.
0
6
1 Introduction and physical motivation
1
:
v
Inmanyproblemsofphysicalinterest,thebasicdynamicvariableisarandomfield. Inadditionto
i
X properstochastic processes,suchasparticlesevolvingaccordingto Brownianmotion,the random
r fieldcoulddescribeforinstanceadensityofparticlesofaHamiltoniansystemwithrandominitial
a
data or after time-averaging.
One particular instance of the second kind is the Green-Kubo formula which connects the
transportcoefficients,suchasthermalconductivity,tointegralsoverequilibrium time-correlations
of the current observable of the relevant conserved quantity, for instance, of the energy current.
The equilibrium time-correlations are cumulants of current fields between time zero and some
later time. The current fields are generated by distributing the initial data according to some
fixed equilibrium measure and then solving the evolution equations: this yields a random field,
even when the time-evolution itself is deterministic.
Hence,thecontrolofcorrelationfunctions,i.e.,cumulants,ofrandomfieldsisacentralproblem
fora rigorousstudy oftransportproperties. One approach,whichhasbeen usedbothinpractical
∗E-mail: [email protected]
†E-mail: [email protected]
‡E-mail: [email protected]
1
applications and in direct mathematical studies, is given by Boltzmann transport equations. It is
usually derived from the microscopic system by using moment hierarchies, such as the BBGKY
hierarchy, and then ignoring higher order moments to close the hierarchy of evolution equations.
Although apparently quite powerful a method, it has not been possible to give any meaningful
general estimates for the accuracy or for regions of applicability of such closure approximations.
The present work arose as part of a project aiming at a rigorous derivation of a Boltzmann
transportequationinthekineticscalinglimitoftheweaklynonlineardiscreteScho¨dingerequation
(DNLS). This system describes the evolution of a complex lattice field ψ (x), with x ∈ Zd and
t
t≥0, by requiring that is satisfies the Hamiltonian evolution equations
i∂ ψ (x)= α(x−y)ψ (y)+λ|ψ (x)|2ψ (x), (1.1)
t t t t t
yX∈Zd
where the function α determines the “hopping amplitudes” and λ > 0 is a coupling constant.
A kinetic scaling limit with a suitably chosen closure assumption predicts that the Green-Kubo
correlation function of the energy density of this system satisfies a linearized phonon Boltzmann
equation in the limit; the explicit form of the Boltzmann collision operator and discussion about
the approximations involved is given in Sections 5 and 6 in [1], and we refer to [2, 3] for more
details about the linearization procedure.
Themethodusedinthederivationofthetransportpropertiesinsuchweakcouplinglimitsare,
naturally,basedonperturbationexpansions. Advanceshaverecentlybeenmadeincontrollingthe
related oscillatory integrals (see for instance [4, 5, 6]), but for nonlinear evolution equations of
the present type a major obstacle has been the lack of useful a priori bounds for the correlation
functions. For instance, Schwarz inequality estimates of moments in the “remainder terms” of
finitely expanded moment hierarchies has been used for this purpose for time-stationary initial
datain[5],whichwasinspiredbytheboundsfromunitarityofthetime-evolutionofcertainlinear
evolutionequationsfirstemployedfortherandomSchr¨odingerequationin[4]andlaterextendedto
othersimilarmodelssuchasthe Andersonmodel[7]andaclassicalharmoniclattice withrandom
massperturbations[8]. However,asarguedin[1],using momentsinsteadofcumulantsto develop
the hierarchy could lead to loss of an important decay property which is valid for cumulants but
not for moments; we shall discuss this point further in Section 2.
In the present contribution we derive a generic result which allows to bound joint correlations
of two random fields in terms of estimates involving only the decay properties of each of the
fields separately. These estimates are immediately applicable for obtaining uniform in time a
priori bounds for time-correlation functions of time-stationary fields. In particular, they imply
that if the initial state of the field is distributed according to an equilibrium measure which is
ℓ -clustering, then all time-correlations are ℓ -summable. The precise assumptions are described
1 2
in Section 2 and the result in Theorem 2.1 there.
If both fields are Gaussian and translation invariant, more direct estimates involving discrete
Fourier-transform become available. We use this in Section 3 to give an explicit example which
shows that ℓ -clustering of the fields does not always extend to their joint correlations, hence
1
showingthattheincreaseofthepowerfromℓ -clusteringtoℓ -summabilityofthejointcorrelations
1 2
in the main theorem is not superfluous.
The result is a corollary of a bound which proves summability of cumulants of any observable
with finite variance with ℓ -clustering fields, for p = 1 and p = 2. The p = 2 case is more
p
involved than the p = 1 case, since the present bound requires taking the sum in a weighted
ℓ -space. The precise statements and all proofs are given in Section 4. The proof is based on
2
representation of cumulants using Wick polynomials. We rely on the results and notations of [1],
andfor convenienceofthe readerwe havesummarizedthe relevantitems inAppendix A.We also
present a few immediate applications of these bounds and discuss possible further applications in
Section 5.
2
Acknowledgements
We thank Antti Kupiainen, Sergio Simonella, and Herbert Spohn for useful discussions on the
topic. The work has been supported by the Academy of Finland via the Centre of Excellence
in Analysis and Dynamics Research (project 271983) and from an Academy Project (project
258302). The work and the related discussions have partially occurred in workshops supported
by the French Ministry of Education through the grant ANR (EDNHS) and also by the Erwin
Schr¨odinger Institute (ESI), Vienna, Austria.
2 Notations and mathematical setting
We consider here complex lattice fields ψ : Z → C where Z is any nonempty countable index
set. We focus on this particular setup since it is the one most directly relevant for physical
applications: common examples would be Z =Zd and Z =Zd×I, where I denotes a finite set of
possible “internaldegreesoffreedom” suchas spin. The setup canalsocovermoreabstractindex
sets, such as the sequence of coefficients in the Karhunen–Lo`eve decomposition of a stochastic
process [9, 10, 11], or distribution-valued random fields evaluated at suitably chosen sequence of
test-functions (details about the definition and properties of general random fields can be found
for instance in [12, 13] and in other sources discussing the Bochner–Minlos theorem).
We also assume that the field is closed under complex conjugation: to every x ∈ Z there is
some x ∈ Z for which ψ(x)∗ = ψ(x ). If needed, this can always be achieved by replacing the
∗ ∗
original index set Z by Z ×{−1,1} and defining a new field Ψ by setting Ψ(x,1) = ψ(x) and
Ψ(x,−1)=ψ(x)∗. Thisprocedurewasinfactusedin[1,5]tostudytheDNLSexamplementioned
above, resulting in the choice Z =Zd×{−1,1}.
A random lattice field on Z is then a collection of random variables ψ(x), x ∈ Z, on the
probability space (Ω,M,µ), where Ω denotes the sample space, M the σ-algebra of measurable
events, and µ the probability measure. We consider here two random fields ψ and φ which are
defined on the same probability space. We denote the expectation over the measure µ by E.
Then:thconnectedcorrelation function u ofthefieldψisamapu :Zn →Cwhichisdefined
n n
asthecumulantofthenrandomvariablesobtainedbyevaluatingthefieldattheargumentpoints;
explicitly,
u (x):=κ[ψ(x ),...,ψ(x )], x∈Zn. (2.1)
n 1 n
We employ here the notations and basic results for cumulants and the related Wick polynomials,
as derived in [1]: a summary of these is also included in Appendix A.
Inphysics,oneoftenencountersrandomfields definedonthe d-dimensionalcubic lattice, with
Z = Zd. One could then study the decay properties of such functions as |x| → ∞ by using the
standard ℓ -norms over (Zd)n. However, this is typically too restrictive for physical applications:
p
it would imply in particular that both the first and the second cumulant, i.e., the mean and the
variance,oftherandomvariableψ(x)decayas|x|→∞,andthusthefieldwouldbealmostsurely
“asymptoticallyzero”atinfinity. Instead,manystationarymeasuresarisingfromphysicalsystems
arespatially translation invariant: the expectationvalues remaininvariantif allof the fields ψ(x)
are replaced by ψ(x+x ) for any given x ∈Zd. Since this implies also translation invariance of
0 0
all correlation functions, they cannot decay at infinity then, unless the field is almost surely zero
everywhere.
To cover also such nondecaying stationary states, one uses instead of the direct ℓ -norms of
p
the function u , the so-calledℓ -clustering norms of the field ψ defined as follows: for 1≤p<∞
n p
and n∈N we set
+
1/p
kψk(n) := sup |κ[ψ(x ),ψ(x ),...,ψ(x )]|p , (2.2)
p 0 1 n−1
x0∈Z(cid:20)x∈XZn−1 (cid:21)
and define analogously kψk(1) := sup |E[ψ(x )]| (note that κ[X] = E[X] for any random
p x0∈Zd 0
variable X). We shall also use the correspondingp=∞ norms,which coincide with the standard
3
sup-norms of u , namely, kψk(n) = sup |κ[ψ(x ),ψ(x ),...,ψ(x )]| = ku k . Since the
n ∞ x∈Zn 1 2 n n ∞
norms concern Lp-spaces over a counting measure, they are decreasing in p, i.e., kψk(n) ≥kψk(n)
p p′
if p≤p′. (This follows from the bound |u (x)|≤kψk(n), valid for all x and p.)
n p
For a translation invariant measure with Z = Zd, we can translate x to the origin in the
0
definition (2.2), and, by a change of variables x =x +y , obtain the simpler expression
n 0 n
1/p
kψk(n) = |κ[ψ(0),ψ(y ),...,ψ(y )]|p . (2.3)
p 1 n−1
(cid:20)y∈(XZd)n−1 (cid:21)
Thesummationheregoesoverthedisplacements y oftheargumentx fromthereferenceposition
i i
x =0. The definition is tailored for random fields which become asymptotically independent for
0
farapartregionsofthelattice,i.e.,when|y |→∞above. Fortranslationnon-invariantmeasures,
i
finiteness ofthe norm(2.2)yields a uniformestimate for the speed ofasymptoticindependence of
thefield. Letususetheopportunitytostressthatitiscrucialtousethecumulants,notmoments,
above: similar moments of the field would not decay as the separation grows, even if the field
values would become independent (see [1] for more discussion about this point).
Wenowcallarandomfieldψℓ -clustering ifkψk(n) <∞foralln=1,2,.... Inparticular,this
p p
requires that all of the cumulants, which define the connected correlation functions u , need to
n
exist. FromtheiterativedefinitionofcumulantsmentionedintheAppendix,orfromtheinversion
formula expressing cumulants in terms of moments, it clearly suffices that E[|ψ(x)|n] <∞ for all
x ∈ Z. We also say that the field ψ is ℓ -clustering up to order m if kψk(n) < ∞ for all n ≤ m.
p p
For such a field, we use the following constants to measure its “magnitude”: we set
1 1/n
M (ψ;p):= max kψk(n) . (2.4)
N 1≤n≤N n! p
(cid:18) (cid:19)
Clearly, the definition yields an increasing sequence in N up to the same order in which the field
is ℓ -clustering. We use the constants M to control the increase of the clustering norms. It
p N
is conceivable that in special cases other choices beside (2.4) could be used with the estimates
belowtoarriveatsharperboundsthanthosestatedinthetheorems. However,theabovechoiceis
convenient for our purposes since it leads to simple combinatorial estimates, increasing typically
only factorially in the degree of the cumulant. It is possible to think of the numbers M as
N
measuring the range of values the field can attain. For instance, if Z = {0} and ψ(0) is a
random variable which almost surely belongs to the interval [−R,R] with R > 0, then M (ψ;p)
n
is independent of p (since there is only one point 0) and M = c R where c remains order one,
n n n
uniformly in n (see, for instance, Lemma C.1 in [8]).
After these preliminaries, we are ready to state the main result:
Theorem 2.1 Suppose ψ and φ are random lattice fields which are closed under complex con-
jugation and defined on the same probability space. Assume that φ is ℓ -clustering and ψ is
1
ℓ -clustering, both up to order 2N for some N ∈ N . Then their joint cumulants satisfy the
∞ +
following ℓ -estimate for any n,m∈N for which n,m≤N,
2 +
1/2
sup κ[ψ(x′),...,ψ(x′ ),φ(x ),...,φ(x )] 2 ≤(M γm)n+m(n+m)!, (2.5)
1 m 1 n m,n
x′∈Zm(cid:20)x∈Zn (cid:21)
X (cid:12) (cid:12)
(cid:12) (cid:12)
where M := max(M (ψ;∞),M (φ;1)) and γ = 2e ≈ 5.44. In particular, all of the above
m,n 2m 2n
sums are then finite.
Loosely speaking, one can say that an ℓ -clustering random field can have at worst ℓ -summable
1 2
joint correlations. We have stated the result in a form which assumes that the field ψ is ℓ -
∞
clustering. Asmentionedabove,theclusteringnormsaredecreasingintheindex: hence,theabove
resultalsoholdsifψ isℓ -clusteringforany1≤q <∞. Onecouldthenalsoreplacetheconstants
q
4
M using the corresponding ℓ -clustering norms, max(M (ψ;q),M (φ;1)). However, these
m,n q 2m 2n
constants are always larger than M and thus can only worsen the bound.
m,n
This result is a consequence of a more general covariance bound given in Theorem 4.1. There
we also give a version of the estimate for fields φ which are merely ℓ -clustering. The price to
2
pay for the relaxation of the norms is an appearance of a weight factor in the ℓ -summation,
2
see Theorem 4.3 for the precise statement. Before going into the details of the proofs, let us go
through a special case clarifying the assumptions and the result.
3 An example: translation invariant Gaussian lattice fields
Inthissection,weconsiderrealvaluedGaussianrandomfieldsψ andφonZ =Zandassumethat
both fields have a zero mean and are invariant under spatial translations. Their joint measure is
then determined by giving three functions F , F , G∈ℓ (Z,R) for which
1 2 2
hψ(x)ψ(y)i=F (x−y), hφ(x)φ(y)i =F (x−y), hψ(x)φ(y)i =G(x−y). (3.1)
1 2
The covariance operator needs to be positive semi-definite. By first using Parseval’s theorem and
then computing the eigenvalues of the remaining 2×2 -matrix, we find that this is guaranteedby
requiring that the Fourier-transformsof the above functions, all of which belong to L2(T), satisfy
almost everywhere
F (k)≥0, F (k)≥0, |G(k)|2 ≤F (k)F (k). (3.2)
1 2 1 2
ThesethreeconditionshencesufficefortheexistenceofauniqueGaussianmeasureondistributions
b b b b b
on Z satisfying (3.1); details about such constructions are given for instance in [13, 14].
The last condition restricts the magnitude of the correlations, and it implies that if each of
the above fields is ℓ -clustering, then their correlations are ℓ -summable (simply because then
2 2
G(k) ∈ L2(T), and thus its inverse Fourier transform gives a function G ∈ ℓ (Z)). Hence, one
2
might wonder if the main theorem could, in fact, be strengthened to show that ℓ -clustering of
1
tbhe fields implies ℓ -summability of the joint correlations. The following example shows that this
1
is not the case.
3.1 ℓ -clustering fields whose joint correlations are not ℓ -summable
1 1
Letusconsidertwoi.i.d.Gaussianfieldsψandφwhosecorrelationsaredeterminedbythefunction
1 π 1
G(x)= sin x , x6=0, G(0)= . (3.3)
πx 2 2
(cid:16) (cid:17)
For such i.i.d. fields F (x) = (x = 0) = F (x) which is equivalent to F (k) = 1 = F (k) for all
1 2 1 2
k ∈T. Now for all x∈Z, clea1rly
b b
1/4
G(x)= dkei2πxk, (3.4)
Z−1/4
and thus G(k) = (|k| < 1) ≤ 1 = F (k)F (k). Therefore, such G indeed defines a possible
1 4 1 2
correlation between the fields ψ and φq.
For sucbh Gaussian fields, all cumulanbts ofborder different from n = 2 are zero. We also have
sup |F (x−y)| = 1, and, as F = F , both fields are ℓ -clustering, with kψk(2) = 1 =
x∈Z y∈Z 1 2 1 1 1
kφk(2) and kψk(n) = 0 = kφk(n) for any other n. However, their joint correlations satisfy for any
1 P 1 1
x′ ∈Z
∞ ∞
1 1 π 1 2 1
|κ[ψ(x′),φ(x)]| = |G(y)|= +2 sin y = + =∞. (3.5)
2 πy 2 2 π 2n+1
xX∈Z Xy∈Z Xy=1 (cid:12) (cid:16) (cid:17)(cid:12) nX=0
(cid:12) (cid:12)
Thus the joint correlations are not ℓ -summable. (cid:12) (cid:12)
1
In contrast, sup |κ[ψ(x′),φ(x)]|2 <∞, since it is equal to |G(y)|2 and G∈ℓ2(Z).
x′ x y
P P
5
4 ℓ -summability of joint correlations of ℓ -clustering fields
2 p
Theorem 4.1 Consider a random lattice field φ on a countable set Z, defined on a probability
space (Ω,M,µ) and closed under complex conjugation. Suppose that φ is ℓ -clustering up to order
p
2N for some N ∈ N , and let M (φ;p) be defined as in (2.4). Suppose also X ∈ L2(µ), i.e., X
+ N
is a random variable with finite variance.
1. If p=1 and n≤N, we have a bound
1/2
κ[X,φ(x ),...,φ(x )] 2 ≤ Cov(X∗,X)M (φ;1)nen (2n)!. (4.1)
1 n 2n
(cid:20)x∈Zn (cid:21)
X (cid:12) (cid:12) p p
(cid:12) (cid:12)
2. If p=2 and n≤N, we have a bound
1/2
sup |Φ (x′,x)| κ[X,φ(x ),...,φ(x )] 2 ≤ Cov(X∗,X)M (φ;2)2ne2n(2n)!,
n 1 n 2n
x′∈Zn(cid:20)x∈Zn (cid:21)
X (cid:12) (cid:12) p
(cid:12) (cid:12) (4.2)
where Φ (x′,x):=E[:φ(x′)∗φ(x′)∗···φ(x′ )∗::φ(x )φ(x )···φ(x ):].
n 1 2 n 1 2 n
The key argument in the proof uses Wick polynomial representation of the above cumulants.
Namely,adirectconsequenceofthetruncatedmoments-to-cumulantsformulagiveninProposition
A.1 in the Appendix, is that
κ[X,φ(x ),...,φ(x )]=E[X:φ(x )φ(x )···φ(x ):]=E[:X::φ(x )φ(x )···φ(x ):]. (4.3)
1 n 1 2 n 1 2 n
ThePropositioncanbe appliedheresincenowE[|X| n |φ(x )|]<∞bythe Schwarzinequality
i=1 i
estimate E[|X| n |φ(x )|]2 ≤ E[|X|2]E[ n |φ(x )|2] where the first factor is finite since X ∈
L2(µ), and the seic=o1nd facitor is finite since φi=i1s assuimQed to be ℓ -clustering up to order 2n.
p
Q Q
Applying Schwarz inequality in (4.3) yields a bound
|κ[X,φ(x ),...,φ(x )]|2 ≤E[|:X:|2]E[|:φ(x )φ(x )···φ(x ):|2]=Cov(X∗,X)Φ (x,x). (4.4)
1 n 1 2 n n
Hence, the theorem is obviously true if Φ (x,x) decreases sufficiently rapidly with “increasing”
n
x. However, this is typically too restrictive: since Φ (x,x) =E[|:φ(x):|2]=Var(φ(x)), this would
1
requirethat the field φ becomes asymptoticallydeterministic. The proofbelow combines suitably
chosen test functions with the above Schwarz estimate and results in bounds which only require
summability of Φ (x′,x) in x for a fixed x′. Such summability is guaranteed by the ℓ -clustering
n p
of the field, andthe rest of the proofconsists of controllingthe combinatorialfactors which relate
these two concepts together, cf. Lemma 4.2.
Let us stress that the above result is typically not true if moments are used there instead of
cumulants. TheaboveSchwarzinequalityestimateswouldbestraightforwardformoments;infact,
such a Schwarz estimate was a key method in [5] to separate time-evolved fields from their time-
zero counterparts in products of these fields. However,the functions resulting from such Schwarz
estimates are of the type E[ n |φ(x )|2] and for these to be summable in x the field not only
i=1 i
hasto become asymptoticallydeterministic, but ithasto evenvanish. Cumulantsofℓ -clustering
p
Q
fields would, on the other hand, be summable, but there is no obvious way of generalizing the
Schwarz inequality bounds for cumulants. The missing ingredient is here provided by the Wick
polynomial representation (4.3).
Proof: There is a naturalHilbert space structure associatedwith correlationsofthe presenttype.
We begin with test-functions f : Zn → C which have a finite support, and define for them a
(semi-)norm by the formula
2
kfk2 :=E f(x):φ(x)Jn: = f(x′)∗f(x)Φ (x′,x),
φ,n n
(cid:20)(cid:12)xX∈Zn (cid:12) (cid:21) x′,Xx∈Zn
(cid:12) (cid:12)
Φn(x′,x):=E(cid:12) :φ∗(x′)Jn′::φ(x)J(cid:12)n: , (4.5)
h i
6
where Jn = {1,2,...,n} = Jn′, and thus we have φ(x)Jn := φ(x1)φ(x2)···φ(xn), φ∗(x′)Jn′ :=
φ(x′)∗φ(x′)∗···φ(x′ )∗. For the definition, we do not yet need any summability properties of
1 2 n
the field φ, it suffices that all the expectations in Φ (x′,x) are well-defined for all x′,x. By the
n
truncated moment-to-cumulants expansion of Wick polynomials, as given in Proposition A.1 in
the Appendix, we have here
Φ (x′,x)= (κ[φ∗(x′) ,φ(x) ] (A′ 6=∅,A6=∅)) , (4.6)
n A′ A 1 A′=S|Jn′,A=S|Jn
π∈PX(Jn′+Jn)SY∈π
where the notation S|J refers to the subsequence composed out of the indices belonging to J
n n
in the cluster S of the partition π of J′ + J .1 The additional restrictions A′,A 6= ∅ in the
n n
product arise from the fact that if either of them is violated, then the corresponding cluster S is
contained entirely in either J′ or J , and vice versa. The partitions containing such a cluster are
n n
preciselythosewhicharemissingfromthemomentstocumulantsformulabytheWickpolynomial
construction. Therefore, Φ is finite, as soon as all cumulants up to order 2n are finite. On the
n
other hand, this is already guaranteedby the assumed ℓ -clustering of the field φ. For notational
p
simplicity, let us drop the name of the field φ from the norm kfk .
φ,n
The norm can be associated with a scalar product using the polarization identity, and we
can then use it to define a Hilbert space H by completion and dividing out the functions with
n
zero norm, if the above formula gives only a semi-norm. The elements of H are thus functions
n
f : Zn → C with kfk< ∞ (or their equivalence classes in the semi-norm case when every f and
g with kf −gk = 0 needs to be identified). However, since we do not use these Hilbert spaces
directly, let us skip the details of the construction.
WebeginwithjointcorrelationsofthetypeG(x):=E[Y :φ(x)Jn:]whereY ∈L2(µ)isarandom
variable. Here G(x) is well defined due to the Schwarz inequality estimate E[|Y||:φ(x)Jn:|]2 ≤
E |Y|2 Φ (x,x). If f :Zn →C has a finite support, we define
n
(cid:2) (cid:3)
Λ[f]:= G(x)f(x)=E Y :φ(x)Jn:f(x) . (4.7)
" #
x∈Zn x∈Zn
X X
Applying the Schwarz inequality as above yields an upper bound
2
|Λ[f]|2 ≤E |Y|2 E f(x):φ(x)Jn: =E |Y|2 kfk2. (4.8)
n
(cid:2) (cid:3) (cid:20)(cid:12)(cid:12)xX∈Zn (cid:12)(cid:12) (cid:21) (cid:2) (cid:3)
Here, by the definition of the norm,(cid:12) we obtain an unw(cid:12)eighted ℓ -estimate by using Ho¨lder’s
2
inequality as follows
kfk2 ≤ |f(x′)||f(x)||Φ (x′,x)|≤ |f(x′)|2|Φ (x′,x)| |f(x)|2|Φ (x′,x)|
n n n n
x′,x∈Zn sx′,x∈Zn sx′,x∈Zn
X X X
≤ |f(x)|2 sup |Φ (x′,x)|, (4.9)
n
x∈Zn x′∈Znx∈Zn
X X
where we have used the obvious symmetry property Φ (x′,x)∗ = Φ (x,x′). As shown be-
n n
low, in Lemma 4.2, ℓ -clustering of the field φ in fact implies that there is c < ∞ such that
1 n
sup |Φ (x′,x)|≤c (theexplicitdependenceofc ontheclusteringnormsisgivenin
x′∈Zn x∈Zn n n n
the Lemma). Hence, we can conclude that |Λ[f]|≤ c E[|Y|2]kfk . Thus, thanks to the Riesz
P n ℓ2
representation theorem, Λ can be extended into a unique functional belonging to the dual of the
p
Hilbert space ℓ (Zn), and hence there is a vector Ψ∈ℓ (Zn) such that Λ[f]= Ψ(x)∗f(x)
2 2 x∈Zn
1IfonehasdistinctlabelsinJn′ andJn,achievable alwaysbyrelabellingofoneofthesetsP, onecansafelytake
hereJn′ +Jn=Jn′ ∪Jn,P(Jn′ +Jn)equaltotheordinarypartitionsofthesetJn′ ∪Jn,andalsoS|Jn=S∩JN.
However,suchrelabellingsleadtounnecessarilyclumsynotationsinthepresentcase,andwehaveoptedtousethe
abovenotations from[1].
7
and kΨk ≤ c E[|Y|2]. Then necessarily G(x) =Ψ(x)∗ for all x, and thus G∈ℓ (Zn) as well,
ℓ2 n 2
with a bound
p
|G(x)|2 ≤ c E[|Y|2]. (4.10)
n
sx∈Zn
X p
If Y = :X:, we have G(x) = E[:X::φ(x)Jn:] = κ[X,φ(x1),...,φ(xn)] as explained in (4.3), and
also E |Y|2 = E[:X∗::X:] = κ[X∗,X]= Cov(X∗,X). Hence, (4.10) implies the bound stated in
the first item.
(cid:2) (cid:3)
Fortheweightedresult,weapply(4.8)forspeciallyconstructedtestfunctionsf. LetF beany
finite subset of Zn and choose an arbitrary point y ∈ Zn. Then f(x) = (x∈F)|Φ (y,x)|G(x)∗
n
1
has finite support and
Λ[f]= |G(x)|2|Φ (y,x)|≤ E[|Y|2]kfk <∞. (4.11)
n n
x∈F
X p
On the other hand, we obtain the following estimate for kfk
n
kfk2 = G(x)∗G(x′)|Φ (y,x′)||Φ (y,x)|Φ (x′,x)
n n n n
x′,x∈F
X
≤ |G(x)|2|Φ (y,x)||Φ (y,x′)||Φ (x′,x)| |G(x′)|2|Φ (y,x′)||Φ (y,x)||Φ (x′,x)|
n n n n n n
sx′,x∈F sx′,x∈F
X X
2
≤ |G(x)|2|Φ (y,x)| sup |Φ (x′,x)|2 , (4.12)
n n
x∈F x′∈Znsx∈F 
X X
 
where we have used Φ (x′,x)∗ =Φ (x,x′) and the Schwarz inequality in the last estimate.
n n
As shown below, in Lemma 4.2, ℓ -clustering of the field φ implies |Φ (x′,x)|2 ≤
2 x∈Zn n
c′ < ∞ where the explicit dependence of c′ on the clustering norms is given in the Lemma.
Tnherefore, Λ[f] ≤ c′ E[|Y|2] Λ[f]. Since n0 ≤ Λ[f] < ∞ for any subpsetPF, we can conclude
n
that the estimate Λ[f] ≤ c′ E[|Y|2] also holds. Thus by using subsets F = F , which are
p np R
constructed by choosing the first R elements from a fixed enumeration of Zn, and then taking
p p
R→∞, we obtain that
|G(x)|2|Φ (y,x)|≤c′ E[|Y|2]<∞, (4.13)
n n
sx∈Zn
X p
for all y ∈Zn. This implies the statement in the second item. (cid:3)
Lemma 4.2 Suppose that the field φ is closed under complex conjugation and ℓ -clustering up to
p
order 2n, for some p∈[1,∞] and n≥1. Then, for any x′ ∈Zn,
kΦ (x′,·)k ≤ kφk(|S|) ≤M (φ;p)2ne2n(2n)!, (4.14)
n ℓp p 2n
π∈PX(J2n)SY∈π
where J ={1,2,...,2n} and P(J ) denotes the collection of its partitions.
2n 2n
Proof: Let us consider some fixed x′ ∈ Zn. We apply the Minkowski inequality to (4.6), as a
function of x, and conclude that
kΦ (x′,·)k ≤ kF(x′,·;π)k (4.15)
n ℓp ℓp
π∈PX(Jn′+Jn)
8
where
F(x′,x;π):= (|κ[φ∗(x′) ,φ(x) ]| (A′ 6=∅,A6=∅)) . (4.16)
A′ A 1 A′=S|Jn′,A=S|Jn
S∈π
Y
Let us first consider the case p < ∞. For any π ∈ P(J′ +J ) yielding a nonzero F, the
n n
restrictions of its clusters with J , A = S|J in the above formula, form a partition of J . Let
n n n
us denote this partition by π . Hence, we can use this partition to reorder the summation over
2
x∈Zn into iterative summation over x ∈ZA for A∈π . Applied to (4.16) this yields
A 2
|F(x′,x;π)|p = (A′ 6=∅,A6=∅) |κ[φ∗(x′) ,φ(x) ]|p . (4.17)
A′ A
1
xX∈Zn SY∈π(cid:18) xAX∈ZA (cid:19)A′=S|Jn′,A=S|Jn
Since the field φ is closed under complex conjugation, for each S ∈ π the sum over x is equal
A
p
to κ[φ(y′) ,φ(x) ] where y′ = ((x′) ) . As A′ 6= ∅, we may choose an element
x∈ZA J|A′| A i ∗ i∈A′
j ∈A′. We(cid:12)then denote x =((cid:12)x′) . and estimate the sum with an ℓ -clustering norm as follows
P (cid:12) 0 (cid:12) j ∗ p
(cid:12) (cid:12)
p p
|κ[φ∗(x′) ,φ(x) ]|p ≤ κ[φ(x ),φ(y) ,φ(x) ] ≤ kφk(|A|+|A′|) .
A′ A 0 J|A′|−1 A p
xX∈ZA y∈ZX|A′|−1xX∈ZA(cid:12) (cid:12) (cid:16) (cid:17)
(cid:12) (cid:12)
(cid:12) (cid:12) (4.18)
Since |A′|+|A|=|S|, we can conclude that, if p<∞,
kF(x′,·;π)k ≤ kφk(|S|). (4.19)
ℓp p
S∈π
Y
The corresponding estimate for p= ∞ is a straightforward consequence of |κ[φ∗(x′) ,φ(x) ]|≤
A′ A
kφk(|A′|+|A|) which was discussed in Section 2 after Eq. (2.2).
∞
Therefore, we can now conclude that the first inequality in (4.14) holds. By the definition in
(2.4), we can then apply an upper bound
kφk(m) ≤m!M (φ;p)m ≤m!M (φ;p)m,
p m 2n
for any m≤2n. If π ∈P(J′ +J ), we have |S|≤2n for any S ∈π, and thus
n n
kφk(p|S|) ≤M2n(φ;p)PS∈π|S| |S|!=M2n(φ;p)2n |S|!. (4.20)
S∈π S∈π S∈π
Y Y Y
A combinatorial estimate shows that
|S|!≤(2n)!e2n (4.21)
π∈PX(J2n)SY∈π
(a proofofthe inequalityis availablefor instancein the proofofLemma7.3 in[5]). Therefore,we
have proven also the second inequality in (4.14), concluding the proof of the Lemma. (cid:3)
The following theorem contains the already stated Theorem 2.1 in the item 1. The remarks
afterthe Theorematthe endofSection2holdalsointhis case. Inparticular,itis obviouslyvalid
for any ℓ -clustering field ψ, as long as 1≤q ≤∞.
q
Theorem 4.3 Consider tworandom latticefields φ(x)andψ(x), x∈Z for acountableZ, defined
on the same probability space (Ω,M,µ) and each closed under complex conjugation. Suppose that
φ is ℓ -clustering and ψ is ℓ -clustering up to order 2N for some N ∈ N . Let M be defined
p ∞ + N
as in (2.4). Then their joint cumulants satisfy the following ℓ -estimates for any n,m ∈ N for
2 +
which n,m≤N:
9
1. If p=1, we have a bound
1/2
sup κ[ψ(x′),...,ψ(x′ ),φ(x ),...,φ(x )] 2 ≤(M γm)n+m(n+m)! (4.22)
1 m 1 n m,n
x′∈Zm(cid:20)x∈Zn (cid:21)
X (cid:12) (cid:12)
(cid:12) (cid:12)
where M :=max(M (ψ;∞),M (φ;1)) and γ =2e.
m,n 2m 2n
2. If p=2, we have a bound
1/2
sup |Φ (y,x)| κ[ψ(x′),...,ψ(x′ ),φ(x ),...,φ(x )] 2
n 1 m 1 n
x′∈Zm,y∈Zn(cid:20)x∈Zn (cid:21)
X (cid:12) (cid:12)
≤(M γm)2(n+m)((n(cid:12) +m)!)2 (cid:12) (4.23)
m,n
where Φ (y,x) := E[:φ(y )∗φ(y )∗···φ(y )∗::φ(x )φ(x )···φ(x ):], and we set M :=
n 1 2 n 1 2 n m,n
max(M (ψ;∞),M (φ;2)) and γ =2e.
2m 2n
Proof: We will proceed by induction over m. Let us recall the above definition of Φ and define
n
analogously Ψ (y′,x′):=E[:ψ(y′)∗ψ(y′)∗···ψ(y′ )∗::ψ(x′)ψ(x′)···ψ(x′ ):]. In particular, then
m 1 2 m 1 2 m
we can apply Theorem 4.1 with X = ψ(x′). By Lemma 4.2, then E[|:X:|2] = Ψ (x′,x′) ≤
1 1 1 1
M (ψ;∞)2e22!, and thus, say for γ = 2e, both items 1 and 2 can be seen to hold for m = 1 and
2
any n≤N thanks to Theorem 4.1 and the estimate (2n)!≤((2n)!!)2 =22n(n!)2.
As an induction hypothesis, we consider some 1 < m ≤ N and assume that the thesis holds
for values up to m−1 with any n ≤ N. We also give the details only for the first ℓ -clustering
1
case, i.e., with p=1.
Let us decompose the cumulant using Proposition A.1. Namely, consider
P(x′,x):=E[:ψ(x′)···ψ(x′ ): :φ(x )···φ(x ):],
1 m 1 n
for which
κ[ψ(x′),...,ψ(x′ ),φ(x ),...,φ(x )]=P(x′,x)−Q(x′,x),
1 m 1 n
with
Q(x′,x):= (|π|>1) (κ[ψ(x′) ,φ(x) ] (A′ 6=∅,A6=∅)) .
1 A′ A 1 A′=S|Jm′ ,A=S|Jn
π∈PX(Jm′ +Jn) SY∈π
Then we can conclude from the Minkowski inequality that
1/2
κ[ψ(x′),...,ψ(x′ ),φ(x ),...,φ(x )] 2 ≤kP(x′,·)k +kQ(x′,·)k . (4.24)
1 m 1 n ℓ2 ℓ2
(cid:20)x∈Zn (cid:21)
X (cid:12) (cid:12)
(cid:12) (cid:12)
WenowestimatekP(x′,·)k usingitem1inTheorem4.1withX =:ψ(x′)···ψ(x′ ):. Clearly,
ℓ2 1 m
then
P(x,x′)=E[:X::φ(x )···φ(x ):]=κ[X,φ(x ),...,φ(x )], (4.25)
1 n 1 n
and, by applying Theorem 4.1 and Lemma 4.2
kP(x′,·)k ≤ Ψ (x′,x′)M (φ;1)nen (2n)!≤M (ψ;∞)mM (φ;1)nen+m (2m)!(2n)!.
ℓ2 m 2n 2m 2n
(4.26)
p p p
Note that (2n)!≤22n(n!)2 and n!m!≤(n+m)!, so, recalling the definition of M , we have
m,n
kP(x′,·)k ≤(M 2e)n+m(n+m)!. (4.27)
ℓ2 m,n
10