Power Series Representations
for Bosonic Effective Actions
Tadeusz Balaban
Rutgers University
Joel Feldman
University of British Columbia
Horst Knörrer, Eugene Trubowitz
ETH-Zürich
(P)reprints: http://www.math.ubc.ca/e feldman/
P1
Abstract
We develop a power series representation and estimates
for an effective action of the form
R F (ψ,ϕ)
e
dµ(ϕ)
ln R f (ϕ,0)
e
dµ(ϕ)
Here, F (ψ, ϕ) is an analytic function of the real fields
ϕ(x), ψ(x) indexed by x in a finite set X, and dµ(ϕ) is
a compactly supported product measure. Such effective
actions occur in the small field region for a renormalization group analysis. The customary way to analyze
them is a cluster expansion, possibly preceded by a decoupling expansion. Using methods similar to a polymer
expansion, we estimate the power series of the effective
action without introducing an artificial decomposition of
the underlying space into boxes.
Outline
I
I
I
I
I
Motivation
The Main Theorem
Outline of the Proof - Algebraic Part
Norms
Outline of the Proof - Bounds
P2
Motivation – Renormalization Group Construction
Protocol
I
I
Express all quantities of interest as functional integrals
like
R A(Ψ,Φ)
e
dµ(Φ)
G(Ψ) = ln R A(0,Φ)
e
dµ(Φ)
Q∞
Factor the measure dµ(Φ) = `=1 dµ` (ϕ` ), with the
least important degrees of freedom having index `
small, to express
R A(Ψ,ϕ ,ϕ ,···) Q∞
1
2
e
`=1 dµ` (ϕ` )
G(Ψ) = ln R A(0,ϕ ,ϕ ,···) Q∞
1
2
e
`=1 dµ` (ϕ` )
I
Do the integrals one at a time. Define the “effective
action at scale n” to be
An (Ψ, ϕn+1 , ϕn+2 , · · ·)
R A(Ψ,ϕ ,ϕ ,···) Qn
1
2
e
`=1 dµ` (ϕ` )
R
Q
= ln
eA(0,ϕ1 ,···,ϕn ,0,···) n`=1 dµ` (ϕ` )
Then
R A (ψ,ϕ)
e n−1
dµn (ϕ)
An (ψ) = ln R A (0,ϕ)
e n−1
dµn (ϕ)
where ϕ = ϕn and ψ = (Ψ, ϕn+1 , ϕn+2 , · · ·).
P3
The Main Theorem
Let X(= space) be a finite set. Let dµ0 (t) be a normalized measure on IR that is supported in |t| ≤ r for some
constant r. We endow IRX with the ultralocal product
measure
Q
dµ(ϕ) =
dµ0 ϕ(x)
x∈X
Theorem III.4 Let w and W be weight systems for 1
and 2 fields, respectively, that obey
W (~x, ~y) ≥ (4r)n(~y) w(~x)
1
, then there is a real
If F (ψ, ϕ) obeys kF kW < 16
analytic function f (ψ) such that
and
R F (ψ,ϕ)
e
dµ(ϕ)
f (ψ)
R
=
e
eF (0,ϕ) dµ(ϕ)
kf kw ≤
(III.1)
kF kW
1−16kF kW
P4
Notation
x ∈ X = space, a finite set
[
~x ∈ X = multispace =
Xn
=
n≥0
(x1 , · · · , xn ) ∈ X
n≥0
n
For ~x = (x1 , · · · , xn ) ∈ X n , ~y = (y1 , · · · , ym ) ∈ X m
and ϕ : X → IR,
n(~x) = n
~x ◦ ~y = x1 , · · · , xn , y1 , · · · , ym ) ∈ X n+m
ϕ(~x) = ϕ(x1 )ϕ(x2 ) · · · ϕ(xn )
supp (~x) = {x1 , · · · , xn } ⊂ X
If F , f are real analytic on a neighbourhood of the origin,
then there are unique expansions
X
A(~x, ~y) ψ(~x)ϕ(~y)
F (ψ, ϕ) =
~
x,~
y∈X
f (ψ) =
X
a(~x) ψ(~x)
~
x∈X
with A(~x, ~y), a(~x) invariant under permutations of the
components of ~x and under permutations of the components of ~y.
P5
Outline of the Proof – Algebra
P
A(~x, ~y) ψ(~x)ϕ(~y).
Write F (ψ, ϕ) =
~
x,~
y∈X
B
Set
α(~y) =
X
A(~x, ~y) ψ(~x)
~
x∈X
With this notation
F (ψ, ϕ) =
X
α(~y) ϕ(~y)
~
y∈X
By factoring eF (ψ,0) out of the integral in the numerator
of (III.1), we may assume that F (ψ, 0) = 0. Expanding
the exponential
e
F (ψ,ϕ)
=
∞
P
`=0
=1+
∞
X
`=1
1
`!
`
1
F
(ψ,
ϕ)
`!
X
α(~y1 ) · · · α(~y` ) ϕ(~y1 ) · · · ϕ(~y` )
~
y1 ,···,~
y` ∈X
P6
Define the incidence graph G(~y1 , · · · , ~y` ) to be the labelled graph with
B vertices {1, · · · , `} and
B an edge between i 6= j when supp ~
yi ∩ supp ~yj 6= ∅.
For a subset of Z ⊂ X, denote by C(Z) the set of all
ordered tuples (~y1 , · · · , ~yn ) such that
B Z = supp ~
y1 ∪ · · · ∪ supp ~yn .
B G(~
y1 , · · · , ~yn ) is connected.
We call such a tuple a connected cover of Z.
So
X
α(~y1 ) · · · α(~y` ) ϕ(~y1 ) · · · ϕ(~y` )
~
y1 ,···,~
y` ∈X
=
X̀
n=1
1
n!
X
Z1 ,···,Zn ⊂X
pairwise disjoint
nonempty
X
X
~
y1 ,···,~
y`
I1 ∪···∪In ={1,···,`}
I1 ,···,In pairwise disjoint (~
yi )i∈I ∈C(Zj )
j
α(~y1 ) · · · α(~y` ) ϕ(~y1 ) · · · ϕ(~y` )
P7
Fix, for the moment, pairwise disjoint nonempty subsets
Z1 , · · · , Zn of X and ` ≥ n. Then
X
X
α(~y1 ) · · · ϕ(~y` )
~
y1 ,···,~
y`
I1 ∪···∪In ={1,···,`}
I1 ,···,In disjoint
(~
yi , i∈Ij )∈C(Zj )
X
=
X
X
α(~y1 ) · · · ϕ(~y` )
k1 ,···,kn ≥1 I1 ,···,In ⊂{1,···,`}
~
y1 ,···,~
y`
k1 +···+kn =` I1 ,···,In disjoint (~
yi , i∈Ij )∈C(Zj )
|Ij |=kj
X
=
X
`!
k1 !···kn !
k1 ,···,kn ≥1
k1 +···+kn =`
α(~y1 ) · · · ϕ(~y` )
(~
y1 ,···,~
yk )∈C(Z1 )
1
(~
y`−k +1
n
...
,···,~
y
` )∈C(Zn )
As the measure µ factorizes with each factor normalized,
and the different Zj ’s are disjoint,
Z
ϕ(~y1 ) · · · ϕ(~y` ) dµ(ϕ)
=
n Z
Y
ϕ(~ypj−1 +1 ) · · · ϕ(~ypj ) dµ(ϕ)
j=1
(where p0 = 0 and, for 1 ≤ j ≤ n, pj = k1 + · · · + kj ).
P8
Writing
Z
eF (ψ,ϕ) dµ(ϕ)
=1+
∞
X
X̀
1
`!
n=1
`=1
=1+
∞ X
∞
X
n=1 `=n
=1+
∞
X
n=1
1
n!
X
1
n!
Z1 ,···,Zn ⊂X
pairwise disjoint
nonempty
X
1
n!
Z1 ,···,Zn ⊂X
pairwise disjoint
nonempty
`!
k1 !···kn !
···
k1 ,···,kn ≥1
k1 +···+kn =`
X
Z1 ,···,Zn ⊂X
pairwise disjoint
nonempty
X
X
1
k1 !···kn !
···
k1 ,···,kn ≥1
k1 +···+kn =`
X
1
k1 !···kn !
···
k1 ,···,kn ≥1
we have
Z
∞
X
eF (ψ,ϕ) dµ(ϕ) = 1 +
n=1
1
n!
X
Z1 ,···,Zn ⊂X
pairwise disjoint
n
Q
Φ(Zj )
j=1
where, for ∅ 6= Z ⊂ X,
Φ(Z) =
∞
X
X
Z
α(~y1 ) · · · α(~yk ) ϕ(~y1 ) · · · ϕ(~yk ) dµ(ϕ)
1
k!
k=1 (~
y1 ,···,~
yk )∈C(Z)
(III.2)
and Φ(∅) = 0.
P9
If we define
0
ζ(Z, Z ) =
0 if Z ∩ Z 0 6= ∅
1 if Z and Z 0 are disjoint
and Gn = {i, j} ⊂ IN
1 ≤ i < j ≤ n is the
complete graph on {1, · · · , n}, then
Z
2
eF (ψ,ϕ) dµ(ϕ)
=1+
∞
X
Y
X X
Y
1
n!
n=1
= 1+
X
ζ(Zi , Zj )
Z1 ,···,Zn ⊂X {i,j}∈Gn
∞
X
n
Y
j=1
1
ζ(Zi , Zj )−1
n!
n=1 Z1 ,···,Zn g⊂Gn {i,j}∈g
=1+
∞
X
n=1
1
n!
X
ρ(Z1 , · · · , Zn )
Φ(Zj )
n
Q
n
Y
Φ(Zj )
j=1
Φ(Zj )
j=1
Z1 ,···,Zn ⊂X
where
ρ(Z1 , · · · , Zn ) =

1

P
Q
g⊂Gn {i,j}∈g
ζ(Zi , Zj )−1
if n = 1
if n ≥ 2
P 10
Define

1
P
ρT (Z1 , · · · , Zn ) =

Q
if n = 1
ζ(Zi , Zj )−1 if n ≥ 2
g∈Cn {i,j}∈g
where Cn is the set of connected subgraphs of Gn . By a
standard argument,
Z
ln eF (ψ,ϕ) dµ
=
∞
X
n=1
1
n!
X
Z1 ,···,Zn ⊂X
T
ρ (Z1 , · · · , Zn )
n
Q
Φ(Zj )
j=1
(III.3)
(By “ln” we just mean that the exponential of the right
R F
hand side is e (ψ, ϕ) dµ.)
P 11
Aside: Outline of the standard argument:
Define the value of the graph g ⊂ Gn to be
 P
Φ(Z)
if n = 1


 Z⊂X
n
Val(g) =
P
Q
Q


C(Zi , Zj )
Φ(Zj ) if n > 1

j=1
Z1 ,···,Zn {i,j}∈g
where C(Zi , Zj ) = ζ(Zi , Zj ) − 1. If the connected components of g ∈ Gn are g1 , · · ·, gm , then
Val(g) =
m
Y
Val(gm )
j=1
Consequently,
Z
e
F (ψ,ϕ)
dµ = 1 +
∞
X
n=1
=
∞ Y
Y
X
1
n!
Val(g)
g⊂Gn
1
e n! Val(g)
n=1 g∈Cn
= exp
nP
∞
n=1
1
n!
P
g∈Cn
Val(g)
o
End of aside.
P 12
We now find a, not necessarily symmetric, coefficient sysR F (ψ,ϕ)
tem for ln e
dµ(ϕ). Recall
Φ(Z) =
∞
X
Z
α(~y1 ) · · · α(~yk ) ϕ(~y1 ) · · · ϕ(~yk ) dµ(ϕ)
X
1
k!
k=1 (~
y1 ,···,~
yk )∈C(Z)
(III.2)
and
α(~y) =
X
A(~x, ~y) ψ(~x)
~
x∈X
So, if we set, for each (~x, ~y) ∈ X 2 ,
Ã(~x, ~y) =
∞
X
k=1
1
k!
X
X
~
x1 ,···,~
xk
(~
y1 ,···,~
yk )∈C(supp ~
y)
~
x1 ◦···◦~
xk =~
x
~
y1 ◦···◦~
yk =~
y
A(~x1 , ~y1 ) · · · A(~xk , ~yk )
Then
Φ(Z)(ψ) =
X
Z
ϕ(~y) dµ(ϕ)
(III.5)
Ã(~x, ~y) ψ(~x)
(~
x,~
y)∈X 2
supp ~
y=Z
P 13
Recall that
Z
ln eF (ψ,ϕ) dµ
=
∞
X
X
1
n!
n=1
T
ρ (Z1 , · · · , Zn )
n
Q
Φ(Zj )
j=1
Z1 ,···,Zn ⊂X
(III.3)
Therefore,
ln
Z
e
F (ψ,ϕ)
dµ(ϕ) =
X
a(~x) ψ(~x)
~
x∈X
where, for ~x ∈ X ,
a(~x) =
∞
X
n=1
X
1
n!
~
x1 ,···,~
xn ∈X
~
x1 ◦···◦~
xn =~
x
X
~
y1 ,···,~
yn ∈X
T
ρ (supp ~y1 , · · · , supp ~yn )
n
Q
Ã(~xj , ~yj )
j=1
(III.6)
Also
R F (ψ,ϕ)
X
e
dµ(ϕ)
f (ψ) = ln R F (0,ϕ)
a(~x) ψ(~x)
=
e
dµ(ϕ)
~
x∈X
n(~
x)>0
P 14
Norms
Definition II.6 (Norms for functions of one field)
Let
X
f (ψ) =
a(~x) ψ(~x)
~
x∈X
with a(~x) invariant under permutations of the compo
nents of ~x. We call a = a(~x) ~x ∈ X the symmetric
coefficient system for f .
If w(~x) is a weight system for one field, we define
kf kw = kakw ≡
X
n≥0
max
X
1≤i≤n
~
x∈X n
z∈X
xi =z
w(~x) a(~x)
Here xi is the ith component of the n–tuple ~x. The
term in the above sum with n = 0 is simply w(−) a(−)
where − denotes the 0–tuple.
P 15
Remark II.7 If
f (ψ) =
X
a(~x) ψ(~x)
~
x∈X
with a(~x) not necessarily invariant under permutations
of the components of ~x, then
kf kw ≤ kakw ≡
X
n≥0
max
X
1≤i≤n
~
x∈X n
z∈X
xi =z
w(~x) a(~x)
Definition II.3 (Weight System for One Field) A
weight system for one field is a function w : X → (0, ∞)
that satisfies:
(a) w(~x) is invariant under permutations of the components of ~x.
(b)
w(~x ◦ ~x0 ) ≤ w(~x)w(~x0 )
for all ~x, ~x0 ∈ X with supp (~x) ∩ supp (~x0 ) 6= ∅.
P 16
Example II.4 (Weight Systems)
(i) If κ : X → (0, ∞) (called a weight factor) then
n(~
x)
w(~x) = κ(~x) =
Y
κ(x` )
`=1
is a weight system for one field.
(ii) Let d : X × X → IR≥0 be a metric. For a subset
S ⊂ X, denote by τ (S) the length of the shortest tree
in X whose set of vertices contains S. Then
w(~x) = eτ (supp (~x))
is a weight system for one field.
(iii) If w1 (~x) and w2 (~x) are weight systems for one field,
then so is
w3 (~x) = w1 (~x)w2 (~x)
P 17
Definition II.3 (Weight System for Two Fields) A
weight system for two fields is a function W : X 2 → (0, ∞)
that satisfies:
(a) W (~x, ~y) is invariant under permutations of the components of ~x and is invariant under permutations of
the components of ~y.
(b)
W (~x ◦ ~x0 , ~y ◦ ~y0 ) ≤ W (~x, ~y)W (~x0 , ~y0 )
whenever supp (~x, ~y) ∩ supp (~x0 , ~y0 ) 6= ∅.
Definition II.6 (Norms for functions of two fields)
Let
X
A(~x, ~y) ψ(~x)ϕ(~y)
F (ψ, ϕ) =
(~
x,~
y)∈X 2
with A(~x, ~y) invariant under permutations of the components of ~x and under permutations of the components
of ~y.
If W (~x, ~y) be a weight system for two fields, we define
X
X
kF kW =
max
W (~x, ~y) A(~x, ~y)
n,m≥0
1≤i≤n+m
(~
x,~
y)∈X n ×X m
z∈X
(~
x,~
y)i =z
Here (~x, ~y)i is the ith component of the n + m–tuple
(~x, ~y). The term in the above sum with n = m = 0 is
simply W (−, −) A(−, −).
P 18
Review of the Main Theorem
Let X(= space) be a finite set. Let dµ0 (t) be a normalized measure on IR that is supported in |t| ≤ r for some
constant r. We endow IRX with the ultralocal product
measure
Q
dµ(ϕ) =
dµ0 ϕ(x)
x∈X
Theorem III.4 Let w and W be weight systems for 1
and 2 fields, respectively, that obey
W (~x, ~y) ≥ (4r)n(~y) w(~x)
1
, then there is a real
If F (ψ, ϕ) obeys kF kW < 16
analytic function f (ψ) such that
and
R F (ψ,ϕ)
e
dµ(ϕ)
f (ψ)
R
=
e
eF (0,ϕ) dµ(ϕ)
kf kw ≤
(III.1)
kF kW
1−16kF kW
P 19
Outline of the Proof – Bounds
Step 1 - organizing the sums
Recall from (III.6) that
a(~x) =
∞
X
X
1
n!
n=1
~
x1 ,···,~
xn ∈X
~
x1 ◦···◦~
xn =~
x
T
X
~
y1 ,···,~
yn ∈X
ρ (supp ~y1 , · · · , supp ~yn )
n
Q
Ã(~xj , ~yj )
j=1
The bound
T
ρ (supp ~y1 , · · · , supp ~yn )
≤ # spanning trees in G(~y1 , · · · , ~yn )
is due to Rota.
P 20
Hence
|a(~x)| ≤
∞
X
1
n!
n=1
X
~
x1 ,···,~
xn ∈X
~
x1 ◦···◦~
xn =~
x
X
~
y1 ,···,~
yn ∈X
X
T spanning tree
for G(~
y1 ,···,~
yn )
n Q
Ã(~xj , ~yj )
j=1
≤
∞
X
1
n!
n=1
X
T labelled tree with
vertices 1,···,n
X
|Ã|T (~x, ~y)
~
y∈X
(III.8)
where
|Ã|T (~x, ~y) =
X
~
y1 ,···,~
yn ∈X
~
y=~
y1 ◦···◦~
yn
T ⊂G(~
y1 ,···,~
yn )
X
~
x1 ,···,~
xn ∈X
~
x=~
x1 ◦···◦~
xn
n
Y
|Ã(~x` , ~y` )|
`=1
P 21
Recall that
Ã(~x, ~y) =
∞
X
X
1
k!
k=1
X
~
x1 ,···,~
xk
(~
y1 ,···,~
yk )∈C(supp ~
y)
~
x1 ◦···◦~
xk =~
x
~
y1 ◦···◦~
yk =~
y
A(~x1 , ~y1 ) · · · A(~xk , ~yk )
Z
ϕ(~y) dµ(ϕ)
(III.5)
For each (~y1 , · · · , ~yk ) G(~y1 , · · · , ~yk ) is connected and
hence contains at least one tree. So
|Ã(~x, ~y)| ≤
∞
X
1
k!
X
T labelled tree
with vertices
1,···,k
k=1
X
~
y1 ,···,~
yk ∈X
~
y=~
y1 ◦···◦~
yk
T ⊂G(~
y1 ,···,~
yk )
r n(~y)
n
Y
X
~
x1 ,···,~
xk ∈X
~
x=~
x1 ◦···◦~
xk
|A(~x` , ~y` )|
`=1
=
∞
X
k=1
1
k!
X
r n(~y) |A|T (~x, ~y) (III.80 )
T labelled tree
with vertices
1,···,k
where
|A|T (~x, ~y) =
X
~
y1 ,···,~
yk ∈X
~
y=~
y1 ◦···◦~
yk
T ⊂G(~
y1 ,···,~
yk )
X
~
x1 ,···,~
xn ∈X
~
x=~
x1 ◦···◦~
xk
k
Y
|A(~x` , ~y` )|
`=1
P 22
Step 2 - bound on BT
Lemma III.5 Let ω be an arbitrary weight system for
two fields and define the weight system ω 0 by
ω 0 (~x, ~y) = 2n(~y) ω(~x, ~y)
Let T be a labelled tree with vertices 1, · · · , n and coordination numbers d1 , · · · , dn . Let B be any (not necessarily symmetric) coefficient system for two fields
with B(−, −) = 0. We define a new coefficient system BT by
BT (~x, ~y) =
X
~
y1 ,···,~
yn ∈X
~
y=~
y1 ◦···◦~
yn
T ⊂G(~
y1 ,···,~
yn )
Then
X
~
x1 ,···,~
xn ∈X
~
x=~
x1 ◦···◦~
xn
n
Y
B(~x` , ~y` )
`=1
BT ≤ d1 ! · · · dn ! kBknω0
ω
P 23
Outline of proof:
Ingredient 1:
B For each 1 ≤ ` ≤ n, think of (~
x` , ~y` ) as the locations
of (two species of) stars in a galaxy.
B In computing
X
B T =
ω
max
X
ω(~x, ~y) BT (~x, ~y)
1≤i≤N +M
N,M ≥0 z∈X (~x,~y)∈X N ×X M
(~
x,~
y)i =z
B
B
B
we must hold fixed the location of one star and sum
over the locations of all other stars. Suppose that the
fixed star is in galaxy ` = 1.
View 1 as the root of T .
Then the set of vertices of T is endowed with a natural
partial ordering under which 1 is the smallest vertex.
For each vertex 2 ≤ ` ≤ n, denote by π(`) the predecessor vertex of ` under this partial ordering.
7 3 4 5
π(7) = π(3) = π(4) = 2
2
π(2) = π(5) = 6
6
π(6) = 1
1
P 24
B
B
B
The condition that T ⊂ G(~y1 , · · · , ~yn ) ensures that,
for each 2 ≤ ` ≤ n, the support of ~y` intersects the
support of ~yπ(`) , so that at least one of the n(~y` )
components of ~y` takes the same value (in X) as some
component of ~yπ(`) .
Write n(~y` ) = n` .
The product over 2 ≤ ` ≤ n of the number of choices
of which ~y–star in galaxy ` is at the same location of
which ~y–star in galaxy π(`) is
n
Y
`=2
n` nπ(`) ) =
n
Y
nd` `
=
`=1
≤ d 1 ! · · · dn !
n
Y
`=1
n
Y
d
n` `
d` !
d` !
2 n`
`=1
by using first year calculus and Stirling.
P 25
Ingredient 2:
B Since T is connected,
ω(~x, ~y) ≤
n
Y
`=1
ω ~x` , ~y`
for all ~x1 , · · · , ~xn ∈ X and ~y1 , · · · , ~yn ∈ X under
consideration. So we may absorb each factor ω ~x` , ~y`
into B ~x` , ~y` and it suffices to consider ω = 1.
P 26
Ingredient 3:
B Iteratively apply
X
~
x` ,~
y` ∈X
~
y`,m =~
yπ(`),p
`
`
X
~
x` ∈X
2
n(~
y` )
|B(~x` , ~y` )| ≤ B ω0
starting with the largest `’s, in the partial ordering of
T , and ending with ` = 1. (For ` = 1, substitute
~x1,1 = x for ~y`,m` = ~yπ(`),p` .)
P 27
Step 3 - sum over n and T
Lemma III.6 Let 0 < ε < 18 . Then
∞
X
n=1
1
n!
X
d1 ,···,dn
d1 +···+dn =2(n−1)
X
d1 ! · · · d n ! εn
T labelled tree
with coordination
numbers d1 ,···,dn
≤
ε
1−8ε
P 28
Proof: By Cayley, the number of labelled trees on n ≥ 2
vertices with coordination numbers (d1 , d2 , · · · , dn ) is
Qn(n−2)!
j=1
(dj −1)!
The number of possible choices of coordination numbers
(d1 , d2 , · · · , dn ) ∈ INn subject to the given constraint is
2(n−1)−1
2n−3
2n−3
=
≤
2
n−1
n−1
Therefore
∞
X
n=2
1
n!
X
d1 ,···,dn
d1 +···+dn =2(n−1)
≤
X
T labelled tree
with coordination
numbers d1 ,···,dn
∞
X
n=2
≤
d1 ! · · · d n ! εn
∞
X
X
d1 · · · d n εn
d1 ,···,dn
d1 +···+dn =2(n−1)
22n−3 2n εn =
8ε2
1−8ε
n=2
For n = 1, d1 = 0 and the number of trees is 1, so
the n = 1 term is ε. So the full sum is bounded by
8ε2
ε
ε + 1−8ε
= 1−8ε
.
P 29
Step 4 - bound on kak in terms of kÃk
We introduce, for each σ > 0, the auxiliary weight system
y)
σ n(~
4r
Wσ (~x, ~y) = W (~x, ~y)
Clearly
W4r (~x, ~y) = W (~x, ~y)
and
w(~x) ≤ W1 (~x, ~y)
(III.9)
for all (~x, ~y) ∈ X 2 .
We now prove
kakw ≤
kÃkW2
1 − 8kÃkW2
(III.10)
Recall from (III.8) that
|a(~x)| ≤
∞
X
n=1
1
n!
X
T labelled tree with
vertices 1,···,n
X
|Ã|T (~x, ~y)
~
y∈X
P 30
Therefore, by (III.9) and Lemma III.5, with ω = W1 and
ω 0 = W2 ,
∞
X
a ≤
w
n=1
≤
∞
X
1
n!
∞
X
1
n!
n=1
≤
n=1
X
1
n!
T labelled tree with
vertices 1,···,n
X
d1 ,···,dn
d1 +···+dn =2(n−1)
X
d1 ,···,dn
d1 +···+dn =2(n−1)
|Ã|T X
T labelled tree
with coordination
numbers d1 ,···,dn
X
W1
|Ã|T W1
n
d1 ! · · · dn ! |Ã|W2
T labelled tree
with coordination
numbers d1 ,···,dn
Now apply Lemma III.6 with ε = |Ã|W2 = kÃkW2 to
get
kÃkW2
kakw ≤ 1−8kÃk
W2
P 31
Step 5 - bound on kÃk in terms of kAk
We now prove
kÃkW2 ≤
kAkW
1−8kAkW
=
kF kW
1−8kF kW
(III.11)
Note that combining (III.10) and (III.11) yields the final
bound
kf kw ≤
kÃkW2
≤
1 − 8kÃkW2
1
Recall from (III.8’) that
∞
X
1
|Ã(~x, ~y)| ≤
k!
k=1
kF kW
1−8kF kW
kF kW
− 8 1−8kF
kW
X
=
kF kW
1−16kF kW
r n(~y) |A|T (~x, ~y)
T labelled tree with
vertices 1,···,k
n(~y)
|A|T (~x, ~y) W2 = |A|T W2r .
By construction, r
Hence, by Lemma III.5, with ω = W2r followed by
Lemma III.6,
∞
X
X
1
kÃkW2 ≤
|A|T W2r
k!
k=1
≤
∞
X
k=1
≤
1
k!
T labelled tree with
vertices 1,···,k
X
d1 ,···,dk
d1 +···+dk =2(k−1)
X
d1 ! · · · dk ! kAkkW4r
T labelled tree
with coordination
numbers d1 ,···,dk
kAkW
1−8kAkW
since W4r = W . This gives (III.11).
P 32
References
[C] Camillo Cammarota, Decay of Correlations for Infinite
Range Interactions in Unbounded Spin Systems, Commun. Math. Phys. 85 (1982), 517–528.
[Ri] Vincent Rivasseau, From Perturbative to Constructive
Renormalization, Princeton University Press, 1991.
[Ro] Gian–Carlo Rota, On the foundations of combinatorial
theory, I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie Verw. Gebiete, 2, 340 (1964).
[Sa] Manfred Salmhofer, Renormalization, An Introduction,
Springer–Verlag, 1999.
[Si] Barry Simon, The Statistical Mechanics of Lattice
Gases, Volume 1, Princeton University Press, 1993.
P 33