arXiv:1305.1975v2 [math-ph] 12 Jul 2013 Infinite Volume Limit for Correlation functions in the Dipole Gas Tuan Minh Le ∗ 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 ≥ We investigate long distance properties by a renormalization group analysis We prove that various correlation functions have an infinite volume limit We also get estimates on the decay of correlation functions Introduction 1.1 Overview In this paper we continue to study the classical dipole gas on a unit lattice Zd with d ≥ 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 ∂µ ϕ as ∂µ ϕ(x) = ϕ(x+eµ )−ϕ(x) Let e−µ = −eµ with µ ∈ {1, , d} Then the definition of ∂µ ϕ can be used to define the forward or backward lattice derivative along the unit vector eµ with µ ∈ {±1, , ±d} d d We have that ∂µ and ∂−µ are adjoint to each other and −∆ = 1/2 ±µ=1 ∂µ∗ ∂µ = 1/2 ±µ=1 ∂−µ ∂µ As in [5], the potential energy between unit dipoles (x, p1 ) and (y, p2 ) is (p1 · ∂)(p2 · ∂)C(x − y) (1) where x, y ∈ Zd are positions, p1 , p2 ∈ Sd−1 are moments, ∂ = (∂1 , , ∂d ) and C(x−y) is the Coulomb potential on the unit lattice Zd , which is the kernel of the inverse Laplacian eip·(x−y) C(x, y) = (−∆)−1 (x, y) = (2π)−d [−π,π]d d µ=1 (1 − cos pµ ) dp (2) And the potential energy of n dipoles, including self energy, has the form 1≤k,j≤n (pk · ∂)(pj · ∂)C(xk , xj ) Let ΛN be a box in Rd ΛN = −LN LN , 2 ∗ e-mail (3) d (4) addresses: tuanle@buffalo.edu or leminhtuan912@gmail.com distinguish forward and backward derivatives to facilitate a symmetric decomposition of V (ΛN ) (defined in (9)) into blocks We where L ≥ 2d+3 + is a very large, odd integer For ΛN ∩ Zd , the classical statistical mechanics of a gas of such dipoles with inverse temperature (for convenience) β = and activity (fugacity) z > is given by the grand canonical partition function   n zn −1 ZN = (pk · ∂)(pj · ∂)C(xk , xj ) dpi exp  (5) n! i=1 Sd−1 d n≥0 1≤k,j≤n xi ∈Z ∩ΛN All fundamental objects to study, such as the pressure, truncated correlation functions, etc are derived from ZN and similar objects The model can be equivalently expressed as a Euclidean field theory (due to Kac [13] and Siegert [16]) and is given by 0Z N where ≡ ZN = exp zW (ΛN , φ) dµC (φ) W (ΛN , φ) = x∈ΛN ∩Zd Sd−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 as given above ã The measure àC : the underlying measure (see section 11.2 [6] and Appendix A [5] for more detail) We discuss about the equivalence of (5) and (6) in appendix A of this paper For investigating the truncated correlation functions, we consider a more general version of (6): f ZN = exp if (φ) + zW (ΛN , φ) dµC (φ) (7) where f (φ) can be: f (φ) = as in [5] m k=1 tk ∂µk φ(xk ) We use this t m t G (x1 , x2 , xm ) ≡ k=1 ∂µk φ(xk ) f (φ) = f (φ) to study the truncated correlation functions m = im ∂t1∂ ∂tm log f Z ′ t1 =0, tm =0 which is nontrivial and previously investigated by Dimock and Hurd [7] f (φ) = m k=1 m k=1 tk exp (i∂µk φ(xk )) (for studying the dipole correlation exp (i∂µk φ(xk )) t m = im ∂t1∂ ∂tm log f Z ′ t1 =0, tm =0 ) 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 by Brydges 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 ≥ For the set {x1 , , xm } ⊂ Zd , let diam(x1 , , xm ) = max1≤i,j≤m dist(xi , xj ) where dist(xi , xj ) is the distance between xi and xj on lattice Zd To get rid of the boundary and study the long distance properties of the system, we would like to take 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 |−1 log Z N (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 For long distance (i.e., when |x − y| large), the potential ∂µ ∂ν C(x − y) behaves like O(|x − y|−d ), that means it is not integrable and we could not use the theory of the Mayer expansion to establish such results To overcome this problem, we use the method of the renormalization group We follow particularly a Renormalization Group approach recently developed by Brydges and Slade [1] and Dimock [5] We generalize Dimock’s framework with an external field and obtain some estimates on the correlation functions as in Dimock and Hurd [7], and Brydges and Keller [2] The main result is the existence of the infinite volume limit for correlations functions, which is new Earlier work using RG approach to the dipole gas can be found in Gawedski and Kupiainen [12], Brydges and Yau [4] Besides the dipole gas papers mentioned above, we would like to cite some other papers on the Coulomb gas in d = which has a dipole phase There are the works of Dimock and Hurd [8], Falco [9] and Zhao [17] 1.2 The main result For our RG approach we follow the analysis of Brydges’ lecture [1] Instead of (7), we use a different finite volume approximation First, we add an extra term (1 − ε)V (ΛN , φ) where < ε is closed to and d V (ΛN , φ) = (∂µ φ(x))2 (9) d ±µ=1 x∈ΛN ∩Z −1 By replacing the covariance C by ε C, this extra term will be partially compensated Hence instead of (7) we will consider a new finite volume generating function f ZN ′′ = f Z ′N /ZN (10) where ′ f ZN = eif (φ) exp zW (ΛN , φ) − (1 − ε)V (ΛN , φ) dµε−1 C (φ) (11) and ′′ ZN = exp − (1 − ε)V (ΛN , φ) dµε−1 C (φ) (12) We have f ZN ′′ = f Z ′N /ZN = eif (φ) exp (zW (ΛN , φ)) (Z”N )−1 exp (−(1 − ε)V (ΛN , φ)) dµε−1 C (φ) (13) When N → ∞, exp (−(1 − ε)V (ΛN , φ)) formally becomes exp (1/2(1 − ε)(φ, −∆φ)), and dµε−1 C (φ) = (1/2(ε)(φ, −∆φ)) dφ So the bracketed expression formally converges to (const.) (1/2(φ, −∆φ)) dφ = (const.)dµC (φ) when N → ∞ Formally this new f ZN gives the same limit as (7) This result holds for d any choice of ε By definition (9), the extra term (1−ε)V (ΛN , φ) = (1−ε) 41 x∈ΛN ∩Zd ±µ=1 (∂µ φ(x))2 Therefore the choice of ε is a choice of how much (∂φ)2 one is putting in the interaction and how much in the measure Similarly to the Theorem in [5], our main theorems are: Theorem For |z| and maxk |tk | sufficiently small there is an ε = ε(z) close to so that −1 log f Z N has a limit as N → ∞.2 f pN = |ΛN | Using f (φ) = m k=1 tk ∂µk φ(xk ), we achieve some estimate for the correlation functions: Theorem For any small ǫ > 0, with L, A sufficiently large (depending on ǫ), η = min{d/2, 2}, we have: m ∂µk φ(xk ) t k=1 ≤ m! diam−η+ǫ (x1 , xm ) am (14) where a depends on ǫ, L, A 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 function t m exists limN →∞ k=1 ∂µk φ(xk ) When d = or 4, the result in Theorem looks like the result in [7], but here it is obtained with the new method m Using f (φ) = k=1 tk exp (i∂µk φ(xk )), we can obtain Theorems 10 and 11 which are similar to Theorems and 3, just with different f At the end of this paper, we investigate a general form of f and obtain Theorems 12 and 13 Applying theorem 12 with a special f for density of dipoles, we have obtained some estimates for truncated correlation functions of density of dipoles with (m ≥ 2) points (Corollary 1) instead of only points as Theorem 1.1.2 in [2] Then we apply theorem 13 to establish the infinite volume limit for truncated 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 density exp zW − (1 − ε)V likely goes to zero under the renormalization group flow and leaves a measure like µε(z)−1 C to describe the long distance behavior of the system Accordingly ε(z) can be interpreted as a dielectric constant √ Now we rewrite the generating function f ZN First we scale φ → φ/ ε and then let σ = ε−1 − Because ε is closed to , we have σ is near zero We also have ′ f Z N (z, σ) = ′′ ZN (σ) = f Z N (z, σ) √ eif (φ) exp zW (ΛN , + σφ) − σV (ΛN , φ) dµC (φ) exp − σV (ΛN , φ)) dµC (φ) (15) ′′ =f Z ′N (z, σ)/ZN (σ) Then we need to show that with |z| sufficiently small there is a (smooth) σ = σ(z) near zero such that, ′′ |ΛN |−1 log f Z N (z, σ(z)) = |ΛN |−1 log f Z ′N (z, σ(z)) − |ΛN |−1 log ZN (σ(z)) (16) has a limit when N → ∞ And theorem 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 |−1 log ZN (σ) converges as N → ∞ (Theorem 2, [5]) Hence we only need to investigate the first term in (16) The paper is organized as follows: In Theorem 1, f (φ) can be 0, m k=1 tk ∂µk φ(xk ), or m k=1 tk exp (iàk (xk )) ã In section we give some general definitions on the lattice and its properties We also give definitions 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 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 we study the RG flow and find the stable manifold σ = σ(z) • In section we assemble the results and prove the infinite volume limit for |ΛN |−1 log f Z ′N exists • Finally in section 6, by combining all the other estimates, we obtain some estimates for correlation functions and establish the infinite volume limit of correlation functions Preliminaries In this section, we quote all notations and basic result from Dimock [5] At the same time, we introduce some new notations which are useful for this paper 2.1 Multiscale decomposition RG methods are based upon a multiscale decomposition of the basic lattice covariance C into a sequence of more controllable integrals and analyze the effects separately at each stage Especially we choose a decomposition into finite range covariances which is developed by Brydges, Guadagni, and Mitter [3] The decomposition of the lattice covariance C has the form C(x − y) = ∞ j=1 Γj (x − y) (17) such that • Γj (x) is defined on Zd , is positive semi-definite, and satisfies the finite range property: Γj (x) = if |x| ≥ Lj /2 • There is a constant c0 independent of L such that, for all j, x, we have |Γj (x)| ≤ c0 L−(j−1)(d−2) (18) This implies that the series converges uniformly • There are constants cα independent of L such that αµ d ±µ=1 ∂µ |∂ α Γj (x)| ≤ cα L−(j−1)(d−2+|α|) where ∂ α = is a multi-derivative and |α| = converges uniformly to ∂ α C µ (19) |αµ | Thus the differentiated series • (Lemma 2, [5]) There are some constants CL,α such that |∂ α C(x)| ≤ CL,α (1 + |x|)−d+2−|α| (20) For our RG analysis we need to break off pieces of covariance C(x − y) one at a time So we define Ck (x − y) = ∞ j=k+1 Γj (x − y) (21) Hence we have C = C0 and Ck (x − y) = Ck+1 (x − y) + Γk+1 (x − y) (22) 2.2 Renormalization Group Transformation The generating function (15) can be rewritten as ′ f Z N (z, σ) = N f Z (φ)dµC0 (φ) (23) with N f Z (φ) √ = eif (φ) exp zW (ΛN , + σφ) − σV (ΛN , φ) (24) We use the left subscript f as an extra notation for cases at the same time: • f (φ) = as in (Dimock, [5]); • f (φ) = • f (φ) = m k=1 tk ∂µk φ(xk )); m k=1 tk exp (i∂µk φ(xk )) Since C0 = C1 + Γ1 we replace an integral over µC0 by an integral over µΓ1 and µC1 So we have ′ f Z N (z, σ) N f Z (φ = N f Z (φ)dµC1 (φ) + ζ)dµΓ1 (ζ)dµC1 (φ) = (25) We define a new density by the fluctuation integral N f Z (φ) = (µΓ1 ∗ f Z N )(φ) ≡ N f Z (φ + ζ)dµΓ1 (ζ) (26) Because Γ1 , C1 are only positive semi-definite, these are degenerate Gaussian measures.3 By continuing this way, we will have the representation for j = 0, 1, 2, ′ f Z N (z, σ) = N f Z j (φ)dµCj (φ) (27) here the density f Z N j (φ) is defined by N f Z j+1 (φ) = (µΓj+1 ∗ f Z N j )(φ) = N f Z j (φ + ζ)dµΓj+1 (ζ) (28) Our job is to investigate the growth of these densities when j go to ∞ 2.3 Local expansion We will rewrite each density f Z N j (φ) in a form which presents its locality properties known as a polymer representation The localization becomes coarser when j gets bigger First we will give some basic 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 of the center j-blocks Bj0 = {x ∈ Zd : |x| < 1/2(Lj − 1)} (29) by points in the lattice Lj Zd The set of all j-blocks in Λ = ΛN is denoted Bj (ΛN ), Bj (Λ) or just Bj A union of j-blocks X is called a j-polymer Note that Λ is also a j-polymer for ≤ j ≤ N The set of Dimock has discussed these in Appendix A, [5] all j-polymers in Λ = ΛN is denoted Pj (Λ) or just Pj The set of all connected j-polymers is denoted ¯ is the smallest Y ∈ Pj+1 such that X ⊂ Y by Pj,c Let X ∈ Pj , the closure X 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 is denoted by Sj (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 ∈ B1∗ then, using above definition, we also have that B1 ∈ B2∗ Similarly a j-polymer X has a small set neighborhood X ∗ For l ≥ and integer d, we define some constants n1 (d), n2 (d), n3 (d, l) which are bounded and, for every j ≥ 0, we have: n1 (d) ≡ n2 (d) ≡ n3 (d, l) ≡ 1/|X|0 = 1/|X|j X∈Sj ,X⊃Bj0 X∈S0 ,X⊃0 1= X∈Sj ,X⊃Bj0 X∈S0 ,X⊃0 l−|X|0 = |X|0 X∈S0 ,X⊃0 (30) X∈Sj ,X⊃Bj0 l−|X|j |X|j d n3 (d, l) ≤n3 (d, 1) = n1 (d) ≤ n2 (d) ≤ (2d )!(2d)2 Furthermore, with a fixed d, we can get ≤ lim n3 (d, l) ≤ lim l→∞ 2.3.2 l→∞ n1 (d) =0 l (31) Local expansion d Using the same approach as in [5], we rewrite the density f Z N j (φ) for φ : Z → R in the the general form (32) f I(Λ − X)f K(X) f Z = (f I ◦ f K)(Λ) ≡ X∈Pj (Λ) Here f I(Y ) is a background functional which is explicitly known and carries the main contribution to the density The f K(X) is so called a polymer activity It represents small corrections to the background In section we will show that the initial density f I0 has the factor property We want to keep this factor property at all scales Then we can use the analysis of Brydges’ lecture [1] Therefore we assume f I(Y ) always is in the form of f I(Y )= f I(B) (33) B∈Bj :B⊂Y and f I(B, φ) depends on φ only B ∗ , the small set neighborhood of B Moreover we assume f K(X) factors over the connected components C(X) of X f K(X) f K(Y = ) (34) Y ⊂C(X) and that f K(X, φ) only depends on φ in X ∗ As in [5], the background functional f I(B) has a special form: f I(f E, σ, B) = exp(−V (f E, σ, B)) where σµν (B)∂µ φ(x)∂ν φ(x) (35) V (f E, σ, B, φ) = f E(B) + µν x∈B Sums over µ are understood to range over µ = ±1, , ±d, unless otherwise specified for some functions f E, σµν : Bj → R Indeed we usually can take σµν (B) = σδµν for some constant σ Then V (f E, σ, B, φ) becomes σ (∂µ φ(x))2 ≡ f E(B) + σV (B) (36) V (f E, σ, B, φ) = f E(B) + µ x∈B Also in our model, when f = 0, we will have K(X, φ) = K(X, −φ) K(X, φ) = K(X, φ + c) (37) The later holds for any constant c which means that K(X, φ) only depends on derivatives ∂φ 2.4 About norms and their properties 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) = sµν (x)∂µ φ(x)∂ν φ(x) (38) µν x∈B here the norms of functions sµν (x) are defined by s j = sup |B|−1 s B∈Bj 1,B = sup L−dj B∈Bj µν x∈B |sµν (x)| If sµν (x) = σδµν then V (s, B) = σV (B) as defined in (36) and the norm s The following lemmas are some results from Section in [5]: j (39) = 2d σ Lemma (Lemma 3, [5]) For any functional sµν (x), we have V (s, B) V (s, B) ′ s,j s,j ≤ h2 s ≤h s j (40) j The function σ → exp(−σV (B)) is complex analytic and if h2 σ is sufficiently small, we have e−σV (B) ′ s,j −σV (B) s,j e ≤2 ≤2 (41) Let c be a constant such that the function σ → exp(−σV (B)) is analytic in |σ| ≤ ch−2 and satisfies exp(−σV (B)) s,j ≤ on that domain To start the RG transformation, we also need some estimate on the initial interaction When j = 0, B ∈ B0 is just a single site x ∈ Zd , so we consider W (u, B, φ) = Sd−1 dp cos(p · ∂φ(x)u) (42) Lemma (Lemma 4, [5]) W (u, B) is bounded by W (u, B) s,0 ≤ 2e √ dhu (43) We also have that W (u, B) is strongly continuously differentiable in u ezW (u,B) is complex analytic in z and satisfies, for |z| sufficiently small (depending on d, h, u), we have ezW (u,B) s,0 ≤ (44) And ezW (u,B) is also strongly continuously differentiable in u Analysis of the RG Transformation Now we use the Brydges-Slade RG analysis and follow the framework of Dimock [5], but with an external field f 3.1 Coordinates (f I j , f K j ) Continuing to the subsection 2.3.2 (Local Expansion), we suppose that we have f Z(φ) = (f I◦f K)(Λ, φ) with polymers on scale j We rewrite it as fZ ′ (φ′ ) = (µΓj+1 ∗ f Z)(φ′ ) ≡ ′ f Z(φ + ζ)dµΓj+1 (ζ) (45) here we try to put it back to the form fZ ′ (φ′ ) = (f I ′ ◦ f K ′ )(Λ, φ′ ) (46) where the polymers are now on scale (j + 1) Furthermore, supposed that we have chosen f I ′ , we will find f K ′ so the identity holds As explained before, our choice of f I ′ is to have the form fI ′ ′ ˜ f I(B, φ ) (B ′ , φ′ ) = B ′ ∈ Bj+1 B∈Bj ,B⊂B ′ (47) Now we define δ f I(B, φ′ , ζ) =f I(B, φ′ + ζ) − f˜I(B, φ′ ) fK ◦ δ f I ≡ f ˜K(X, φ′ , ζ) = ′ f K(Y, φ ′ f J(B, X, φ ) = (48) Y ⊂X For connected X we write f ˜K(X, φ′ , ζ) in the form ′ ˜ f K(X, φ , ζ) + ζ)δ f I X−Y (φ′ , ζ) + f ˇK(X, φ′ , ζ) (49) B⊂X Given f K and f J the equation (49) would give us a definition of f ˇK(X) for X connected And for any X ∈ Pj , we define ′ ′ ˇ ˇ (50) f K(Y, φ , ζ) f K(X, φ , ζ) = Y ∈C(X) After using the finite range property and making some rearrangements as Proposition 5.1, Brydges [1], we have (46) holds with fK ′ (U, φ′ ) = fJ χ (φ′ )f˜I U−(Xχ ∪X) # (φ′ )f ˇK (X, φ′ ) X,χ→U U ∈ Pj+1 (51) where χ = (B1 , X1 , Bn , Xn ) and the condition X, χ → U means that X1 , Xn , X be strictly disjoint and satisfy (B1∗ ∪ · · · ∪ Bn∗ ∪ X) = U Moreover n fJ χ ′ f J(Bl , Xl , φ ) (φ′ ) = i=1 ˜ fI U−(Xχ ∪X) ′ ˜ f I(B, φ ) ′ (φ ) = (52) B∈U−(Xχ ∪X) As in (Dimock, [5]), J(B, X) will be chosen to depend on K and required J(B, X) = unless X ∈ S , B ⊂ X j f f f and that f J(B, X, φ′ ) depend on φ′ only in B ∗ # with Xχ = ∪i Xi And f ˇK (X, φ′ ) is f ˇK(X, φ′ , ζ) integrated over ζ as (78) in [5] At this point f K ′ is considered as a function of f I, f˜I, f J, f K It vanishes at the point (f I, f˜I, f J, f K) = (1, 1, 0, 0) since χ = ∅ and X = ∅ iff U = ∅ We study its behavior in a neighborhood of this point We have the norm on f K as (75) in [5] and we define f I s,j = sup ˜ ′ f I s,j = sup ′ fJ j B∈Bj B∈Bj = f I(B) s,j ′ ˜ f I(B) s,j sup X∈Sj ,B⊂X (53) ′ f J(X, B) j We also set δf K = f K − 0K (54) Using the same argument as Theorem in [5], we have the following result Theorem Let A be sufficiently large For R > there is a r > such that the following holds for all j If f I − s,j < r, f˜I − ′s,j < r, max{ f J ′j , J ′j } < r and max{ f K j , K j } < r then max{ f K ′ j+1 , K ′ j+1 } < R Furthermore f K ′ is a smooth function of f I, f˜I, f J, f K on this domain with derivatives bounded uniformly in j The analyticity of f K ′ in t1 , , tm still holds when we go from j-scale to (j + 1)-scale If also f J(B, X) =0 (55) X∈Sj :X⊃B then the linearization of f K ′ = f K ′ (f I, f˜I, f J, f K) at (f I, f˜I, f J, f K) = (1, 1, 0, 0) is fK X∈Pj,c X=U # (X) + (f I # (X) − 1)1X∈Bj − (f˜I(X) − 1)1X∈Bj − f J(B, X) (56) B⊂X where fK # (X, φ) = f K(X, φ + ζ)dµΓj+1 (ζ) (57) and J actually is f J at f = 3.2 3.2.1 Choosing J and Estimating L1 , L2 Choosing J For a smooth function g(φ) on φ ∈ RΛ let T2 g denote a second order Taylor expansion: (T2 g)(φ) =g(0) + g1 (0; φ) + g2 (0; φ, φ) (T0 g)(φ) =g(0) (58) With f K # defined in (57), for X ∈ Sj , X ⊃ B, X = B, we pick: f J(B, X) [T2 (0 K # (X)) + T0 (f K # (X)) − T0 (0 K # (X))] |X|j [T2 (0 K # (X)) + T0 (δ f K # (X))] = |X|j = 10 (59) 3.5 Simplifying for the next scale We now pick f˜E(B), σ ˜ so the V terms in (111) cancel We have: σ V # (f E, σ, B, φ) =f E(B) + σ =f E(B) + + σ σ + (∂µ φ(x) + ∂µ ζ(x))2 dµΓj+1 (ζ) x∈B µ ∂µ φ(x)2 dµΓj+1 (ζ) x∈B µ ∂µ ζ(x)dµΓj+1 (ζ) ∂µ φ(x) x∈B µ (114) ∂µ ζ(x) dµΓj+1 (ζ) x∈B µ =f E(B) + σ ∂µ φ(x)2 + + x∈B µ σ ≡V (f E, σ, B, φ) + σ (∂µ Γj+1 ∂µ∗ )(x, x) x∈B µ T r(1B ∂µ Γj+1 ∂µ∗ ) µ because ∂µ ζ(x)dµΓj+1 (ζ) = ∂µ ζ(x)2 dµΓj+1 (ζ) = (ζ, ∂µ∗ δx )(ζ, ∂µ∗ δx )dµΓj+1 (ζ) = (∂µ∗ δx , Γj+1 ∂µ∗ δx ) (115) = (δx , ∂µ Γj+1 ∂µ∗ δx ) = (∂µ Γj+1 ∂µ∗ )(x, x) If we choose f˜E = f˜E(f E, σ, f K) ˜ f E(B) = f E(B) + σ T r(1B ∂µ Γj+1 ∂µ∗ ) + f β(f K, B) (116) µ then the constant terms of (114) will be canceled The second order terms of (114) would be vanish if we define σ ˜=σ ˜ (σ, f K) = σ ˜ (σ, K) by σ ˜ = σ + α(f K) = σ + α(0 K) (117) Here we are canceling the constant term exactly for all B, but for the quadratic term we only cancel the invariant version away from the boundary By composing f K ′ = f K ′ (f˜E, σ ˜ , f E, σ, f K, K) in theorem with newly defined f˜E = f˜E(f E, σ, f K) and σ ˜ =σ ˜ (σ, f K) = σ ˜ (σ, K) we obtain a new map f K ′ = f K ′ (f E, σ, f K, K) We also have new quantities f E ′ (f E, σ, f K) defined by f E ′ (B ′ ) = B⊂B ′ f˜E(B) and σ ′ = σ ′ (σ, f K) = σ ′ (σ, K) defined by σ ′ = σ ˜ = σ + α(f K) = σ + α(0 K) as normal These quantities satisfy (45) µΓj+1 ∗ (f I(f E, σ) ◦ f K) (Λ) = fI ′ (f E ′ , σ ′ ) ◦ f K ′ (Λ) (118) Here we still assume that L is sufficiently large, and that A is sufficiently large depending on L Theorem For R > there is a r > such that the following holds for all j If f E j , |σ|, max{ f K j , K j } < r then f E ′ j+1 , |σ ′ |, max{ f K ′ j+1 , K ′ j+1 } < R Furthermore ′ ′ ′ f E , f K , σ are smooth functions of f E, σ, f K, K on this domain with derivatives bounded uniformly in j The analyticity of f K ′ in t1 , , tm still holds when we go from j-scale to (j + 1)scale 20 The linearization of f K ′ = f K ′ (f E, σ, f K, K) at the origin is the contraction L(f K) where L = L1 + L2 + L′3 + L4 + ∆ (119) Proof For the first part, by combining with theorem 5, it suffices to show that the linear maps f˜E and σ ˜ have norms bounded uniformly in j Using the estimate |α(f K)| = |α(0 K)| ≤ 4(2d)2 n2 (d)h−2 A−1 K j from lemma 5, we have σ ˜ is bounded From lemma we also have the bound on f β(f K) j ≤ 2n2 (d)A−1 f K j For B ∈ Bj , the estimate (19) gives us σ µ T r(1B (∂µ Γj+1 ∂µ∗ ) ≤ dc1,1 |σ| x∈B L−dj ≤ dc1,1 |σ| (120) where c1,1 as in (19) Combining with (116) we have that f˜E = f˜E(f E, σ, f K) satisfies ˜ fE j ≤ fE j + C(|σ| + A−1 f K j) (121) where C = max{dc1,1 , 2n2 (d) The second part follows since the linearization of the new function f K ′ is the linearization of the old function f K ′ in theorem composed with f˜E = f˜E(f E, σ, f K), σ ˜=σ ˜ (σ, f K) = σ ˜ (σ, K) (All of them vanish at zero.) The cancellation gives us only with L(f K) 3.6 Forming RG FLow It is easier for us if we can extract the energy from the other variables Assume that we start with E(B) = in (118) µΓj+1 ∗ (f I(0, σ) ◦ f K) (ΛN ) = f I ′ (f E ′ , σ ′ ) ◦ f K ′ (Λ) (122) where σ ′ = σ ′ (σ, f K) = σ ′ (σ, K) and f K ′ = f K ′ (0, σ, f K) and f E ′ = f E ′ (0, σ, f K) as above Then we remove the f E ′ by making an adjustment in f K ′ µΓj+1 ∗ (f I(0, σ) ◦ f K) (ΛN ) = = fI U∈Pj+1 = U∈Pj+1 = U∈Pj+1 = U∈Pj+1  = exp   (f E ′ , σ ′ )(Λ − U ) fI  B ′ ∈B  B ′ ∈B   j+1 (Λ−U) j+1 (Λ−U)  exp  ′ j+1 (ΛN ) (f E ′ , σ ′ ) ◦ f K ′ (Λ) ′ (0, σ, f K, U )  (f E ′ , σ ′ )(B ′ ) fK ′ (0, σ, f K, U )  exp(f E ′ (B ′ ))[f I ′ (0, σ ′ )(B ′ )] j+1 (Λ−U) + ′ fK fE B ′ ∈B fE B ′ ∈B + ′ fI  (B ′ ) ′  fK ′ (0, σ, f K, U )  (B ′ ) f I ′ (0, σ ′ )(Λ − U ) fI ′ fK ′ (123) (0, σ, f K, U ) (0, σ + ) ◦ f K + (ΛN ) where f E (σ, f K, B ), σ (σ, f K), f K + (σ, f K, U ) are defined as following (U ∈ Pj+1 , B ′ ∈ Bj+1 ) ′ fE + + (σ, f K, B ′ ) ≡f E ′ (0, σ, f K, B ′ ) = + fK + ′ ˜ f E(0, σ, f K, B) B⊂B ′ ′ σ (σ, f K) ≡σ (σ, f K) = σ (σ, K) = σ + α(0 K)   (σ, f K, U ) ≡ exp − fE B ′ ∈Bj+1 (U) 21 + (B ′ ) f K ′ (0, σ, f K, U ) (124) The dynamical variables are now σ + (σ, f K) and f K + (σ, f K) The extracted energy f E + (σ, K) is controlled by the other variables Because everything vanishes at the origin the linearization of + + ˜ and our f K (σ, f K) is still L(f K) The bound (121) on f E would give us an upper bound on f E theorem becomes: Theorem For R > there is a r > such that the following holds for all j If |σ|, max{ f K j , K j } < r then |σ + |, max{ f K + j+1 , K + j+1 } < R Furthermore σ + , f K + are smooth functions of σ, f K on this domain with derivatives bounded uniformly in j The analyticity of f K + in t1 , , tm still holds when we go from j-scale to (j + 1)-scale The extracted energies satisfy fE + (σ, f K) j+1 ≤ C(Ld ) |σ| + A−1 fK j (125) The linearization of K + at the origin is the contraction L The stable manifold Up to now, we have not specialized to the dipole gas, but take a general initial point σ0 , f K corresponding to an integral (f I(0, σ0 )◦ f K )(ΛN )dµC0 We assume K (X, φ) has the lattice symmetries and satisfies the conditions (37) We also assume |σ0 |, max{ f K , K } < r where r is small enough so the theorem holds, say with R = 1, then we can take the first step We apply the transformation (123) for j = 0, 1, 2, and continue as far as we can Then we get a sequence σj , f K N j (X) by + N N + N N = E (σ ) with extracted energies E = K (σ , K ) and K σj+1 = σ + (σj , f K N f j , f K j ) f j+1 f j f f j+1 j j Then we have, for any l, with f I j (σj ) = f I j (0, σj )  (f I (σ0 ) ◦ f K )(ΛN )dµC0 = exp  l j=1 B∈Bj (ΛN )  N  f E j (B) (f I l (σl ) ◦ f K N l )(ΛN )dµCl (126) N The quantities K N j (X) and E j (B) are independent of N and have the lattice symmetries if X, B are away from the boundary ∂ΛN in the sense that they have no boundary blocks These properties are true initially and are preserved by the iteration In these cases we denote these quantities by just K j (X) and E j (B) With our construction α defined in (89),(107) only depends on K j By splitting K + into a linear and a higher order piece the sequence σj , f K N j (X) is generated by the RG transformation σj+1 =σj + α(Kj ) N K j+1 δf K N j+1 N =L(0 K N j ) + g(σj , K j ) = (L1 + L2 ) (δ f K N j ) + N N f g(σj , f K j , K j ) (127) − N g(σj , K j ) This is regarded as a mapping from the Banach space R × (Kj (ΛN ) × Kj (ΛN )) to the Banach space R × (Kj (ΛN ) × Kj (ΛN )) The 2nd equation of (127) defines g which is smooth with derivatives bounded uniformly in j and satisfies g(0, 0) = 0, D(0 g)(0, 0) = The last equation of (127) defines f g which is also smooth with derivatives bounded uniformly in j and satisfies f g(0, 0) = 0, D(f g)(0, 0) = Now we consider the first two equations in (127) Around the origin there are a neutral direction σj and a contracting direction Kj (since L is a contraction.) Hence we expect there is a stable manifold We quote a version of the stable manifold theorem due to Brydges [1], as applied in Theorem in Dimock [5] 22 Theorem (Theorem 7, Dimock [5]) Let L be sufficiently large, A sufficiently large (depending on L), and r sufficiently small (depending on L, A) Then there is < τ < r and a smooth real-valued function σ0 = h(0 K ), h(0) = 0, mapping N K 0 < τ into |σ0 | < r such that with these start values the sequence σj , K j is defined for all ≤ j ≤ N and N −j (128) |σj | ≤ r2−j K j j ≤ r2 Furthermore the extracted energies satisfy N E j+1 j+1 ≤ 2C(Ld )r2−j (129) Remark Using the Lemma 10 below, given r > 0, we can always choose z, σ0 and maxk |tk | sufficiently N small then max{ f K , K } ≤ r Now we claim that f K N j j has the same bound as the K j j in the last theorem −k Supposed that at j = k, we have: f K N As in the proof of Theorem in (Dimock, j j ≤ r2 [5]), we can say that L and (L1 + L2 ) is a contraction with norm less than 1/8 and f g(σj , f K N j , 0K j ) is second order Hence there are some constant H such that: N 0K j j + N f Kj j with |σj |, N f K j+1 j+1 N N 0K j j , f K j j N N f g(σj , f K j , K j ) ≤ H |σj |2 + small Then we have: N N N K j j + δ f K j j + H |σj | + K j N −j ≤ 0K N j j + f K j j + 3H r2 3r2−j + 3H r2−j ≤ ≤r2−j−1 ≤ j + N f Kj j (130) for r sufficiently small The bound for f E N j+1 comes from the bound on σj , f K N j j , (125) and A > Combining with the last theorem, for all ≤ j ≤ N we can have: N f Kj j |σj | ≤ r2−j ≤ r2−j (131) and the extracted energies satisfy N f E j+1 j+1 5.1 ≤ 2C(Ld )r2−j (132) The dipole gas The initial density Now we consider the generating function: f Z N (z, σ) eif (φ) exp zW (ΛN , = √ + σφ) − σV (ΛN , φ) dµC0 (φ) (133) When f = 0, it becomes Z N (z, σ) = √ exp zW (ΛN , + σφ) − σV (ΛN , φ) dµC0 (φ) 23 (134) √ For B ∈ B0 , we define: W0 (B) = zW ( + σ0 , B) as in (42) and V0 (B) = σ0 V (B) as in (36) Then we follow with a Mayer expansion to put the density in the form we want N f Z0 eif (φ)+W0 (B)−V0 (B) = B⊂ΛN = B⊂ΛN = e−V0 (B) + (eif (φ)+W0 (B) − 1)e−V0 (B) f I (σ0 , ΛN X⊂ΛN (135) − X)f K (X) =(f I (σ0 ) ◦ f K )(ΛN ) where I0 (σ0 , B) = e−V0 (B) and f K (X) = f K (z, σ0 , X) is given by f K (X) = B⊂X when f (φ) = if (φ) B = or m k=1 tk exp (i∂µk φ(xk )), if (φ) f K (X) = B⊂X m k=1 tk ∂µk φ(xk ), when f (φ) = f (φ) = if (φ) B = or f K (X) (eif (φ)|B +W0 (B) − 1)e−V0 (B) B (136) = tk exp (i∂µk φ(xk )) if B = {xk } for some k, otherwise (eif (φ)|B +W0 (B) − 1)e−V0 (B) if (φ) B = B⊂X (137) = tk ∂µk φ(xk ) if B = {xk } for some k, otherwise (eW0 (B) − 1)e−V0 (B) (138) when f (φ) = Note that, when f = , K actually is the K0 in lemma 12, [5] We also can prove the same result for f K Lemma 10 Given > r > , there are some sufficiently small a(r), b(r) and c(r) such that if maxk |tk | ≤ a(r), |z| ≤ b(r) and |σ0 | ≤ c(r) then f K (z, σ0 ) ≤ r Furthermore f K is a smooth function of (z, σ0 ), and analytic in tk for all k = 1, , m Proof *When f = 0, using lemma 12 [5], we have some b0 (r), c0 (r) such that K (z, σ0 ) |z| ≤ b0 (r) and |σ0 | ≤ c0 (r) m *In the case f (φ) = k=1 tk ∂µk φ(xk ), using ([5], (95)), for φ = φ′ + ζ, we have: (eif (φ)|B +W0 (B) − 1) = (eif (φ)|B +zW ( ∞ √ ≤ zW ( + σ0 , B) + if (φ)|B n! n=1 ≤ ≤ ∞ n! n=1 ∞ n! n=1 √ zW ( + σ0 , B) 2|z|eh √ d(1+σ0 ) √ 1+σ0 ,B) − 1) (139) n n + max |tk |h−1 φ 24 ≤ r if n + if (φ)|B k Φ0 (B ∗ ) We can assume that maxk |tk |h−1 ≤ Applying lemma in [5], we get e−V0 (B) f K (B) = sup φ′ ,ζ ′ f K (B, φ ≤ sup (eif (φ)|B +W0 (B) − 1) φ′ ,ζ ≤ e−V0 (B) s,0 + ζ) G0 (X, φ′ , ζ)−1 e−V0 (B) ≤ sup exp 2|z|eh √ d(1+σ0 ) + max |tk |h−1 φ′ + ζ k √ d(1+σ0 ) − exp max |tk |h−1 φ′ + ζ + sup exp max |tk |h−1 φ′ + ζ k ≤ sup exp 2|z|eh √ d(1+σ0 ) φ′ ,ζ k Φ0 (B ∗ ) Φ0 (B ∗ ) exp(− φ′ Φ0 (B ∗ ) Φ0 (B ∗ ) z,σ0 →0 φ′ Φ0 (B ∗ ) ) − ζ e− Φ0 (B ∗ ) − e− exp 2|z|eh lim Gs,0 (X, φ′ , ζ)−1 Φ0 (B ∗ ) (140) − Gs,0 (X, φ′ , ζ)−1 φ′ + ζ − exp + sup exp max |tk |h−1 φ′ + ζ φ′ + ζ − Gs,0 (X, φ′ , ζ)−1 Φ0 (B ∗ ) k φ′ ,ζ Because exp ′ −2 Gs,0 (X, φ , ζ) φ′ ,ζ φ′ ,ζ φ′ ,ζ ≤ sup (eif (φ)|B +W0 (B) − 1) Gs,0 (X, φ′ , ζ)−1 ≤ sup exp 2|z|eh φ′ ,ζ s,0 √ d(1+σ0 ) Φ0 (B ∗ ) − φ′ Φ0 (B ∗ ) − ζ Φ0 (B ∗ ) Φ0 (B ∗ ) ζ is bounded and −1 =0 (141) there exist some sufficiently small b1 (r), c1 (r) > such that we have √ ′ 2 sup exp 2|z|eh d(1+σ0 ) − exp φ′ + ζ Φ0 (B ∗ ) e− φ Φ0 (B∗ ) − ζ Φ0 (B ∗ ) φ′ ,ζ ≤ r 4A (142) for all |z| ≤ b1 (r) and |σ0 | ≤ c1 (r) For other part, we have: sup exp max |tk |h−1 φ′ + ζ k φ′ ,ζ ≤ sup exp φ′ ,ζ φ′ Φ0 (B ∗ ) + ζ − exp(− φ′ Φ0 (B ∗ ) Φ0 (B ∗ ) − exp(− φ′ We also can find some sufficiently large H such that if φ′ φ′ exp Φ0 (B ∗ ) + ζ Φ0 (B ∗ ) − exp(− φ′ Φ0 (B ∗ ) Φ0 (B ∗ ) Φ0 (B ∗ ) Φ0 (B ∗ ) − ζ + ζ − ζ − ζ k Φ0 (B ∗ ) − exp(− φ′ Φ0 (B ∗ ) Φ0 (B ∗ ) Φ0 (B ∗ ) ) − ζ (143) Φ0 (B ∗ ) ) For φ′ Φ0 (B ∗ ) + ζ Φ0 (B ∗ ) ≤ H, we have φ′ + ζ Φ0 (B ∗ ) ≤ φ′ Φ0 (B ∗ ) + ζ maxk |tk | ≤ a1 (r) sufficiently small and φ′ Φ0 (B ∗ ) + ζ Φ0 (B ∗ ) ≤ H, exp max |tk |h−1 φ′ + ζ Φ0 (B ∗ ) ) ≥ H then ≤ r 4A Φ0 (B ∗ ) Φ0 (B ∗ ) ) ≤ (144) ≤ H So with r 4A (145) In summary we can always choose sufficiently small a(r), b(r), c(r) such that if maxk |tk | ≤ a1 (r), |z| ≤ b1 (r), and |σ0 | ≤ c1 (r) then f K (B) ≤2 r r = 4A 2A 25 ∀B ∈ B0 (146) For those a1 (r), b1 (r), c1 (r), maxk |tk | ≤ a1 (r), |z| ≤ b1 (r), and |σ0 | ≤ c1 (r) we have f K0 = sup X∈P0,c |X|0 f K (X) A ≤ sup X∈P0,c ≤ sup X∈P0,c m k=1 tk * In the last case, f (φ) = (e if (φ)|B +W0 (B) A|X|0 f K (B) (147) B⊂X r 2A |X|0 r 0, we can always find A, L sufficiently large such that: (L1 + L2 + L′3 + L4 ) (0 K) j+1 ≤ 0K j 4Lη−ǫ (161) 1 ( δ K ) ≤ ( K + K ) (L1 + L2 ) (δ f K) j+1 ≤ f j j f j 4Lη−ǫ 4Lη−ǫ with j ≥ by using the explicit upper bounds in Lemmas 3, 4, 6, and Then we can replace µ = 1/2 in Theorem by µ = 1/M for M = Lη−ǫ ≥ We still have −j d −j |σj | ≤ rM −j and f K N and f E N with maxk |tk | < a sufficiently j j ≤ rM j+1 j+1 ≤ O(L )rM # (X, 0) is analytic, using Cauchy’s bound and (79) in [5], we small and ≤ j ≤ N − Because f K N j have: ∂m N# (X, 0) f Kj ∂t1 ∂tm t1 =0, tm =0 28 m! am A −|X|j m! ≤ m a A −|X|j ≤ N f Kj j (162) rM −j Then ∂m ∂t1 ∂tm ≤ ≤ X∈Sj ,X⊃B N# (X, 0) f Kj |X|j t1 =0, tm =0 X∈Sj ,X⊃B ∂m N# (X, 0) f Kj |X|j ∂t1 ∂tm X∈Sj ,X⊃B m! |X|j am ≤ n3 (d, A −|X|j t1 =0, tm =0 (163) rM −j A m!rM −j ) am So |f FN | ≤ N −1 j=0 n3 (d, B∈Bj (ΛN ) B ⊃{x1 ,x2 , xm } A m!rM −j ) am (164) ∗ By (153)-(157), we have: log + (f I N (σN ) − + f K N N )(ΛN )dµCN ≤ log (1 + F (ΛN ) (165) N) ≤ log + 2[4c−1 h2 + A−1 ]r2−N Using the Cauchy’s bound as above, we obtain: ∂m log + (f I N (σN ) − + f K N N )(ΛN )dµCN ∂t1 ∂tm m! ≤ m log + 2[4c−1 h2 + A−1 ]r2−N a t1 =0, tm =0 (166) So lim N →∞ ∂m log + ∂t1 ∂tm (f I N (σN ) − + f K N N )(ΛN )dµCN t1 =0, tm =0 =0 (167) Now let j0 be the smallest integer such that ∃B ∈ Bj0 : B ∗ ⊃ {x1 , x2 , xm } Without loosing the generality, we can assume that |x1 − x2 | = diam(x1 , xm ) For every j ≥ j0 , let Bj1 ∈ Bj be the unique j-block that contains {x1 } For any B ∈ Bj , j ≥ j0 ∗ with B ∗ ⊃ {x1 , x2 , xm }, B must be in Bj1 We have |f FN | ≤ = N −1 j=0 n3 (d, B∈Bj (ΛN ) B ∗ ⊃{x1 ,x2 , xm } N −1 j=j0 B∈Bj (ΛN ) B ∗ ⊃{x1 ,x2 , xm } 29 A m!rM −j ) am A m!rM −j n3 (d, ) am (168) Since M ≥ 2, the last part of (168) is bounded by N −1 j=j0 N −1 B∈Bj (ΛN ) B ∗ ⊃{x1 ,x2 , xm } A m!rM −j n3 (d, ) ≤ am j=j B∈Bj (ΛN ) ∗ B∈Bj1 ≤ n3 (d, N −1 (2d 2)d n3 (d, j=j0 ≤ 2d(d+1) n3 (d, Therefore, we have: |f FN | ≤ 2d(d+1) n3 (d, A m!rM −j ) am A m!rM −j ) am (169) A m!rM −j0 )2 am A m!rM −j0 )2 am (170) By the definition of j0 , we have: |x1 − x2 | ≤ d2d+1 Lj0 Because M = Lη−ǫ , we get M −j0 = L−j0 (η−ǫ) ≤ (d2d+1 )η |x1 − x2 |−η+ǫ = (d2d+1 )η diam−η+ǫ (x1 , , xm ) (171) Hence, we have: |f FN | ≤ 2d(d+1) n3 (d, A m!r diam−η+ǫ (x1 , xm ) dη 2η(d+1) )2 am (172) Using this with (167), we obtain: ∂m log f Z ′ ∂t1 ∂tm t1 =0, tm =0 Combining with (31), we get n3 (d, sufficiently large A, we have: ≤ 2d(d+1) 4n3 (d, A m!r diam−η+ǫ (x1 , xm ) dη 2η(d+1) ) am A d(d+1) 4r(d2d+1 )η )2 m G t (x1 , x2 , xm ) = ∂µk φ(xk ) t = k=1 (173) ≤ with sufficiently large A Therefore, with ∂m log f Z ′ ∂t1 ∂tm t1 =0, tm =0 (174) m! ≤ m diam−η+ǫ (x1 , xm ) a We complete the proof of Theorem Remark Actually for any N − ≥ q ≥ j0 , similarly to (169), we have N −1 j=q ≤ B∈Bj (ΛN ) B ∗ ⊃{x1 ,x2 , xm } ∂m ∂t1 ∂tm N −1 j=q n3 (d, B∈Bj (ΛN ) B ∗ ⊃{x1 ,x2 , xm } ≤ 2d(d+1)n3 (d, X∈Sj ,X⊃B A m!rM −j ) am A m!rM −q )2 am 30 N# (X, 0) f Kj |X|j t1 =0, tm =0 (175) 6.1.2 Proof of Theorem Now we fix the set {x1 , x2 , xm } Let j1 be the smallest integer such that Bj01 ⊃ {x1 , x2 , xm } Then j1 is the smallest integer which is greater than logL maxi xi ∞ We also have: j0 ≤ j1 Let q be any number such that q ≥ j1 + ≥ j0 + And let N1 , N2 be any integers such that N2 ≥ N1 > q Using the definition of j0 , we have q−1 = f F N1 j=j0 + B∈Bj (ΛN ) B ∗ ⊃{x1 ,x2 , xm } ∂m ∂t1 ∂tm N1 −1 j=q B∈Bj (ΛN ) B ⊃{x1 ,x2 , xm } ∂m ∂t1 ∂tm X∈Sj ,X⊃B N2 # (X, 0) f Kj |X|j X∈Sj ,X⊃B ∗ q−1 f F N2 = j=j0 + B∈Bj (ΛN ) B ∗ ⊃{x1 ,x2 , xm } ∂m ∂t1 ∂tm N2 −1 j=q B∈Bj (ΛN ) B ∗ ⊃{x1 ,x2 , xm } ∂m ∂t1 ∂tm t1 =0, tm =0 N2 # (X, 0) f Kj |X|j t1 =0, tm =0 (176) X∈Sj ,X⊃B N2 # (X, 0) f Kj |X|j X∈Sj ,X⊃B t1 =0, tm =0 N2 # (X, 0) f Kj |X|j t1 =0, tm =0 We also notice that: q−1 j=j0 B∈Bj (ΛN ) B ∗ ⊃{x1 ,x2 , xm } ∂m ∂t1 ∂tm q−1 = j=j0 B∈Bj (ΛN ) B ⊃{x1 ,x2 , xm } X∈Sj ,X⊃B ∂m ∂t1 ∂tm N2 # (X, 0) f Kj |X|j t1 =0, tm =0 (177) X∈Sj ,X⊃B ∗ N1 # (X, 0) f Kj |X|j t1 =0, tm =0 # because for ≤ j ≤ q − 1, f K N (X, 0) only depend on φ within X ∗ and X ∗ ⊂ Λq which is the center j q-block of ΛN1 ⊂ ΛN2 Therefore, |f F N − f F N | ≤ + N2 −1 j=q B∈Bj (ΛN ) B ∗ ⊃{x1 ,x2 , xm } N1 −1 j=q B∈Bj (ΛN ) B ∗ ⊃{x1 ,x2 , xm } ∂m ∂t1 ∂tm X∈Sj ,X⊃B N2 # (X, 0) f Kj |X|j X∈Sj ,X⊃B N1 # (X, 0) f Kj |X|j t1 =0, tm =0 (178) ∂m ∂t1 ∂tm t1 =0, tm =0 Then using (175) with µ = 1/2 instead of µ = 1/M = L−η+ǫ , we obtain: N2 −1 j=q B∈Bj (ΛN ) B ∗ ⊃{x1 ,x2 , xm } ∂m ∂t1 ∂tm X∈Sj ,X⊃B ≤ 2d(d+1) n3 (d, N2 # (X, 0) f Kj |X|j A m!r2−q )2 am 31 t1 =0, tm =0 (179) and N1 −1 j=q B∈Bj (ΛN ) B ∗ ⊃{x1 ,x2 , xm } ∂m ∂t1 ∂tm X∈Sj ,X⊃B ≤ 2d(d+1) n3 (d, N1 # (X, 0) f Kj |X|j t1 =0, tm =0 (180) A m!r2−q )2 am That means we have: |f F N2 − f F N1 | ≤ 2d(d+1) n3 (d, A m!r2−q →0 )4 am when q → ∞ Combining this with (158) and (167), we can conclude that limN →∞ m k=1 (181) ∂µk φ(xk ) t exists Remark We have N -uniformly boundedness on correlation functions and limN →∞ G t (x1 , x2 , xm ) exists Therefore the bounds are held for infinite volume limit 6.2 When f (φ) = m k=1 tk exp (i∂µk φ(xk )) Using exactly the same argument as the above subsection, we obtain these following results: Theorem 10 For any small ǫ > 0, with L, A sufficiently large (depending on ǫ), let η = min{d/2, 2} we have: m exp (i∂µk φ(xk )) t k=1 ≤ m! diam−η+ǫ (x1 , x2 ) am (182) where a depends on ǫ, L, A Theorem 11 With L, A sufficiently large, the infinite volume limit of the truncated correlation funct m tion limN →∞ exists k=1 exp (i∂µk φ(xk )) 6.3 Other cases We can consider f (φ) = m k=1 tk fk (φ)(xk ) with * tk ∈ C * xk ∈ Zd are different points * fk is bounded in the sense that there are some Mk , mk ≥ such that fk ({xk }, φ) ≤ Mk φ Φ0 + mk (183) With the same argument as above cases, we have: Theorem 12 For any small ǫ > 0, with L, A sufficiently large (depending on ǫ), let η = min{d/2, 2} we have: m fk (φ)(xk ) k=1 t ≤ m! diam−η+ǫ (x1 , x2 ) am (184) where a depends on ǫ, L, A Theorem 13 With L, A sufficiently large, the infinite volume limit of the truncated correlation funct m exists tion limN →∞ k=1 fk (φ)(xk ) 32 m In the case f = k=1 tk W0 ({xk }), with W0 ({xk }) = zW (1, {xk }) as in (42) Using the Lemma (or the lemma in [5] ), these W0 ({xk }) satisfy those above conditions The W0 ({xk }) are actually the density of the dipoles at xk used in [2] Applying theorems 12 and 13, we obtain these results: Corollary For any small ǫ > 0, with L, A sufficiently large (depending on ǫ), let η = min{d/2, 2} we have: m k=1 W0 ({xk }) t ≤ m! diam−η+ǫ (x1 , x2 ) am (185) This result somehow looks like the theorem (1.1.2) in [2] However it gives estimates for truncated correlation functions of (p ≥ 2) points instead of some estimate for only points Corollary With L, A sufficiently large, the infinite volume limit of the truncated correlation function t m exists limN →∞ k=1 W0 ({xk }) m Remark We can consider the more general form f (φ) = k=1 tk fk (φ) with * tk ∈ C * Ak ≡ suppfk are pairwise disjoint and |Ak | < ∞ * fk is bounded in the sense that there are some Mk , mk ≥ such that fk (Ak , φ) ≤ Mk φ Φ0 + mk (186) Then we still get similar results as in Theorems 12 and 13 A Kac-Siegert Transformation By expanding the exponential in (6) and carrying out the Gaussian integrals, we can rewrite Z N as   n n z  dpi (eipi ·∂φ(xi ) + e−ipi ·∂φ(xi ) )/2 dµC (φ) 0Z N = n! i=1 d−1 S n≥0 xi ∈ΛN ∩Zd   n n z dpi eipi ·∂φ(xi )  dµC (φ) =  n! i=1 d−1 S n≥0 xi ∈ΛN ∩Zd (187) n n n z = dpi ei k=1 pk ·∂φ(xk ) dµC (φ) n! i=1 d−1 S n≥0 xi ∈ΛN ∩Zd   n −1 zn (pk · ∂)(pj · ∂)C(xk , xj ) dpi exp  = n! i=1 Sd−1 d n≥0 1≤k,j≤n xi ∈ΛN ∩Z which is exactly the same as the grand canonical partition function (5) Acknowledgments This work was in partial fulfillment of the requirements for the Ph.D degree at the University at Buffalo, State University of New York The author owes deep gratitude to his Ph.D advisor Jonathan D Dimock for his continuing help and support Dimock’s prior investigations of infinite volume limit for the dipole gas [5] have served as a framework for much of the current work 33 References [1] D.C.Brydges, Lectures on the renormalisation group, in Statistical Mechanics, IAS/Park City Math Ser., volume 16 (American Mathematical Society, Providence, RI, 2009), pp 7-93 [2] D Brydges and G Keller, Correlation functions of general observables in dipole-type systems, Helv Phys Acta 67, 43-116 (1994) [3] D.C.Brydges, G Guadagni, P.K Mitter, Finite range decomposition of Gaussian processes, J Stat Phys 115, 415-449, (2004) [4] D.C Brydges, H.T Yau., Grad φ perturbations of massless Gaussian fields, Commun Math Phys 129, 351–392, (1990) [5] J Dimock, Infinite volume limit for the dipole gas, Journal of Statistical Physics, 35, 393-427, (2009) [6] J Dimock, Quantum mechanics and quantum field theory A mathematical primer., Cambridge University Press, Cambridge, 2011 [7] J.Dimock, T.R Hurd A renormalization group analysis of correlation functions for the dipole gas, J Stat Phys 66, 1277–1318, (1992) [8] J Dimock, T Hurd, Sine-Gordon revisited, Ann Henri Poincar´e 1, 499-541 (2000) [9] P Falco, Kosterlitz-Thouless transition line for the two dimensional Coulomb gas, Comm Math Phys 312, no 2, 559-609 (2012) [10] J Frohlich, Y.M Park, Correlation inequalities and the thermodynamic limit for classical and quantum continuous systems, Commun Math Phys 59, 235-266 (1978) [11] J Frohlich, T Spencer, On the statistical mechanics of classical Coulomb and dipole gases, J Stat Phys 24 , 617-701 (1981) [12] K Gawedzki and A Kupiainen, Lattice dipole gas and (∇φ)4 models at long distance, decay of correlations and scaling limit, Commun Math Phys 92, 531 (1984) [13] M Kac, On the partition function of one dimensional gas, Phys Fluids, vol 2, (1959) [14] M Reed, B Simon, Methods of modern mathematical physics, Vol IV, Academic Press (1978) [15] B Simon, Functional integration and quantum physics, Academic Press (1979) [16] A F J Siegert, Partition functions as averages of functionals of Gaussian random functions, Physica, Vol 26, 530-535, (1960) [17] G Zhao, Dipole-Dipole Correlations for the sine-Gordon Model, arXiv:1108.3232 34 ... 1) instead of only points as Theorem 1.1.2 in [2] Then we apply theorem 13 to establish the infinite volume limit for truncated correlation functions of density of dipoles (Corollary 2) For the. .. and obtain some estimates on the correlation functions as in Dimock and Hurd [7], and Brydges and Keller [2] The main result is the existence of the infinite volume limit for correlations functions, ... study the RG flow and find the stable manifold σ = σ(z) • In section we assemble the results and prove the infinite volume limit for |ΛN |−1 log f Z ′N exists • Finally in section 6, by combining