Infinite volume limit for correlation functions in the dipole gas

arXiv:1305.1975v2 [math-ph] 12 Jul 2013Infinite Volume Limit for Correlation functions in the Dipole Gas Department of Mathematics SUNY at Buffalo Buffalo, NY 14260 January 10, 2014 Abst

arXiv:1305.1975v2 [math-ph] 12 Jul 2013

Infinite Volume Limit for Correlation functions

in the Dipole Gas

Department of Mathematics SUNY at Buffalo Buffalo, NY 14260 January 10, 2014

Abstract

We study a classical lattice dipole gas with low activity in dimension d ≥ 3 We investigatelong distance properties by a renormalization group analysis We prove that various correlationfunctions have an infinite volume limit We also get estimates on the decay of correlation functions

In this paper we continue to study the classical dipole gas on a unit lattice Zd with d≥ 3 Each dipole

is described by its position coordinate x∈ Zd and a unit polarization vector (moment) p∈ Sd −1.Let{e1, , ed} be the standard basic for Zd For ϕ : Zd→ R and µ ∈ {1, , d} we define ∂µϕ as

∂µϕ(x) = ϕ(x+eµ)−ϕ(x) Let e−µ=−eµwith µ∈ {1, , d} Then the definition of ∂µϕ can be used

to define the forward or backward lattice derivative along the unit vector eµ with µ ∈ {±1, , ±d}

We have that ∂µ and ∂−µare adjoint to each other and−∆ = 1/2Pd

X

1≤k,j≤n

(pk· ∂)(pj· ∂)C(xk, xj) (3)Let ΛN be a box in Rd

∗ e-mail addresses: tuanle@buffalo.edu or leminhtuan912@gmail.com

1 We distinguish forward and backward derivatives to facilitate a symmetric decomposition of V (Λ N ) (defined in (9)) into blocks.

where L≥ 2 + 1 is a very large, odd integer For ΛN ∩ Z , the classical statistical mechanics of agas of such dipoles with inverse temperature (for convenience) β = 1 and activity (fugacity) z > 0 isgiven by the grand canonical partition function

x ∈Λ N ∩Z d

Z

S d−1

dp cos(p· ∂φ(x)) (6)with

• dp : the standard normalized rotation invariant measure on Sd −1

• The fields φ(x) : a family of Gaussian random variables (on some abstract measure space)indexed by x∈ Zd with mean zero and covariance C(x, y) which is a positive definite function

4 Other general form which will be discussed at the end of this paper The general form can

be applied for truncated correlations of density of the dipoles which also has been studied byBrydges and Keller [2] We think that this general form has more applications

Here xk ∈ Zd are different points; µk ∈ {±1, , ±d} and tk small and complex; m≥ 2 For the set{x1, , xm} ⊂ Zd, let diam(x1, , xm) = max1 ≤i,j≤mdist(xi, xj) where dist(xi, xj) is the distancebetween xi and xj on lattice Zd

To get rid of the boundary and study the long distance properties of the system, we would like totake the thermodynamic limit for these quantities, i.e the limit as N → ∞ which is so called infinite

volume limit Actually ZN is not expected to have a limit as N → ∞ In [5], Dimock has established

an infinite volume limit for the pressure defined by

pN =|ΛN|−1log0ZN (8)(with|z| sufficiently small) Such infinite volume limits have also been obtained by Frohlich and Park[10] and by Frohlich and Spencer [11] They used a method of correlation inequalities

In this paper, we continue the study of the long distance properties of the dipole gas model Forlong distance (i.e., when |x − y| large), the potential ∂µ∂νC(x− y) behaves like O(|x − y|−d), thatmeans it is not integrable and we could not use the theory of the Mayer expansion to establish suchresults To overcome this problem, we use the method of the renormalization group

We follow particularly a Renormalization Group approach recently developed by Brydges andSlade [1] and Dimock [5] We generalize Dimock’s framework with an external field and obtain someestimates on the correlation functions as in Dimock and Hurd [7], and Brydges and Keller [2] Themain result is the existence of the infinite volume limit for correlations functions, which is new Earlierwork using RG approach to the dipole gas can be found in Gawedski and Kupiainen [12], Brydges andYau [4]

Besides the dipole gas papers mentioned above, we would like to cite some other papers on theCoulomb gas in d = 2 which has a dipole phase There are the works of Dimock and Hurd [8], Falco[9] and Zhao [17]

For our RG approach we follow the analysis of Brydges’ lecture [1] Instead of (7), we use a differentfinite volume approximation First, we add an extra term (1− ε)V (ΛN, φ) where 0 < ε is closed to 1and

V (ΛN, φ) = 1

4X

By replacing the covariance C by ε−1C, this extra term will be partially compensated Hence instead

of (7) we will consider a new finite volume generating function

ZN′′ =

Zexp− (1 − ε)V (ΛN, φ)dµε −1 C(φ) (12)

Therefore the choice of ε is a choice of how much (∂φ) one is putting in the interaction and how much

in the measure

Similarly to the Theorem 1 in [5], our main theorems are:

Theorem 1 For |z| and maxk|tk| sufficiently small there is an ε = ε(z) close to 1 so that

fpN =|ΛN|−1log fZN
has a limit as N → ∞.2

Using f (φ) =Pm

k=1tk∂µ kφ(xk), we achieve some estimate for the correlation functions:

Theorem 2 For any small ǫ > 0, with L, A sufficiently large (depending on ǫ), η = min{d/2, 2}, wehave:

And we also can obtain the existence of infinite volume limit for correlation functions

Theorem 3 With L, A sufficiently large, the infinite volume limit of truncated correlation functionlimN →∞ mk=1∂µkφ(xk)t

existsWhen d = 3 or 4, the result in Theorem 2 looks like the result in [7], but here it is obtained with thenew method

2 points as Theorem 1.1.2 in [2] Then we apply theorem 13 to establish the infinite volume limit fortruncated correlation functions of density of dipoles (Corollary 2)

For the proof of Theorem 1, we will show that, with a suitable choice of ε = ε(z), the densityexp zW − (1 − ε)V
likely goes to zero under the renormalization group flow and leaves a measurelike µε(z) −1 Cto describe the long distance behavior of the system Accordingly ε(z) can be interpreted

fZN(z, σ) =fZ′N(z, σ)/ZN′′(σ)

(15)

Then we need to show that with|z| sufficiently small there is a (smooth) σ = σ(z) near zero such that,

|ΛN|−1logfZN(z, σ(z)) =|ΛN|−1logfZ′N(z, σ(z))− |ΛN|−1log ZN′′(σ(z)) (16)has a limit when N → ∞ And theorem 1 is proved just by putting ε(z) = (1 + σ(z))−1 back.Dimock has proved that, for small real σ with|σ| < 1, we have |ΛN|−1log Z′′

N(σ) converges as N → ∞(Theorem 2, [5]) Hence we only need to investigate the first term in (16)

The paper is organized as follows:

2 In Theorem 1, f (φ) can be 0, P m

k=1 t k ∂ µkφ(x k ), or P m

k=1 t k exp (i∂ µkφ(x k )).

• In section 2 we give some general definitions on the lattice and its properties We also givedefinitions about the norms we use together with their crucial properties and estimates Then

we define the basic Renormalization Group transformation as in ([5])

• In section 3 we accomplish the detailed analysis of the Renormalization Group transformation

to isolate the leading terms Then we simplify them for the next scale

• In section 8 we study the RG flow and find the stable manifold σ = σ(z)

• In section 5 we assemble the results and prove the infinite volume limit for |ΛN|−1logfZ′N exists

• Finally in section 6, by combining all the other estimates, we obtain some estimates for correlationfunctions and establish the infinite volume limit of correlation functions

• There are constants cαindependent of L such that

|∂αΓj(x)| ≤ cαL−(j−1)(d−2+|α|) (19)where ∂α=Qd

2.2 Renormalization Group Transformation

The generating function (15) can be rewritten as

fZ′N(z, σ) =

Z

fZN0(φ)dµC 0(φ) (23)with

fZ′N(z, σ) =

Z

fZNj (φ)dµC j(φ) (27)here the densityfZN

j (φ) is defined by

fZNj+1(φ) = (µΓ j+1∗fZNj )(φ) =

Z

fZNj (φ + ζ)dµΓ j+1(ζ) (28)Our job is to investigate the growth of these densities when j go to∞

We will rewrite each density fZN

j (φ) in a form which presents its locality properties known as apolymer representation The localization becomes coarser when j gets bigger First we will give somebasic definitions on the lattice Zd

2.3.1 Basic definitions on the lattice Zd

For j = 0, 1, 2, we partition Zd into j-blocks B These blocks have side Lj and are translates ofthe center j-blocks

Bj0={x ∈ Zd:|x| < 1/2(Lj

by points in the lattice LjZd The set of all j-blocks in Λ = ΛN is denoted Bj(ΛN),Bj(Λ) or justBj

A union of j-blocks X is called a j-polymer Note that Λ is also a j-polymer for 0≤ j ≤ N The set of

3 Dimock has discussed these in Appendix A, [5].

all j-polymers in Λ = ΛN is denotedPj(Λ) or justPj The set of all connected j-polymers is denoted

byPj,c Let X∈ Pj, the closure ¯X is the smallest Y ∈ Pj+1such that X⊂ Y

For a j-polymer X, let |X|j be the number of j-blocks in X We call j-polymer X a small set

if it is connected and contains no more than 2d j-blocks The set of all small set j-polymers in Λ isdenoted bySj(Λ) or just Sj A j-block B has a small set neighborhood B∗ =∪{Y ∈ Sj : Y ⊃ B}.Note: If B1, B2 are j-blocks and B2 ∈ B∗

1 then, using above definition, we also have that B1 ∈ B∗

2.Similarly a j-polymer X has a small set neighborhood X∗

For l≥ 1 and integer d, we define some constants n1(d), n2(d), n3(d, l) which are bounded and, forevery j≥ 0, we have:

Using the same approach as in [5], we rewrite the densityfZN

j (φ) for φ : Zd → R in the the generalform

fZ = (fI◦fK)(Λ)≡ X

X ∈P j (Λ)

Here fI(Y ) is a background functional which is explicitly known and carries the main contribution

to the density The fK(X) is so called a polymer activity It represents small corrections to thebackground

In section 5 we will show that the initial density fI0 has the factor property We want to keepthis factor property at all scales Then we can use the analysis of Brydges’ lecture [1] Therefore weassumefI(Y ) always is in the form of

and that fK(X, φ) only depends on φ in X∗

As in [5], the background functionalfI(B) has a special form: fI(fE, σ, B) = exp(−V (fE, σ, B))where 4

V (fE, σ, B, φ) =fE(B) + 1

4X

for some functionsfE, σµν :Bj → R Indeed we usually can take σµν(B) = σδµν for some constant σ.Then V (fE, σ, B, φ) becomes

V (fE, σ, B, φ) =fE(B) +σ

4X

x ∈B

X

µ

(∂µφ(x))2 ≡fE(B) + σV (B) (36)Also in our model, when f = 0, we will have

0K(X, φ) = 0K(X,−φ) 0K(X, φ) = 0K(X, φ + c) (37)The later holds for any constant c which means that 0K(X, φ) only depends on derivatives ∂φ

In this paper we use exactly the same norms and notations as in Dimock [5] Now we consider potential

V (s, B, φ) of the form

V (s, B, φ) = 1

4X

x ∈B

X

µν

sµν(x)∂µφ(x)∂νφ(x) (38)here the norms of functions sµν(x) are defined by

If sµν(x) = σδµν then V (s, B) = σV (B) as defined in (36) and the normkskj= 2d σ

The following lemmas are some results from Section 3 in [5]:

Let c be a constant such that the function σ→ exp(−σV (B)) is analytic in |σ| ≤ ch−2and satisfies

k exp(−σV (B))ks,j≤ 2 on that domain

To start the RG transformation, we also need some estimate on the initial interaction When j = 0,

B ∈ B0is just a single site x∈ Zd, so we consider

We also have that W (u, B) is strongly continuously differentiable in u

2 ezW (u,B) is complex analytic in z and satisfies, for|z| sufficiently small (depending on d, h, u),

3 Analysis of the RG Transformation

Now we use the Brydges-Slade RG analysis and follow the framework of Dimock [5], but with anexternal field f

fZ′(φ′) = (fI′◦fK′)(Λ, φ′) (46)where the polymers are now on scale (j + 1) Furthermore, supposed that we have chosenfI′, we willfindfK′ so the identity holds As explained before, our choice offI′ is to have the form

fI′(B′, φ′) = Y

B ∈B j ,B ⊂B ′

˜

fI(B, φ′) B′∈ Bj+1 (47)Now we define

δfI(B, φ′, ζ) =fI(B, φ′+ ζ)− ˜fI(B, φ′)

fK◦ δfI≡ ˜fK(X, φ′, ζ) = X

Y ⊂X

fK(Y, φ′+ ζ)δfIX−Y(φ′, ζ) (48)For connected X we write ˜fK(X, φ′, ζ) in the form5

with Xχ=∪iXi And ˇfK (X, φ′) is ˇfK(X, φ′, ζ) integrated over ζ as (78) in [5].

At this pointfK′is considered as a function offI, ˜fI,fJ,fK It vanishes at the point (fI, ˜fI,fJ,fK) =(1, 1, 0, 0) since χ =∅ and X = ∅ iff U = ∅ We study its behavior in a neighborhood of this point

We have the norm onfK as (75) in [5] and we define

Using the same argument as Theorem 3 in [5], we have the following result

Theorem 4 Let A be sufficiently large

1 For R > 0 there is a r > 0 such that the following holds for all j IfkfI−1ks,j< r,k ˜fI−1k′

where

fK#(X, φ) =

Z

fK(X, φ + ζ)dµΓ j+1(ζ) (57)and0J actually isfJ at f = 0

3.2.1 ChoosingJ

For a smooth function g(φ) on φ∈ RΛ let T2g denote a second order Taylor expansion:

(T2g)(φ) =g(0) + g1(0; φ) +1

2g2(0; φ, φ)(T0g)(φ) =g(0)

and choosefJ(B, B) so that (55) is satisfied Otherwise, we letfJ(B, X) = 0.

B ⊂B ′V (B) = V (B′), we have

fI′(B′) =fI(fE′, σ′, B′) = exp(−V (fE′, σ′, B)) (62)with

And the theorem 4 becomes:

Theorem 5 Let A be sufficiently large

1 For R > 0 there is a r > 0 such that the following holds for all j If k ˜fEkj ,|˜σ|, kfEkj, |σ|,max{kfKkj,k0Kkj} < r then max{kfK′kj+1,k0K′kj+1} < R Furthermore fK′ is a smoothfunction of ˜fE, ˜σ,fE, σ,fK,0K on this domain with derivatives bounded uniformly in j Theanalyticity of fK′ in t1, , tmstill holds when we go from j-scale to (j + 1)-scale

2 The linearization offK′ at the origin has the form

L1(fK) +L2(fK) +L3(fE, σ, ˜fE, ˜σ,fK,0K) (65)where



V ( ˜E, ˜σ, B)− V#(E, σ, B)

+ X

¯ B=U

Proof The new map actually is the composition of the mapfK′ =fK′(fI, ˜fI,fJ,fK) of theorem 4with the mapsfI =fI(fE, σ), ˜fI =fI( ˜fE, ˜σ),fJ =fJ(fK,0K) Thus it suffices to establish uniformbounds and smoothness for the latter.

In the case f = 0, we’ve already have the proof in Theorem 4, Dimock [5] So we only consider thecase f6= 0

ForfI =fI(fE, σ), ˜fI =fI( ˜fE, ˜σ) the proof is the same as the proof for I, I′ in (Theorem 4, [5])

3.2.2 EstimatingL1,L2 - the first two linearization parts

Next we make some estimates on the linearization’s parts First we estimateL1 which is the tion on the large j-polymers

lineariza-Lemma 3 Let A be sufficiently large depending on L Then the operatorL1 is a contraction with anorm which goes to zero as A→ ∞

Proof We use the same proof as in (Dimock, [5], Lemma 5), but with updated notations We estimate

Now we estimate and find an explicit upper bound forL2

Lemma 4 Let L be sufficiently large Then the operatorL2 is a contraction with a norm which goes

to zero as L→ ∞

Proof For f = 0, we have the Lemma 6, in [5]

For f 6= 0, we can write

Using property (64) in [5], we have



Gj+1( ¯X, φ′, ζ) (79)and also using (79) in [5], we obtain:

If we also average over z∈ B (86) becomes

and we can sayL′

3(δfK)(U ) = 0 Next we define

=− X

B=U

fβ(B) +1

4X

First we find some explicit upper bounds for α andfβ

Lemma 5 (Estimatesfβ and α)

Remark The normkαkj agrees with the normkskj in (39) if sµν(x) = αµν(B) for x∈ B

Proof By (70) and (79) in [5], with A very large, we have:

Now we give some estimate forL′

3

Lemma 6 Let L be sufficiently large Then the operator L′

3 is a contraction with arbitrarily smallnorm

k0HX(U, φ)kj+1≤ 18d222d(L−d−2)kK2#(X, 0)kj(1 +kφk2Φ j+1 (U ∗ )) (102)But for φ = φ′+ ζ

(1 +kφk2

Φ j+1 (U ∗ ))≤ Gs,j+1(U, φ, 0)≤ G2

s,j+1(U, φ′, ζ)≤ Gj+1(U, φ′, ζ) (103)Also using (94) we can get:

kHX(U )kj+1≤ 72d222d(L−d−2)A−1kKkj (104)which yields to

kL′3K(U )kj+1≤ n1(d) X

¯ B=U

3.4 Identifying invariant parts and estimating the others

N0w we investigate the 1st term of (92) We notice that αµν(B) = αµν(fK, B) = αµν(0K, B) isindependent from f (φ) and 0E(B),0K(X, φ) actually is the same as E(B), K(X, φ) in lemma 9 [5].Therefore we have the same result as lemma 9 [5]

Lemma 7 (Lemma 9, Dimock [5])

Suppose 0E(B),0K(X, φ) are invariant under lattice symmetries away from the boundary of ΛN

and ˜0E(B) is invariant for B∗ away from the boundary Then

1 0E′(B′),0K′(U, φ) are invariant for B′, U away from the boundary

2 If B∗is away from the boundary then0β(B), αµν(B) are independent of B and αµν(B) = ˆαµν(B)defined for all B by

ˆ

αµν(B) = α

2 (δµν− δµ, −ν) (107)where α is a constant

For all B∈ Bj we define

α′µν(B) = α δµν (108)and write, for any U∈ Bj+1

(111)

Remark BecauseL4(fK) =L4(0K) and ∆(fK) = ∆(0K) are independent from f , we will have thesame results as Lemma 10 and Lemma 11 in (Dimock, [5]) Moreover, by using Lemma 5 above, wecan obtain some explicit upper bounds forL4(0K) and ∆(0K)

Lemma 8 Let L be sufficiently large Then the operatorL4 is a contraction with

kL4(0K)kj+1≤ 4(2d)3n2(d)L−(j+1)k0Kkj (112)Lemma 9 Let L be sufficiently large Then the operator ∆ is a contraction with

k∆(0K)k ≤ 4(2d)52dn2(d)L−1k0Kkj (113)

And we also can obtain the existence of infinite volume limit for correlation functions

Theorem With L, A sufficiently large, the infinite volume limit of truncated correlation functionlimN... 4, the result in Theorem looks like the result in [7], but here it is obtained with thenew method

2 points as Theorem 1.1.2 in [2] Then we apply theorem 13 to establish the infinite volume. .. estimates, we obtain some estimates for correlationfunctions and establish the infinite volume limit of correlation functions

• There are constants cαindependent of L such