Power Series Representations for Complex Bosonic Effective Actions. III. Substitution and Fixed Point Equations
Tadeusz Balaban, Joel Feldman, Horst Knörrer, Eugene Trubowitz
aa r X i v : . [ m a t h - ph ] S e p Power Series Representations for Complex BosonicEffective Actions.
III. Substitution and Fixed Point Equations
Tadeusz Balaban , Joel Feldman ∗ , Horst Kn¨orrer , and EugeneTrubowitz Department of MathematicsRutgers, The State University of New [email protected] ∼ feldman/ ∼ knoerrer/ October 11, 2018
Abstract
In [3, 4, 5] we developed a polymer–like expansion that applies when the (ef-fective) action in a functional integral is an analytic function of the fields beingintegrated. Here, we develop methods to aid the application of this techniquewhen the method of steepest descent is used to analyze the functional integral.We develop a version of the Banach fixed point theorem that can be used to ∗ Research supported in part by the Natural Sciences and Engineering Research Council ofCanada and the Forschungsinstitut f¨ur Mathematik, ETH Z¨urich. onstruct and control the critical fields, as analytic functions of external fields,and substitution formulae to control the change in norms that occurs whenone replaces the integration fields by the sum of the critical fields and thefluctuation fields. ontents Introduction
In [3, 4, 5], we developed a power series representation, norms and estimates for aneffective action of the form ln R e f ( α , ··· ,α s ; z ∗ ,z ) dµ ( z ∗ , z ) R e f (0 , ··· , z ∗ ,z ) dµ ( z ∗ , z )Here, f ( α , · · · , α s ; z ∗ , z ) is an analytic function of the complex fields α ( x ), · · · , α s ( x ), z ∗ ( x ), z ( x ) indexed by x in a finite set X , and dµ ( z ∗ , z ) is a compactlysupported product measure. This framework has been used in [6].In [8, 9] we combine these power series methods with the technique of the blockspin renormalization group for functional integrals [12, 1, 11, 2, 10] to see, for amany particle system of weakly interacting Bosons in three space dimensions, theformation of a potential well of the type that typically leads to symmetry breaking inthe thermodynamic limit. (For an overview, see [7].) A basic ingredient of this blockspin/functional integral approach is a stationary phase argument for the effectiveactions. For this, it is necessary to construct and analyze “critical fields” at eachstep. These critical fields are themselves functions of some external fields. The“background fields” of the block spin approach arise as compositions of critical fieldsat several renormalization group steps and are also functions of some external fields.In our construction [8, 9], the “background fields” and “critical fields” are analyticmaps that are defined on a neighbourhood of the origin in an appropriate Hilbertspace of fields and that take values in another Hilbert space of fields. We call suchobjects “field maps”. See Definition 2.3, where we also generalize the definition ofthe norm of a (complex valued) analytic function of fields [4, Definition 2.6] to fieldmaps.In § h ( α , · · · , α s ) = h (cid:16) A ( α , · · · , α s ) , · · · , A r ( α , · · · , α s ) (cid:17) in terms of bounds on h and the A j ’s. Here, h is a function of r fields and A , · · · , A r are field maps. See Proposition 3.2 and Corollary 3.3.The critical fields for each block spin renormalization group transformation arecritical configurations for some action. The equations that determine these criticalconfigurations can be expressed as (systems of) implicit equations of the type γ = F ( α , · · · , α s ; γ )which have to be solved for γ as a function α , · · · , α s . In §
4, we prove the existenceand uniqueness of, and bounds on, solutions to systems of equations of that type.See Proposition 4.1. 4
Field Maps
For an abstract framework, we consider analytic functions f ( α , · · · , α s ) of the com-plex fields α , · · · , α s (none of which are “history” or source fields, in the terminologyof [4]) on a finite set X . Here are some associated definitions and notation from [4]. Definition 2.1 ( n –tuples) . (a) Let n ∈ Z with n ≥ ~ x = ( x , · · · , x n ) ∈ X n be an ordered n –tuple ofpoints of X . We denote by n ( ~ x ) = n the number of components of ~ x . Set α ( ~ x ) = α ( x ) · · · α ( x n ) . If n ( ~ x ) = 0, then α ( ~ x ) = 1.(b) For each s ∈ N , we denote X ( s ) = [ n , ··· ,n s ≥ X n × · · · × X n s If ( ~ x , · · · , ~ x s − ) ∈ X ( s − then ( ~ x , · · · , ~ x s − , − ) denotes the element of X ( s ) having n ( ~ x s ) = 0. In particular, X = {−} and α ( − ) = 1.(c) We define the concatenation of ~ x = ( x , · · · , x n ) ∈ X n and ~ y = ( y , · · · , y m ) ∈ X m to be ~ x ◦ ~ y = (cid:0) x , · · · , x n , y , · · · , y m ) ∈ X n + m For ( ~ x , · · · , ~ x s ) , ( ~ y , · · · , ~ y s ) ∈ X ( s ) ( ~ x , · · · , ~ x s ) ◦ ( ~ y , · · · , ~ y s ) = ( ~ x ◦ ~ y , · · · , ~ x s ◦ ~ y s ) Definition 2.2 (Coefficient Systems) . (a) A coefficient system of length s is a function a ( ~ x , · · · , ~ x s ) which assigns acomplex number to each ( ~ x , · · · , ~ x s ) ∈ X ( s ) . It is called symmetric if, for each1 ≤ j ≤ s , a ( ~ x , · · · , ~ x s ) is invariant under permutations of the components of ~ x j .(b) Let f ( α , · · · , α s ) be a function which is defined and analytic on a neighbour-hood of the origin in C s | X | . Then f has a unique expansion of the form f ( α , · · · , α s ) = X ( ~ x , ··· ,~ x s ) ∈ X ( s ) a ( ~ x , · · · , ~ x s ) α ( ~ x ) · · · α s ( ~ x s ) We distinguish between X n × · · · × X n s and X n + ··· + n s . We use X n × · · · × X n s as the setof possible arguments for α ( ~ x ) · · · α s ( ~ x s ), while X n + ··· + n s is the set of possible arguments for α ( ~ x ◦ · · · ◦ ~ x s ), where ◦ is the concatenation operator of part (c). a ( ~ x , · · · , ~ x s ) a symmetric coefficient system. This coefficient system iscalled the symmetric coefficient system of f .We assume that we are given a metric d on a finite set X and constant weightfactors κ , · · · , κ s . In this environment [4, Definition 2.6], for the norm of the function f ( α , · · · , α s ) = X ( ~ x , ··· ,~ x s ) ∈ X ( s ) a ( ~ x , · · · , ~ x s ) α ( ~ x ) · · · α s ( ~ x s )with a ( ~ x , · · · , ~ x s ) a symmetric coefficient system, simplifies to k f k w = (cid:12)(cid:12) a ( − ) (cid:12)(cid:12) + X n , ··· ,ns ≥ n ··· + ns ≥ max x ∈ X max ≤ j ≤ snj =0 max ≤ i ≤ n j X ~ x ℓ ∈ Xnℓ ≤ ℓ ≤ s ( ~ x j ) i = x (cid:12)(cid:12) a ( ~ x , · · · , ~ x s ) (cid:12)(cid:12) κ n · · · κ n s s e τ d ( ~ x , ··· ,~ x s ) (2.1)where τ d ( ~ x , · · · , ~ x s ) denotes the length of the shortest tree in X whose set of verticescontains all of the points in the ~x j ’s. The family of functions w ( ~ x , · · · , ~ x s ) = κ n ( ~ x )1 · · · κ ( ~ x s ) s e τ d ( ~ x , ··· ,~ x s ) is called the weight system with metric d that associates the weight factor κ j to thefield α j .We need to extend these definitions to functions A ( α , · · · , α s ) that take val-ues in C X , rather than C . That is, which map fields α , · · · , α s to another field A ( α , · · · , α s ). A trivial example would be A ( α )( x ) = α ( x ). Definition 2.3. (a) An s –field map kernel is a function A : ( x ; ~ x , · · · , ~ x s ) ∈ X × X ( s ) A ( x ; ~ x , · · · , ~ x s ) ∈ C which obeys A ( x ; − , · · · , − ) = 0 for all x ∈ X .(b) If A is an s –field map kernel, we define the “ s –field map” ( α , · · · , α s ) A ( α , · · · , α s ) by A ( α , · · · , α s )( x ) = X ( ~ x , ··· ,~ x s ) ∈ X ( s ) A ( x ; ~ x , · · · , ~ x s ) α ( ~ x ) · · · α s ( ~ x s )6c) We define the norm ||| A ||| w of the s –field map kernel A by ||| A ||| w = X n , ··· ,ns ≥ n ··· + ns ≥ (cid:13)(cid:13) A (cid:13)(cid:13) w ; n , ··· ,n s where (cid:13)(cid:13) A (cid:13)(cid:13) w ; n , ··· ,n s = max (cid:8) L ( A ; w ; n , · · · , n s ) , R ( A ; w ; n , · · · , n s ) (cid:9) and L ( A ; w ; n , · · · , n s ) = max x ∈ X X ~ x ℓ ∈ Xnℓ ≤ ℓ ≤ s (cid:12)(cid:12) A ( x ; ~ x , · · · , ~ x s ) (cid:12)(cid:12) κ n · · · κ n s s e τ d ( x ,~ x , ··· ,~ x s ) R ( A ; w ; n , · · · , n s ) = max x ′ ∈ X max ≤ j ≤ snj =0 max ≤ i ≤ n j X x ∈ X X ~ x ℓ ∈ Xnℓ ≤ ℓ ≤ s ( ~ x j ) i = x ′ (cid:12)(cid:12) A ( x ; ~ x , · · · , ~ x s ) (cid:12)(cid:12) κ n · · · κ n s s e τ d ( x ,~ x , ··· ,~ x s ) We also denote the norm of the corresponding s –field map A ( α , · · · , α s ) by ||| A ||| w . Remark 2.4.
We associate to each s –field map kernel A the analytic function f A ( β ; α , · · · , α s ) = X x ∈ X β ( x ) A ( α , · · · , α s )( x )= X ( ~ x , ··· ,~ x s ) ∈ X ( s ) x ∈ X A ( x ; ~ x , · · · , ~ x s ) β ( x ) α ( ~ x ) · · · α s ( ~ x s )Denote by ˆ w the weight system with metric d that associates the weight factor κ j to α j , for each 1 ≤ j ≤ s , and the weight factor 1 to β . Then k f A k ˆ w = ||| A ||| w Lemma 2.5 (Young’s Inequality) . Let d , · · · , d s ≥ be integers.(a) Let f ( α , · · · , α s ) be a function which is defined and analytic on a neighbourhoodof the origin in C s | X | and is of degree at least d i in the field α i . Furthermore let p , · · · , p s ∈ (0 , ∞ ] be such that s P j =1 d j p j = 1 . Then, for all fields α , · · · , α s suchthat | α j ( x ) | ≤ κ j for all x ∈ X and ≤ j ≤ s , (cid:12)(cid:12) f ( α , · · · , α s ) (cid:12)(cid:12) ≤ k f k w s Q j =1 (cid:0) κ j k α j k p j (cid:1) d j where k α k p = (cid:0) P x ∈ X | α ( x ) | p (cid:1) /p denotes the L p norm of α . b) Let ( α , · · · , α s ) A ( α , · · · , α s ) be an s –field map which is of degree at least d i in the field α i . Furthermore let p, p , · · · , p s ∈ (0 , ∞ ] be such that s P j =1 d j p j = p .Then, for fields α , · · · , α s such that | α j ( x ) | ≤ κ j for all x ∈ X and ≤ j ≤ s ,the L p norm of the field A ( α , · · · , α s ) is bounded by (cid:13)(cid:13) A ( α , · · · , α s ) (cid:13)(cid:13) p ≤ ||| A ||| w s Q j =1 (cid:0) κ j k α j k p j (cid:1) d j In particular max x ∈ X (cid:12)(cid:12) A ( α , · · · , α s )( x ) (cid:12)(cid:12) ≤ ||| A ||| w Proof. (a) By the definition (2.1) of k f k w , we may assume that f is of the form f ( α , · · · , α s ) = X ~ x ℓ ∈ Xnℓ ≤ ℓ ≤ s a ( ~ x , · · · , ~ x s ) α ( ~ x ) · · · α s ( ~ x s )with a symmetric coefficient a and n ℓ ≥ d ℓ . Now apply Lemma A.1 with K = a s Q j =1 κ d j j , where we use the L p j norm for the first d j components of the variable ~x j ,and the L ∞ norm for the last n j − d j components of this variable.(b) As in Remark 2.4 set f A ( β ; α , · · · , α s ) = X x ∈ X β ( x ) A ( α , · · · , α s )( x )As in [13, Theorem 4.2] choose β ( x ) = e − iθ ( x ) | A ( α , · · · , α s )( x ) | p/p ′ where θ ( x ) is defined by A ( α , · · · , α s )( x ) = e iθ ( x ) | A ( α , · · · , α s )( x ) | and p + p ′ = 1.By part (a) and Remark 2.4 (cid:13)(cid:13) A ( α , · · · , α s ) (cid:13)(cid:13) pp = (cid:12)(cid:12) f A ( β ; α , · · · , α s ) (cid:12)(cid:12) ≤ ||| A ||| w k β k p ′ s Q j =1 (cid:0) κ j k α j k p j (cid:1) d j = ||| A ||| w (cid:13)(cid:13) A ( α , · · · , α s ) (cid:13)(cid:13) p/p ′ p s Q j =1 (cid:0) κ j k α j k p j (cid:1) d j emark 2.6. A linear map L : C X → C X can be thought of as a 1–field map kernel.The relation between the norm ||| L ||| w as a field map kernel and the norm ||| L ||| as in[4, Definition A.1] is ||| L ||| w = κ ||| L ||| The field L ( α ) is L ( α )( x ) = X y ∈ X L ( x , y ) α ( y ) Remark 2.7.
In Definition 2.3, we have assumed, for simplicity, that the field map A maps fields α , · · · , α s on a set X to a field A ( α , · · · , α s ) on the same set X . Wewill apply this definition and the results later in this paper when the input fields α , · · · , α s are defined on a subset X ⊂ X and the output field A ( α , · · · , α s ) isdefined on a, possibly different, subset X ⊂ X . We extend Definition 2.3 and the re-sults later in this paper to cover this setting by viewing α , · · · , α s and A ( α , · · · , α s )to be fields on X — set α , · · · , α s to zero on X \ X and A ( α , · · · , α s ) to zero on X \ X . 9 Substitution
We now proceed to prove bounds on compositions like˜ h ( α , · · · , α s ) = h (cid:16) A ( α , · · · , α s ) , · · · , A r ( α , · · · , α s ) (cid:17) in terms of bounds on h and the A j ’s. Lemma 3.1.
Let λ , · · · , λ s be constant weight factors and let w δ be the weightsystem with metric d that associates the weight factor κ j to α j and λ j to a field δ j .Fix any σ ≥ and let w σ be the weight system with metric d that associates theweight factor κ j + σλ j to α j .(a) Let f ( α , · · · , α s ) be an analytic function on a neighbourhood of the origin in C s | X | . Set δf (cid:0) α , · · · , α s , δ , · · · , δ s (cid:1) = f (cid:0) α + δ , · · · , α s + δ s (cid:1) − f (cid:0) α , · · · , α s (cid:1) Then k δf k w δ ≤ σ k f k w σ More generally, if p ∈ N and δf ( ≥ p ) (cid:0) α , · · · , α s , δ , · · · , δ s (cid:1) is the part of δf thatis of degree at least p in (cid:0) δ , · · · , δ s (cid:1) , then k δf ( ≥ p ) k w δ ≤ σ p k f k w σ (b) Let A be an s –field map and define the s –field map δA by δA (cid:0) α , · · · , α s , δ , · · · , δ s (cid:1) = A (cid:0) α + δ , · · · , α s + δ s (cid:1) − A (cid:0) α , · · · , α s (cid:1) Then ||| δA ||| w δ ≤ σ ||| A ||| w σ Proof.
Let a ( ~ x , · · · , ~ x s ) be a symmetric coefficient system for f . Since a is invariantunder permutation of its ~ x j components, f (cid:0) α + δ , · · · , α s + δ s (cid:1) = X ( ~ x , ··· ,~ x s ) ∈ X ( s ) a ( ~ x , · · · , ~ x s ) ( α + δ )( ~ x ) · · · ( α s + δ s )( ~ x s )= X ( ~ x , ··· ,~ x s ) ∈ X ( s )( ~ y , ··· ,~ y s ) ∈ X ( s ) a ( ~ x ◦ ~ y , · · · , ~ x s ◦ ~ y s ) s Y j =1 (cid:0) n ( ~ x j )+ n ( ~ y j ) n ( ~ y j ) (cid:1) α j ( ~ x j ) δ j ( ~ y j )10o that δa p ( ~ x , · · · x s ; ~ y , · · · , y s )= χ (cid:0) n ( ~ y ) + · · · + n ( ~ y s ) ≥ p (cid:1) a ( ~ x ◦ ~ y , · · · , ~ x s ◦ ~ y s ) s Y j =1 (cid:0) n ( ~ x j )+ n ( ~ y j ) n ( ~ y j ) (cid:1) is a symmetric coefficient system for δf ( ≥ p ) . Of course δf = δf ( ≥ . By definition k δf ( ≥ p ) k w δ = X k , ··· ,ks ≥ ℓ , ··· ,ℓs ≥ ℓ ··· + ℓs ≥ p max x ∈ X max ≤ j ≤ skj + ℓj =0 max ≤ i ≤ k j + ℓ j X ~ x m ∈ Xkm~ y m ∈ Xℓm ( ~ x j ◦ ~ y j ) i = x (cid:12)(cid:12) δa ( ~ x , · · · x s ; ~ y , · · · , y s ) (cid:12)(cid:12) e τ d ( ~ x ◦ ~ y , ··· ,~ x s ◦ ~ y s ) s Y j =1 κ k j j λ ℓ j j = X k , ··· ,ks ≥ ℓ , ··· ,ℓs ≥ ℓ ··· + ℓs ≥ p ω ( k + ℓ , · · · , k s + ℓ s ) s Y j =1 (cid:0) k j + ℓ j ℓ j (cid:1) s Y j =1 κ k j j λ ℓ j j = X n , ··· ,nsn ··· + ns ≥ p ω ( n , · · · , n s ) c p ( n , · · · , n s )where ω ( n , · · · , n s ) = max ≤ j ≤ snj =0 max ≤ i ≤ n j X ~ z p ∈ Xnp ( ~ z j ) i = x (cid:12)(cid:12) a ( ~ z , · · · , ~ z s ) (cid:12)(cid:12) e τ d ( ~ z , ··· ,~ z s ) and c p ( n , · · · , n s ) = X kj,ℓj ≥ kj + ℓj = njℓ ··· + ℓs ≥ p s Y j =1 (cid:0) n j ℓ j (cid:1) κ k j j λ ℓ j j ≤ σ p s Y j =1 ( κ j + σλ j ) n j For the last inequality, apply the binomial expansion to each ( κ j + σλ j ) n j and comparethe two sides of the inequality term by term. This proves part (a). Part (b) followsby Remark 2.4. Proposition 3.2.
Let h ( γ , · · · , γ r ) be an analytic function on a neighbourhood ofthe origin in C r | X | , and let A j , δA j , ≤ j ≤ r be s –field maps. Furthermore let λ , · · · , λ r be constant weight factors and let w λ be the weight system with metric d that associates the weight factor λ j to the field γ j . a) Set ˜ h ( α , · · · , α s ) = h (cid:16) A ( α , · · · , α s ) , · · · , A r ( α , · · · , α s ) (cid:17) Assume that ||| A j ||| w ≤ λ j for each ≤ j ≤ r . Then k ˜ h k w ≤ k h k w λ (b) Assume that there is a σ ≥ such that ||| A j ||| w + σ ||| δA j ||| w ≤ λ j for all ≤ j ≤ r . Set f δh ( α , · · · , α s )= h (cid:16) A ( α , · · · , α s )+ δA ( α , · · · , α s ) , · · · , A r ( α , · · · , α s )+ δA r ( α , · · · , α s ) (cid:17) − h (cid:16) A ( α , · · · , α s ) , · · · , A r ( α , · · · , α s ) (cid:17) More generally, if p ∈ N and δh ( ≥ p ) is the part of δh ( γ , · · · , γ r ; δ , · · · , δ r ) = h ( γ + δ , · · · , γ r + δ r ) − h ( γ , · · · , γ r ) that is of degree at least p in ( δ , · · · , δ r ) , set f δh ( ≥ p ) ( α , · · · , α s )= δh ( ≥ p ) (cid:16) A ( α , · · · , α s ) , · · · , A r ( α , · · · , α s ) ; δA ( α , · · · , α s ) , · · · , δA r ( α , · · · , α s ) (cid:17) Then k f δh k w ≤ σ k h k w λ (cid:13)(cid:13)f δh ( ≥ p ) (cid:13)(cid:13) w ≤ σ p k h k w λ Proof. (a) Let a ( ~ y , · · · , ~ y r ) be a symmetric coefficient system for h . Define, for each n ( ~ x i ) = n i ≥
0, 1 ≤ i ≤ s ,˜ a ( ~ x , · · · , ~ x s )= X m , ··· ,m r ≥ X ni,j,k ≥ ≤ i ≤ s, ≤ j ≤ r, ≤ k ≤ mj with Σ j,kni,j,k = ni X ~ y ∈ Xm ... ~ y r ∈ Xmr a ( ~ y , · · · , ~ y r ) r Y j =1 h m j Q k =1 A j (( ~ y j ) k ; ~ x ,j,k , · · · , ~ x s,j,k ) i ~ y j ) k is the k th component of ~ y j and the ~ x ijk ’s are determined by the conditionsthat n ( ~ x ijk ) = n ijk and ~ x i = ◦ j,k ~ x ijk = ~ x i ◦ ~ x i ◦ · · · ◦ ~ x i m ◦ ~ x i ◦ · · · ◦ ~ x i m ◦ · · · ◦ ~ x irm r (3.1)Then ˜ a ( ~ x , · · · , ~ x s ) is a (not necessarily symmetric) coefficient system for ˜ h . Since τ d (cid:0) supp( ~ x , · · · , ~ x s ) (cid:1) ≤ τ d (cid:0) supp( ~ y , · · · , ~ y s ) (cid:1) + X ≤ j ≤ r ≤ k ≤ mj τ d (cid:0) supp(( ~ y j ) k , ~ x ,j,k , · · · , ~ x s,j,k ) (cid:1) we have w ( ~ x , · · · , ~ x s ) (cid:12)(cid:12) ˜ a ( ~ x , · · · , ~ x s ) (cid:12)(cid:12) ≤ X m , ··· ,m r ≥ X ni,j,k ≥ ≤ i ≤ s, ≤ j ≤ r, ≤ k ≤ mj with Σ j,kni,j,k = ni X ~ y ∈ Xm ... ~ y r ∈ Xmr w λ ( ~ y , · · · , ~ y r ) (cid:12)(cid:12) a ( ~ y , · · · , ~ y r ) (cid:12)(cid:12) r Y j =1 h m j Q k =1 B j (( ~ y j ) k ; ~ x ,j,k , · · · , ~ x s,j,k ) i (3.2)where B j ( y ; ~ x ′ , · · · , ~ x ′ s ) = λ j | A j ( y ; ~ x ′ , · · · , ~ x ′ s ) | κ n ( ~ x ′ )1 · · · κ n ( ~ x ′ s ) s e τ d (supp( y ,~ x ′ , ··· ,~ x ′ s )) We first observe that when ~ x = · · · = ~ x s = − , we have ˜ a ( − , · · · , − ) = a ( − , · · · , − ) so that the corresponding contributions to k ˜ h k w and k h k w λ are identical.Therefore we may assume, without loss of generality, that h (0 , · · · ,
0) = 0.We are to bound k ˜ h k w = X n , ··· ,ns ≥ n ··· + ns ≥ max x ∈ X max ≤ ¯ ≤ sn ¯ =0 max ≤ ¯ ı ≤ n ¯ X ( ~ x , ··· ,~ x s ) ∈ Xn ×···× Xns ( ~ x ¯ )¯ ı = x w ( ~ x , · · · , ~ x s ) (cid:12)(cid:12) ˜ a ( ~ x , · · · , ~ x s ) (cid:12)(cid:12) First fix any n , · · · , n s ≥ n + · · · + n s ≥
1. We claim thatmax x ∈ X max ≤ ¯ ≤ sn ¯ =0 max ≤ ¯ ı ≤ n ¯ X ( ~ x , ··· ,~ x s ) ∈ Xn ×···× Xns ( ~ x ¯ )¯ ı = x w ( ~ x , · · · , ~ x s ) (cid:12)(cid:12) ˜ a ( ~ x , · · · , ~ x s ) (cid:12)(cid:12) ≤ X m , ··· ,m r ≥ k w λ a (cid:13)(cid:13) m , ··· ,m r X ni,j,k ≥ ≤ i ≤ s, ≤ j ≤ r, ≤ k ≤ mj with Σ j,kni,j,k = ni Y ≤ j ≤ r ≤ k ≤ mj h λ j (cid:13)(cid:13) A j (cid:13)(cid:13) w ; n ,j,k , ··· ,n s,j,k i (3.3)13ere, as in [4, Definition 2.6], k b k m , ··· ,m r = max y ∈ X max ≤ j ≤ rmj =0 max ≤ i ≤ m j X ~ y ℓ ∈ Xmℓ ≤ ℓ ≤ r ( ~ y j ) i = y (cid:12)(cid:12) b ( ~ y , · · · , ~ y r ) (cid:12)(cid:12) To prove (3.3), fix any x ∈ X and assume, without loss of generality that n ≥ = ¯ ı = 1. By (3.2), (the meaning of the ˆ , ˆ k introduced after the “=” below isexplained immediately following this string of inequalities) X ( ~ x , ··· ,~ x s ) ∈ Xn ×···× Xns ( ~ x x w ( ~ x , · · · , ~ x s ) (cid:12)(cid:12) ˜ a ( ~ x , · · · , ~ x s ) (cid:12)(cid:12) ≤ X ( ~ x , ··· ,~ x s ) ∈ Xn ×···× Xns ( ~ x x X m , ··· ,m r ≥ X ni,j,k ≥ ≤ i ≤ s, ≤ j ≤ r, ≤ k ≤ mj with Σ j,kni,j,k = ni X ~ y ∈ Xm ... ~ y r ∈ Xmr w λ ( ~ y , · · · , ~ y r ) (cid:12)(cid:12) a ( ~ y , · · · , ~ y r ) (cid:12)(cid:12) r Y j =1 h m j Q k =1 B j (( ~ y j ) k ; ~ x ,j,k , · · · , ~ x s,j,k ) i = X m , ··· ,m r ≥ X ni,j,k ≥ ≤ i ≤ s, ≤ j ≤ r, ≤ k ≤ mj with Σ j,kni,j,k = ni X ~ x i,j,k ∈ Xni,j,k for1 ≤ i ≤ s, ≤ j ≤ r, ≤ k ≤ mj with ( ~ x , ˆ , ˆ k )1= x X ~ y ∈ Xm ... ~ y r ∈ Xmr w λ ( ~ y , · · · , ~ y r ) (cid:12)(cid:12) a ( ~ y , · · · , ~ y r ) (cid:12)(cid:12) r Y j =1 h m j Q k =1 B j (( ~ y j ) k ; ~ x ,j,k , · · · , ~ x s,j,k ) i ≤ X m , ··· ,m r ≥ X ni,j,k ≥ ≤ i ≤ s, ≤ j ≤ r, ≤ k ≤ mj with Σ j,kni,j,k = ni X ~ x i, ˆ , ˆ k ∈ Xni, ˆ , ˆ k for1 ≤ i ≤ s with ( ~ x , ˆ , ˆ k )1= x X ~ y ∈ Xm ... ~ y r ∈ Xmr w λ ( ~ y , · · · , ~ y r ) (cid:12)(cid:12) a ( ~ y , · · · , ~ y r ) (cid:12)(cid:12) B ˆ (( ~ y ˆ ) ˆ k ; ~ x , ˆ , ˆ k , · · · , ~ x s, ˆ , ˆ k ) Y ≤ j ≤ r ≤ k ≤ mj ( j,k ) =(ˆ , ˆ k ) h λ j L (cid:0) A j ; w ; (cid:8) n i,j,k (cid:9) ≤ i ≤ s (cid:1)i ≤ X m , ··· ,m r ≥ X ni,j,k ≥ ≤ i ≤ s, ≤ j ≤ r, ≤ k ≤ mj with Σ j,kni,j,k = ni X ~ x i, ˆ , ˆ k ∈ Xni, ˆ , ˆ k for1 ≤ i ≤ s with ( ~ x , ˆ , ˆ k )1= x X y ∈ X k w λ a (cid:13)(cid:13) m , ··· ,m r B ˆ ( y ; ~ x , ˆ , ˆ k , · · · , ~ x s, ˆ , ˆ k ) Y ≤ j ≤ r ≤ k ≤ mj ( j,k ) =(ˆ , ˆ k ) h λ j L (cid:0) A j ; w ; (cid:8) n i,j,k (cid:9) ≤ i ≤ s (cid:1)i X m , ··· ,m r ≥ k w λ a (cid:13)(cid:13) m , ··· ,m r X ni,j,k ≥ ≤ i ≤ s, ≤ j ≤ r, ≤ k ≤ mj with Σ j,kni,j,k = ni λ ˆ R (cid:0) A ˆ ; w ; (cid:8) n i, ˆ , ˆ k (cid:9) ≤ i ≤ s (cid:1)Y ≤ j ≤ r ≤ k ≤ mj ( j,k ) =(ˆ , ˆ k ) h λ j L (cid:0) A j ; w ; (cid:8) n i,j,k (cid:9) ≤ i ≤ s (cid:1)i ≤ X m , ··· ,m r ≥ k w λ a (cid:13)(cid:13) m , ··· ,m r X ni,j,k ≥ ≤ i ≤ s, ≤ j ≤ r, ≤ k ≤ mj with Σ j,kni,j,k = ni Y ≤ j ≤ r ≤ k ≤ mj h λ j (cid:13)(cid:13) A j (cid:13)(cid:13) w ; n ,j,k , ··· ,n s,j,k i Here, for each (cid:8) n ,j,k (cid:9) ≤ j ≤ r ≤ k ≤ mj , the pair (ˆ , ˆ k ) is the first ( j, k ), using the lexicographicalordering of (3.1), for which n ,j,k = 0. a A j m ~ y m j ( ~ y j ) k m r ~ x jk n ijk ~ x ijk ~ x sjk Having completed the proof of (3.3), we now have, recalling the hypothesis thateach ||| A j ||| w ≤ λ j , k ˜ h k w ≤ X n , ··· ,ns ≥ n ··· + ns ≥ X m , ··· ,m r ≥ k w λ a (cid:13)(cid:13) m , ··· ,m r X ni,j,k ≥ ≤ i ≤ s, ≤ j ≤ r, ≤ k ≤ mj with Σ j,kni,j,k = ni Y ≤ j ≤ r ≤ k ≤ mj h λ j (cid:13)(cid:13) A j (cid:13)(cid:13) w ; n ,j,k , ··· ,n s,j,k i ≤ X m , ··· ,m r ≥ k w λ a (cid:13)(cid:13) m , ··· ,m r X ni,j,k ≥ ≤ i ≤ s, ≤ j ≤ r, ≤ k ≤ mj Y ≤ j ≤ r ≤ k ≤ m j h λ j (cid:13)(cid:13) A j (cid:13)(cid:13) w ; n ,j,k , ··· ,n s,j,k i ≤ X m , ··· ,m r ≥ k w λ a (cid:13)(cid:13) m , ··· ,m r = k h k w λ w δ be the weight system with metric d that associates the weight factor ||| A j ||| w to γ j and the weight factor ||| δA j ||| w to δ j . By part (a) of Lemma 3.1, with f → hs → rα j with weight κ j → γ j with weight ||| A j ||| w δ j with weight λ j → δ j with weight ||| δA j ||| w we have k δh k w δ ≤ σ k h k w λ k δh ( ≥ p ) k w δ ≤ σ p k h k w λ Now f δh and f δh ( ≥ p ) are obtained from δh and δh ( ≥ p ) , respectively, by the substitutions γ j = A j ( α , · · · , α s ) δ j = δA j ( α , · · · , α s )and the statement follows by part (a). Corollary 3.3.
Let B be an r –field map and let A j , ≤ j ≤ r , be s –field maps.Define the s –field map ˜ B by ˜ B ( α , · · · , α s ) = B (cid:16) A ( α , · · · , α s ) , · · · , A r ( α , · · · , α s ) (cid:17) Furthermore let λ , · · · , λ r be constant weight factors and let w λ be the weight systemwith metric d that associates the weight factor λ j to the j th field of B . Assume that ||| A j ||| w ≤ λ j for each ≤ j ≤ r . Then ||| ˜ B ||| w ≤ ||| B ||| w λ Proof.
This follows from Proposition 3.2 and Remark 2.4.
Definition 3.4.
Denote by w κ,λ the weight system with metric d that associates theconstant weight factor κ i to the field α i and the constant weight factor λ j to the field γ j . Let B ( ~α, ~γ ) be an ( s + r )–field map with ||| B ||| w κ,λ < ∞ .(a) Set, for each r–tuple of nonnegative integers n s +1 , · · · , n s + r , B n s +1 , ··· ,n s + r ( x ; ~ x , · · · , ~ x s + r )= ( B ( x ; ~ x , · · · , ~ x s + r ) if n ( ~ x s + j ) = n s + j for all 1 ≤ j ≤ r B = X n s +1 , ··· ,n s + r ≥ B n s +1 , ··· ,n s + r and ||| B ||| w κ,λ = X n s +1 , ··· ,n s + r ≥ ||| B n s +1 , ··· ,n s + r ||| w κ,λ B is said to have minimum degree at least d min and maximum degree at most d max ≤ ∞ in its last r arguments if B n s +1 , ··· ,n s + r = 0 unless d min ≤ n s +1 + · · · + n s + r ≤ d max Set ||| B ||| ′ w κ,λ = X n s +1 , ··· ,n s + r ≥ (cid:0) n s +1 + · · · + n s + r (cid:1) ||| B n s +1 , ··· ,n s + r ||| w κ,λ Think of ||| B ||| ′ w κ,λ as a bound on the derivative of B ( ~α, ~γ ) with respect to ~γ .See Lemma 3.7.(b) Denote by B the Banach space of all r –tuples ~ Γ = (Γ , · · · , Γ r ) of s –field mapswith the norm k ~ Γ k = max ≤ j ≤ r λ j ||| Γ j ||| w Also, for each ρ >
0, denote by B ρ , the closed ball in B of radius ρ .(c) For each r –tuple ~ Γ ∈ B , we define the s –field map ˜ B ( ~ Γ) by (cid:0) ˜ B ( ~ Γ) (cid:1) ( ~α ) = B (cid:0) ~α, ~ Γ( ~α ) (cid:1) Remark 3.5.
Let B be an ( s + r )–field map with minimum degree at least d min andmaximum degree at most d max < ∞ in its last r arguments.(a) d min ||| B ||| w κ,λ ≤ ||| B ||| ′ w κ,λ ≤ d max ||| B ||| w κ,λ (b) If d min = d max = 1, B is said to be linear. In this case, for any fixed α , · · · , α s ,the map ( γ , · · · , γ s ) B ( α , · · · , α s , γ , · · · γ r )is linear and ||| B ||| ′ w κ,λ = ||| B ||| w κ,λ Example 3.6.
A simple example with s = 0 and r = 1 is the truncated exponential B (cid:0) γ (cid:1) ( x ) = E n (cid:0) aγ ( x ) (cid:1) where E n ( z ) = ∞ X ℓ = n ℓ ! z ℓ a is a constant. In this example, B is a local function of γ , so that all of thekernels of B are just delta functions. Hence ||| B ||| w κ,λ = ∞ X ℓ = n ℓ ! a ℓ λ ℓ = E n ( aλ ) ≤ a n λ n n ! e aλ ||| B ||| ′ w κ,λ = ∞ X ℓ = n ℓ − a ℓ λ ℓ = aλE n − ( aλ ) ≤ a n λ n ( n − e aλ Lemma 3.7.
Let B be an ( s + r ) –field map with ||| B ||| ′ w κ,λ < ∞ . Assume that B hasminimum degree at least d min in its last r arguments. Then, for each ~ Γ , ~ Γ ′ ∈ B , ||| ˜ B ( ~ Γ) − ˜ B ( ~ Γ ′ ) ||| w ≤ k ~ Γ − ~ Γ ′ k max (cid:8) k ~ Γ k , k ~ Γ ′ k (cid:9) d min − ||| B ||| ′ w κ,λ Proof.
Write B = X ns +1 , ··· ,ns + r ≥ ns +1+ ··· + ns + r ≥ d min B n s +1 , ··· ,n s + r as in Definition 3.4. Since ||| B ||| w κ,λ = X ns +1 , ··· ,ns + r ≥ ns +1+ ··· + ns + r ≥ d min ||| B n s +1 , ··· ,n s + r ||| w κ,λ ||| B ||| ′ w κ,λ = X ns +1 , ··· ,ns + r ≥ ns +1+ ··· + ns + r ≥ d min ||| B n s +1 , ··· ,n s + r ||| ′ w κ,λ we may assume, without loss of generality, that at most one B n s +1 , ··· ,n s + r is nonvan-ishing. By renaming the γ fields and changing the value of r , we may assume that n s +1 = · · · = n s + r = 1. Then B (cid:0) ~α, γ , · · · γ r ) is multilinear in γ , · · · , γ r so that˜ B ( ~ Γ)( ~α ) − ˜ B ( ~ Γ ′ )( ~α ) = B (cid:0) ~α, Γ ( ~α ) , · · · , Γ r ( ~α ) (cid:1) − B (cid:0) ~α, Γ ′ ( ~α ) , · · · , Γ ′ r ( ~α ) (cid:1) = r X j =1 B (cid:16) ~α , Γ ( ~α ) , · · · , Γ j − ( ~α ) , Γ j ( ~α ) − Γ ′ j ( ~α ) , Γ ′ j +1 ( ~α ) , · · · , Γ ′ r ( ~α ) (cid:17) So, by Corollary 3.3, ||| ˜ B ( ~ Γ) − ˜ B ( ~ Γ ′ ) ||| w ≤ r X j =1 (cid:16) j − Y k =1 ||| Γ k ||| w λ k (cid:17) ||| Γ j − Γ ′ j ||| w λ j (cid:16) r Y k = j +1 ||| Γ ′ k ||| w λ k (cid:17) ||| B ||| w κ,λ ≤ r max (cid:8) k ~ Γ k , k ~ Γ ′ k (cid:9) r − k ~ Γ − ~ Γ ′ k ||| B ||| w κ,λ ≤ max (cid:8) k ~ Γ k , k ~ Γ ′ k (cid:9) r − k ~ Γ − ~ Γ ′ k ||| B ||| ′ w κ,λ (cid:8) k ~ Γ k , k ~ Γ ′ k (cid:9) ≤ r ≥ d min . Lemma 3.8 (Product Rule) . Let A ( ~α, ~γ ) and B ( ~α, ~γ ) be ( s + r ) –field maps with ||| A ||| ′ w κ,λ , ||| B ||| ′ w κ,λ < ∞ . Define C ( ~α, ~γ )( x ) = A ( ~α, ~γ )( x ) B ( ~α, ~γ )( x ) Then ||| C ||| ′ w κ,λ ≤ ||| A ||| ′ w κ,λ ||| B ||| w κ,λ + ||| A ||| w κ,λ ||| B ||| ′ w κ,λ Proof.
For convenience of notation, write ~n = ( n s +1 , · · · , n s + r ), | ~n | = n s +1 + · · · + n s + r and ~n ≥ n s +1 , · · · , n s + r ≥
0. Then, in the notation of Definition 3.4.a, C = X ~N ≥ C ~N with C ~N = X vecn,~m ≥ ~n + ~m = ~N A ~n B ~m and ||| C ||| ′ w κ,λ = X ~N ≥ | ~N | ||| C ~N ||| w κ,λ ≤ X ~n, ~m ≥ (cid:0) | ~n | + | ~m | (cid:1) ||| A ~n B ~m ||| w κ,λ So the claim follows from ||| A ~n B ~m ||| w κ,λ ≤ ||| A ~n ||| w κ,λ ||| B ~m ||| w κ,λ Solving Equations
In this section we consider systems of r ≥ γ j = f j ( ~α ) + L j ( ~α, ~γ ) + B j (cid:0) ~α, ~γ (cid:1) (4.1.a)for “unknown” fields γ , · · · , γ r as a function of fields α , · · · , α s . In the aboveequation, ~α = (cid:0) α , · · · , α s (cid:1) , ~γ = (cid:0) γ , · · · , γ r (cid:1) , and for each 1 ≤ j ≤ r , • f j is an s –field map, • L j is an ( s + r )–field map that is linear in its last r arguments, and • B j is an ( s + r )–field map.We write the system (4.1.a) in the shorthand notation ~γ = ~f ( ~α ) + ~L ( ~α, ~γ ) + ~B (cid:0) ~α, ~γ (cid:1) (4.1.b)Example 4.2, below, is of this form and is a simplified version of the kind ofequations that occur as equations for “background fields” and “critical fields” in [8, 9].The following proposition gives conditions under which this system of equations hasa solution ~γ = ~ Γ( ~α ), estimates on the solution, and a uniqueness statement. Proposition 4.1.
Let κ , · · · , κ s and λ , · · · , λ r be constant weight factors forthe fields α , · · · , α s and γ , · · · , γ r , respectively. As in Definition 3.4 set B = (cid:8) ~ Γ (cid:12)(cid:12) k ~ Γ k ≤ (cid:9) where k ~ Γ k = max ≤ j ≤ r λ j ||| Γ j ||| w κ . Let < c < be a contraction factor.Assume that, for each ≤ j ≤ r , the ( s + r ) –field map B j ( ~α ; ~γ ) has minimum degreeat least in its last r arguments (that is, in ~γ ). Also assume that for ≤ j ≤ r ||| f j ||| w κ + ||| L j ||| w κ,λ + ||| B j ||| w κ,λ ≤ λ j ||| L j ||| w κ,λ + ||| B j ||| ′ w κ,λ ≤ c λ j (a) Then there is a unique ~ Γ ∈ B for which ~ Γ( ~α ) = ~f ( ~α ) + ~L (cid:0) ~α, ~ Γ( ~α ) (cid:1) + ~B (cid:0) ~α, ~ Γ( ~α ) (cid:1) That is, which solves (4.1) . Furthermore max j λ j ||| Γ j ||| w ≤ − c max j λ j ||| f j ||| w max j λ j ||| Γ j − f j ||| w ≤ c − c max j λ j ||| f j ||| w (b) Assume, in addition, that ||| f j ||| w ≤ (1 − c ) λ j for all ≤ j ≤ r enote by ~ Γ the solution of part (a) and by ~ Γ (1) the unique element of B thatsolves γ j = f j ( ~α ) + L j ( ~α, ~γ ) for ≤ j ≤ r . Then k ~ Γ (1) k ≤ − c k ~f k k ~ Γ (1) − ~f k ≤ c − c k ~f k and k ~ Γ − ~ Γ (1) k ≤ k ~f k (1 − c ) max ≤ j ≤ r λ j ||| B j ||| w κ,λ ≤ max ≤ j ≤ r λ j ||| B j ||| w κ,λ Proof. (a) Define F ( ~ Γ) by ~F ( ~ Γ) = f + ˜ L ( ~ Γ) + ˜ B ( ~ Γ)... f r + ˜ L r ( ~ Γ) + ˜ B r ( ~ Γ) Recall, from Definition 3.4, that (cid:0) ˜ L j ( ~ Γ) (cid:1) ( ~α ) = L j (cid:0) ~α, ~ Γ( ~α ) (cid:1) and (cid:0) ˜ B j ( ~ Γ) (cid:1) ( ~α ) = B j (cid:0) ~α, ~ Γ( ~α ) (cid:1) By Corollary 3.3 and the hypothesis ||| f j ||| w κ + ||| L j ||| w κ,λ + ||| B j ||| w κ,λ ≤ λ j , ~F maps B into B . By Lemma 3.7 and Remark 3.5.b, k ~F ( ~ Γ) − ~F ( ~ Γ ′ ) k ≤ c k ~ Γ − ~ Γ ′ k so that ~F isa strict contraction. The claims are now a consequence of the contraction mappingtheorem.(b) The first two bounds are special cases of part (a) with B j = 0. Since L j is linearin its last r arguments, δ~ Γ = ~ Γ − ~ Γ (1) obeys δ Γ j ( ~α ) = L j (cid:0) ~α , δ~ Γ( ~α ) (cid:1) + B j (cid:0) ~α , ~ Γ (1) ( ~α ) + δ~ Γ( ~α ) (cid:1) for 1 ≤ j ≤ r . View this a fixed point equation determining ~δ Γ. The equation is ofthe form ~δ = ~G ( ~δ ) where ~G ( ~δ ) = ˜ L ( ~δ ) + ˜ B ( ~ Γ (1) + ~δ )...˜ L r ( ~δ ) + ˜ B r ( ~ Γ (1) + ~δ ) If k ~δ k ≤ c then k ~ Γ (1) + ~δ k ≤
1. Therefore, by Corollary 3.3, ~G maps B c into B c .By Lemma 3.7, ~G is a strict contraction. Apply the contraction mapping theorem.Since G j ( ~
0) = ˜ B j ( ~ Γ (1) ) and k ~ Γ (1) k ≤ − c k ~f k = ⇒ ||| Γ (1) j ||| w ≤ k ~f k − c λ j ≤ j ≤ r and B j is of degree at least two in its last r arguments we have ||| ˜ B j ( ~ Γ (1) ) ||| w ≤ (cid:0) k ~f k − c (cid:1) ||| B j ||| w κ,λ so that k ~G ( ~ k ≤ (cid:0) k ~f k − c (cid:1) max ≤ j ≤ r λ j ||| B j ||| w κ,λ . Thereforethe fixed point ~δ = δ~ Γ obeys k δ~ Γ k ≤ − c k ~G ( ~ k ≤ k ~f k (1 − c ) max ≤ j ≤ r λ j ||| B j ||| w κ,λ ≤ (1 − c ) max ≤ j ≤ r λ j ||| B j ||| w κ,λ Example 4.2.
We assume that X is a finite lattice of the form X = L / L , where L is a lattice in R d and L is a sublattice of L of finite index. The Euclidean distanceon R d induces a distance | · | on X .Let W , W : X → C and set, for complex fields φ , φ on X W ( φ , φ )( x ) = X y,z ∈ X W ( x, y, z ) φ ( y ) φ ( z ) W ( φ , φ )( x ) = X y,z ∈ X W ( x, y, z ) φ ( y ) φ ( z )Aso let S and S be two invertible operators on L ( X ). Pretend that S − and S − are “differential operators”. Suppose that we are interested in solving S − φ + W ( φ , φ ) = α S − φ + W ( φ , φ ) = α (4.2)for φ , φ as functions of complex fields α , α . Suppose further that we are thinkingof the W j ’s as small. We would like to write the solution as a perturbation of the W = W = 0 solution φ = S α , φ = S α . So we substitute φ = S (cid:0) α + γ (cid:1) φ = S (cid:0) α + γ (cid:1) into (4.2), giving γ + W (cid:0) S ( α + γ ) , S ( α + γ ) (cid:1) = 0 γ + W (cid:0) S ( α + γ ) , S ( α + γ ) (cid:1) = 022his is of the form (4.1) with ~f ( ~α ) = (cid:20) −W (cid:0) S α , S α (cid:1) −W (cid:0) S α , S α (cid:1)(cid:21) ~L ( ~α, ~γ ) = (cid:20) −W (cid:0) S γ , S α (cid:1) − W (cid:0) S α , S γ (cid:1) −W (cid:0) S γ , S α (cid:1) − W (cid:0) S α , S γ (cid:1)(cid:21) ~B ( ~α, ~γ )( u ) = (cid:20) −W (cid:0) S γ , S γ (cid:1) −W (cid:0) S γ , S γ (cid:1)(cid:21) To apply Proposition 4.1 to Example 4.2, fix any m , k > ||| φ j ||| with metric m | · | and weight factors k to measure analytic maps like φ j ( α , α ). SeeDefinition 2.3.c. The weight factor k is used for both α and α . Like in [3, § IV]and [4, Definition 4.2] we define, for any linear operator S : L ( X ) → L ( X ), the“weighted” ℓ – ℓ ∞ norm k S k m = max n sup y ∈ X X x ∈ X | S ( x, y ) | e m | y − x | , sup x ∈ X X y ∈ X | S ( x, y ) | e m | y − x | o Proposition 4.1 can be applied to this situation:
Corollary 4.3.
Let
K > . Write ¯ S = max j =1 , k S j k m and ¯ W = max j =1 , k W j k m and assumethat ¯ S ¯ W k < min (cid:8) , K (cid:9) Then there are field maps φ ( ≥ , φ ( ≥ such that φ ( α , α ) = S α + φ ( ≥ ( α , α ) φ ( α , α ) = S α + φ ( ≥ ( α , α ) solves the equations (4.2) of Example 4.2 and obeys (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ ( ≥ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ S ¯ W k Furthermore φ ( ≥ j is of degree at least two in ( α , α ) . The solution is unique in (cid:8) ( φ , φ ) ∈ L ( X ) × L ( X ) (cid:12)(cid:12) ||| S − φ ||| , ||| S − φ ||| ≤ K k (cid:9) roof. In Example 4.2 we wrote the equations (4.2) in the form ~γ = ~f ( ~α ) + ~L ( ~α, ~γ ) + ~B (cid:0) ~α, ~γ (cid:1) (4.3)Now apply Proposition 4.1.a and Remark 3.5.a with r = s = 2 and d max = 2 c = κ = κ = λ = λ = k Since ||| f j ||| w ≤ k S k m k S k m k W j k m κ κ ||| L j ||| w κ,λ ≤ k S k m k S k m k W j k m (cid:0) λ κ + κ λ (cid:1) ||| B j ||| w κ,λ ≤ k S k m k S k m k W j k m λ λ By hypothesis, ||| f j ||| w , ||| L j ||| w κ,λ , ||| B j ||| w κ,λ < λ j and Proposition 4.1.a gives a so-lution ~ Γ( ~α ) to (4.3) that obeys the bound ||| Γ j ||| w ≤ k S k m k S k m k W j k m k Setting φ ( α , α ) = S α + S Γ ( α , α ) φ ( ≥ ( α , α ) = S Γ ( α , α ) φ ( α , α ) = S α + S Γ ( α , α ) φ ( ≥ ( α , α ) = S Γ ( α , α )we have all of the claims, except for uniqueness.We now prove uniqueness. Assume that φ j = S j Φ j and that φ j = S j (Φ j + δ Φ j )both solve (4.2), with ||| Φ j + δ Φ j ||| ≤ K k and with S j Φ j being the solution constructedabove. Then δ Φ j is a solution of δ Φ = −W (cid:0) S (Φ + δ Φ ) , S (Φ + δ Φ ) (cid:1) + W (cid:0) S Φ ∗ , S Φ (cid:1) δ Φ = −W (cid:0) S (Φ + δ Φ ) , S (Φ + δ Φ ) (cid:1) + W (cid:0) S Φ , S Φ (cid:1) Since (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W j (cid:0) S α , S α (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k W j k m ||| S α ||| ||| S α ||| ≤ k W j k m k S k m k S k m ||| α ||| ||| α ||| we have ||| δ Φ ||| ≤ k W k m k S k m k S k m (cid:8) ||| δ Φ ||| ||| Φ + δ Φ ||| + ||| Φ ||| ||| δ Φ ||| (cid:9) ||| δ Φ ||| ≤ k W k m k S k m k S k m (cid:8) ||| δ Φ ||| ||| Φ + δ Φ ||| + ||| Φ ||| ||| δ Φ ||| (cid:9)
24y hypothesis ||| Φ ||| ≤ k + 2 k S k m k S k m k W j k m k ≤ k ||| Φ + δ Φ ||| ≤ K k so that ||| δ Φ ||| + ||| δ Φ ||| ≤ (cid:0) k W k m + k W k m (cid:1) k S k m k S k m max (cid:8) , K (cid:9) k (cid:0) ||| δ Φ ||| + ||| δ Φ ||| (cid:1) ≤ ¯ S ¯ W k (cid:8) , K (cid:9) (cid:0) ||| δ Φ ||| + ||| δ Φ ||| (cid:1) thereby forcing ||| δ Φ ∗ ||| = ||| δ Φ ||| = 0. 25 A Generalisation of Young’s Inequality
Lemma A.1.
Let n ∈ N . For each ≤ ℓ ≤ n , let • ( X ℓ , dµ ℓ ) be a measure space, • f ℓ : X ℓ → C be measureable and • p ℓ ∈ (0 , ∞ ] .Let K : n X ℓ =1 X ℓ → C have finite L – L ∞ norm and assume that n P ℓ =1 1 p ℓ = 1 . Then (cid:12)(cid:12)(cid:12)(cid:12) Z n X ℓ =1 X ℓ K ( x , · · · , x n ) n Q ℓ =1 f ℓ ( x ℓ ) n Q ℓ =1 dµ ℓ ( x ℓ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k K k L − L ∞ n Q ℓ =1 k f ℓ k L pℓ ( dµ ℓ ) Proof.
We’ll use the short hand notations dm ( x , · · · , x n ) = n Q ℓ =1 dµ ℓ ( x ℓ ) and X = n X ℓ =1 X ℓ . By H¨older (with the usual interpretations when some p ℓ = ∞ ), (cid:12)(cid:12)(cid:12)(cid:12) Z X K ( x , · · · , x n ) n Q ℓ =1 f ℓ ( x ℓ ) dm ( x , · · · , x n ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z X n Q ℓ =1 (cid:8)(cid:12)(cid:12) K ( x , · · · , x n ) (cid:12)(cid:12) /p ℓ | f ℓ ( x ℓ ) | (cid:9) dm ( x , · · · , x n )= n Q ℓ =1 (cid:20) Z X (cid:12)(cid:12) K ( x , · · · , x n ) (cid:12)(cid:12) | f ℓ ( x ℓ ) | p ℓ n Q ℓ =1 dµ ℓ ( x ℓ ) (cid:21) /p ℓ ≤ n Q ℓ =1 (cid:20) k K k L − L ∞ Z X ℓ | f ℓ ( x ℓ ) | p ℓ dµ ℓ ( x ℓ ) (cid:21) /p ℓ = k K k L − L ∞ n Q ℓ =1 k f ℓ k L pℓ ( dµ ℓ ) eferences [1] T. Balaban. The Ultraviolet Stability Bounds for Some Lattice σ –Models andLattice Higgs–Kibble Models. In Proc. of the International Conference on Math-ematical Physics, Lausanne, 1979 , pages 237–240. Springer, 1980.[2] T. Balaban. A low temperature expansion and “spin wave picture” for classical N -vector models. In Constructive Physics (Palaiseau, 1994), Lecture Notes inPhysics, 446 , pages 201–218. Springer, 1995.[3] T. Balaban, J. Feldman, H. Kn¨orrer, and E. Trubowitz. Power Series Represen-tations for Bosonic Effective Actions.
Journal of Statistical Physics , 134:839–857, 2009.[4] T. Balaban, J. Feldman, H. Kn¨orrer, and E. Trubowitz. Power Series Represen-tations for Complex Bosonic Effective Actions. I. A Small Field RenormalizationGroup Step.
Journal of Mathematical Physics , 51:053305, 2010.[5] T. Balaban, J. Feldman, H. Kn¨orrer, and E. Trubowitz. Power Series Represen-tations for Complex Bosonic Effective Actions. II. A Small Field Renormaliza-tion Group Flow.
Journal of Mathematical Physics , 51:053306, 2010.[6] T. Balaban, J. Feldman, H. Kn¨orrer, and E. Trubowitz. The Temporal Ultravi-olet Limit for Complex Bosonic Many-body Models.
Annales Henri Poincar´e ,11:151–350, 2010.[7] T. Balaban, J. Feldman, H. Kn¨orrer, and E. Trubowitz. Complex BosonicMany–body Models: Overview of the Small Field Parabolic Flow. Preprint,2016.[8] T. Balaban, J. Feldman, H. Kn¨orrer, and E. Trubowitz. The Small FieldParabolic Flow for Bosonic Many–body Models: Part 1 — Main Results andAlgebra. Preprint, 2016.[9] T. Balaban, J. Feldman, H. Kn¨orrer, and E. Trubowitz. The Small FieldParabolic Flow for Bosonic Many–body Models: Part 2 — Fluctuation Inte-gral and Renormalization. Preprint, 2016.[10] J. Dimock. The renormalization group according to Balaban – I. small fields.
Reviews in Mathematical Physics , 25:1–64, 2013.2711] K. Gawedzki and A. Kupiainen. A rigorous block spin approach to masslesslattice theories.
Comm. Math. Phys. , 77:31–64, 1980.[12] L.P. Kadanoff. Scaling laws for Ising models near T c . Physics , 2:263, 1966.[13] E. H. Lieb and M. Loos.