A Non–Gaussian Renormalization Group Fixed Point for Hierarchical Scalar Lattice Field Theories.

Hans Koch^{1} and Peter Wittwer^{2}^{,} ^{3}

Rutgers University, Department of Mathematics New Brunswick, NJ 08903

Abstract. A rigorous method is developed to handle the “large field problems” in the Wilson–Kadanoff renormalization group approach to critical lattice systems of unbounded spins. We use this method to study in a hierarchical approximation the non–Gaussian renormalization group fixed point which governs the infrared behaviour of critical lattice field theories in three dimensions. The method is an improvement of the analyticity techniques of Gawedzki and Kupiainen: using Borel summation techniques we are able to incorporate the “large field region” into “the perturbative region” so that the theory is completely described in terms of convergent expansions.

1 Supported in part by the National Science Foundation under Grant No. DMS–8540879

2 Supported in part by the National Science Foundation under Grant No. DMR81–14726

3 Part of this work has been carried out during a visit of the author at the Courant Institute of Mathematical Sciences, NYU.

1. Introduction

A major obstacle in the analysis of critical statistical mechanics systems is the lack of small expansion parameters. For example very little is known about scalar lattice spin systems with non–Gaussian critical long distance behavior

∫ dμ(ϕ) ϕ_{i} ϕ_{j} ∼ \frac{1}{|i − j|^{d−2+η}} , |i − j| → ∞,  η ≥ 0.  (1.1)

Here, dμ is a translation invariant Gibbs measure on configurations ϕ: ^{/ /} Z^{d} → IR. Although renormalization group (RG) concepts provide a powerful framework to think about such problems, it has been impossible so far to convert them into rigorous results, except for certain limit cases [10–12].

The methods presented in this paper are not restricted to such cases. However, we restrict our analysis to a class of hierarchical models. Such models have a long history in the testing of RG ideas [13–20, 25, 26]. We prove here the existence of a non–Gaussian hierarchical fixed point for a Wilson–Kadanoff RG transformation, in d = 3 dimensions. This type of transformation acts on a measure in two steps. First, some short range degrees of freedom (those which are thought to be irrelevant) are integrated out. Then the system is rescaled in a way that is consistent with a long distance behavior as described in equation (1.1). In a suitable space of measures, the RG transformation N is believed to behave as shown in Fig.1.

Here, dμ^{∗} is a non–Gaussian hyperbolic fixed point for N, with an unstable manifold W_{u} of dimension one. The codimension one stable manifold W_{s} through dμ^{∗} represents the set of critical measures. These measures approach dμ^{∗} under iteration of N, and from this setup it follows that they all have the same long distance behavior. If such a picture is correct, a critical point β_{c} will be observed in smooth one parameter families β ↦ dμ^{β} passing near dμ^{∗}. Furthermore, if the linearization DN of N at dμ^{∗} is analyzed, it is possible to make precise predictions about the rate of convergence or divergence (critical indices) of certain physical quantities as β approaches β_{c}. These rates are universal (i.e. independent of the particular one parameter family considered), since every family intersecting W_{s} is mapped arbitrarily close to W_{u} under iteration of N, so that the behavior near β_{c} is controlled by the local properties of N near dμ^{∗}. We do not attempt here to prove the correctness of Fig.1, except for the existence of the fixed point dμ^{∗}.

Consider now probability measures of the form

dμ(ϕ) = dμ_{C}(ϕ)F (ϕ),  (1.2)

where dμ_{C} denotes the Gaussian measure on the space D^{∗} of configurations ϕ: ^{/ /} Z^{d} → IR, with covariance C. Here, C is some fixed nonnegative operator on ℓ_{2}( ^{/ /} Z^{d}), as for example the inverse lattice Laplacean.

We specify a RG transformation by choosing an “averaging operator” B and a “scaling parameter” α > 0 such that

Γ = C − α^{2} B^{∗} C B  (1.3)

is positive. Given such a choice, we define the RG operator N by the equation

N (dμ) = dμ̃ ≡ dμ_{C}( .) F̃( .) ,  (1.4)

where

F̃(ϕ) = ∫ dμ_{Γ}(ψ) F (αB^{∗}ϕ + ψ) .  (1.5)

This definition realizes, for suitable choices of B and α, the above mentioned desiderata for a RG transformation. It is motivated by the identity

∫ dμ_{C}(ϕ) F (ϕ) = ∫ dμ_{C}(ϕ) (∫ dμ_{Γ}(ψ) F (α B^{∗}ϕ + ψ)) ,  (1.6)

which shows that the image measure dμ̃, which we get by integrating out fluctuations (integration with respect to dμ_{Γ}) and by rescaling, is again a probability measure.

The following choice of B is tailored to the hierarchical model which is defined below. We fix L to be an integer larger than one, and we set

B = L^{\frac{d}{2}} A^{∗}A^{2},  (1.7)

for

(Aϕ)_{i} = L^{− \frac{d}{2}} ∑_{j:[L^{−1}j]=i} ϕ_{j} .  (1.8)

Here, [x] is the point in ^{/ /} Z^{d} obtained from x ∈ IR^{d} by taking the integer part of its coordinates. Note that A^{∗}A is the projection in ℓ_{2}( ^{/ /} Z^{d}) onto functions which are constant on blocks of size L^{d}, and that AA^{∗} = Id..

Fix now

ℓ = L^{d} ,  α = L^{−(d−2)/2} .  (1.9)

Then our class of hierarchical models is defined by choosing the covariance C to be

C_{ij} = \frac{ℓ}{2} (δ_{ij} + L^{−2} ∑_{n=2}^{∞} α^{2n} δ_{[L^{−n}i],[L^{−n}j]}) .  (1.10)

This covariance does not describe a very “realistic” spin–spin–interaction, since it is not translation invariant. However, it mimicks the long distance behavior of (−Δ)^{−1}. For the fluctuation covariance (1.3) we obtain from equations (1.7) and (1.8)

Γ = \frac{ℓ}{2} · Id. .  (1.11)

In order to get a first impression of this particular RG transformation, let us apply N to a measure

dμ(ϕ) = dμ_{C}(ϕ) exp (−\frac{1}{2}(ϕ, Mϕ) − const.) ,  (1.12)

for M a nonnegative bounded operator on ℓ_{2}( ^{/ /} Z^{d}). A short exercise in Gaussian integrals shows that dμ̃ is again of the form (1.12), but with M replaced by

M̃ = α^{2}BM(1 + ΓM)^{−1}B^{∗} .  (1.13)

By using this relation, the domain of N can be extended to include measures which are not necessarily continuous with respect to dμ_{C}. Consider e.g. the family of Gaussians, formally corresponding to

M = λ · 2A^{∗}A ,  λ ≥ 0.  (1.14)

Inserting (1.11) and (1.14) into (1.13), we get

M̃ = λ̃ · 2A^{∗}A ,  λ̃ = \frac{L^{2}λ}{λℓ + 1} · 2A^{∗}A .  (1.15)

The action of the map induced by N in λ–space is shown in Fig.2.

It is an important feature of this RG transformation that the high temperature fixed point has a finite mass (λ_{HT} < ∞ in Fig.2). It reflects the fact that the “irrelevant” degrees of freedom integrated out in (1.5) are not just the fluctuations of wavelength < L.

In the search for a nontrivial fixed point we will now focus on a class of measures of the form

dμ(ϕ) = dμ_{G}(ϕ) ∏_{j} g_{B} (((A^{∗}Aϕ)_{j})^{2}) ,  (1.16)

where

G^{−1} = C^{−1} + M ,  (1.17)

with M given by (1.14) for some fixed λ > 0. To simplify the notation we have omitted at this point any reference to an infrared cutoff. The functions g_{B} in (1.16) are chosen to be entire analytic, with a growth rate restricted in such a way that the mass term (ϕ, Mϕ) dominates for large fields ϕ. Using (1.4), …, (1.11) we obtain

dμ̃(ϕ) = dμ_{G}(ϕ) ∏_{j} g̃_{B}(((A^{∗}Aϕ)_{j})^{2}),  (1.18)

where

\begin{aligned}g̃_{B}(t^{2}) & = exp ((λ − \frac{L^{2}λ}{λℓ + 1}) t^{2}) \\ & · \frac{1}{Z} ∫_{−∞}^{∞} ds exp (− (1 + λℓ) s^{2}) [g_{B} ((\frac{α}{1 + λℓ} t + s)^{2})]^{ℓ}, \end{aligned}  (1.19)

for some normalization constant Z. Our main goal is to prove that the reduced RG transformation g_{B} ↦ g̃_{B} has a non–Gaussian fixed point in d = 3 dimensions, for L = 2 and λ = 1/3.

The remaining part of the paper is organized as follows. In Section 2 we present our main results and a theorem on Borel transformations, which we need in later sections. Section 3 contains an outline of the proofs. In Section 4 we construct the infinite volume limit of our model and analyze the correlation functions. Section 5, finally, contains the details of our proofs in form of a computer program (written in the programming language C [23]).

2. Results

Motivated by the reasons mentioned in the introduction, we consider here the fixed point equation T_{B}h_{B} = h_{B}, for a renormalization transformation of the form

\begin{aligned}(T_{B}h_{B})(t^{2}) & = h̃_{B}(t^{2}) \\ & ≡ exp ((λ − \frac{L^{2}λ}{λℓ + 1}) t^{2}) \\ & · \frac{1}{c} ∫_{−∞}^{∞} ds exp (− (1 + λℓ) s^{2}) [h_{B}((\frac{α}{1 + λℓ} t + s)^{2})]^{ℓ} . \end{aligned}  (2.1)

The independent parameters d (dimension), L (linear block size) and λ (mass counterterm) will be fixed as follows.

d = 3,  L = 2,  λ = \frac{1}{3}.  (2.2)

Then the block volume ℓ and the scaling parameter α take the values

ℓ = L^{d} = 8,  α = L^{−(d−2)/2} = \sqrt{2}/2 .  (2.3)

For convenience later on we choose the normalization constant c in (2.1) to be

c = c(h_{B}) = ∫_{−∞}^{∞} ds exp (−(1 + λℓ)s^{2}) [h_{B}(s^{2})]^{ℓ},  (2.4)

so that h̃_{B}(0) = 1. Note that the fixed point of (2.1) and (1.19) agree up to a constant factor.

In order to complete the definition of T_{B}, we will now specify its domain.

Definition 2.1. For each ρ > 0, we define E_{ρ} to be the Banach space of all functions h_{B} which are entire analytic, real valued on real points, and for which the norm

∥h_{B}∥_{ρ} = sup_{t} |h_{B}(t)| exp (−\frac{|t|}{ρ})  (2.5)

is finite. Furthermore, we define E_{ρ}^{+} = E_{ρ} \ {0}.

Proposition 2.2. Let ρ be a real constant, satisfying 10/3 < ρ < 22. Then there exists some positive constant σ < ρ such that T_{B} is a C^{∞} map from E_{σ}^{+} to E_{ρ}^{+}.

Proof. Let ρ be fixed according to the above conditions. Then for every pair (σ, h_{B}) satisfying

\frac{10}{3} < σ < ρ, h_{B} ∈ E_{σ}^{+},  (2.6)

the integrals (2.1) and (2.4) converge and c(h_{B}) > 0. We shall now determine those choices (2.6) for which h̃_{B} ∈ E_{ρ}. By (2.2) and (2.3) we have

h̃_{B}(t^{2}) = \frac{1}{c(h_{B})} exp (−\frac{1}{33} t^{2}) ∫_{−∞}^{∞} ds exp (−\frac{11}{3} s^{2}) [h_{B}((\frac{3\sqrt{2}}{22} t + s)^{2})]^{8} .  (2.7)

Using the fact that

|h_{B}((r + s)^{2})| ≤ ∥h_{B}∥_{σ} · exp (\frac{|r + s|^{2}}{σ} )

≤ ∥h_{B}∥_{σ} · (exp (\frac{(|r| + s)^{2}}{σ} ) + exp (\frac{(|r| − s)^{2}}{σ} )) ,

(2.8)

for r ∈ ^{|}C and s ∈ IR, a short computation leads to the inequality

|h̃_{B}(t^{2})| exp (−\frac{|t|^{2}}{ρ} ) ≤ const. · exp (|t|^{2} (\frac{1}{σ̃} − \frac{1}{ρ})) ,  (2.9)

with σ̃ given by

\frac{1}{σ̃} = \frac{1}{33} + \frac{9}{242} ( \frac{8}{ρ} + \frac{64}{ρ^{2}} ( \frac{11}{3} − \frac{8}{ρ} )^{−1} ) .  (2.10)

It is now easy to check that σ̃ > ρ for σ sufficiently close to ρ. This implies that T_{B}E_{σ}^{+} ⊂ E_{ρ}^{+}. The same estimates can be used to show that T_{B} is C^{∞}.

Note that E_{ρ} is a subspace of E_{σ} for σ < ρ. In fact the injection E_{ρ} ↦ E_{σ} is compact (see e.g. [22]). As a consequence we have the following result.

Corollary 2.3. Assume ρ satisfies 10/3 < ρ < 22, and let h_{B} be an element of E_{ρ}^{+}. Then the tangent map DT_{B} of T_{B} at h_{B} is a compact linear operator on E_{ρ}.

For more information on DT_{B} see below. We shall now state the main result of this paper.

Theorem 2.4. For ρ = 7/2 there exists a non–Gaussian fixed point h_{B}^{∗} of T_{B} in E_{ρ}^{+}.

Remark. With the above choice of constants λ = 1/3 > 2/7 = 1/ρ, the function

f^{∗}(t) = const. · e^{−λt} h_{B}^{∗}(t)  (2.11)

is exponentially decreasing along the positive real axis, i.e. the mass term dominates the large field behavior of the (finite volume) measure (1.16). In Section 4 we will use this fact to control the thermodynamic limit.

We prove Theorem 2.4 by first designing an algorithm which reduces the assertion to a large but finite set of inequalities ( see Secton 3 ). Then we use a computer ^{∗)} to generate and verify the inequalities. The listing of the computer program is given in Section 5. As a by–product of this method we get very good bounds on the function h_{B}^{∗}. We also have the following \underline{nonrigorous} numerical results (which we believe can be proved by the same methods used here and in [1, 3]).

Numerical Results 2.5. For ρ = 7/2, the spectrum of the tangent map DT_{B} of T_{B} at the fixed point h_{B}^{∗} is contained inside the unit disk, with the exception of a simple eigenvalue δ = 2.9040714988 ….

From these data we get, using the relations given in [18, 19], the following critical indices.

γ = \frac{2 log L}{log δ} = 1.300 …  (γ_{I} ≈ 1.250 …),
ν = \frac{log L}{log δ} = 0.650 …  (ν_{I} ≈ 0.638 …),
η = 0  (η_{I} ≈ 0.041 …).

(2.12)

For a comparision we have listed in parenthesis the alleged values of the critical indices for the Ising model, calculated numerically by various authors [27–36, 38–47].

The main idea behind the proof of Theorem 2.4 is to consider h_{B} and h̃_{B} to be the \underline{Borel transform} [6, 9, 10] of two functions h and h̃, and then to study the fixed point problem for the RG transformation

T : h ↦ h̃ .  (2.13)

Note that this is exactly the opposite of what is normally done in perturbation theory, which is usually carried out around the trivial Gaussian fixed point of T_{B} [8, 9]. There the Borel transforms of h_{B} and h̃_{B} are studied, and not the \underline{inverse Borel transform} as we do. The main advantage of our method is that it eliminates the need to distinguish between large fields and small fields, which has been the principal source of technical difficulties in [11, 12]. In fact the fields in the inverse Borel plane are bounded, and contrary to what one might expect, the new transformation T takes a very simple form.

Since our definition of the Borel transformation differs slightly from the standard one, we now first discuss some domain questions before we explicitly define the action of T . The domain of T will be chosen among the following function spaces.

Definition 2.6. Given a positive number ρ, define A_{ρ} to be the Banach space of all functions h, which are analytic on D_{ρ} = {z ∈ ^{|}C | |z| < ρ}, continuous on the boundary ∂D_{ρ}, real valued on real points, equipped with the norm

∥h∥_{ρ} = sup_{x∈D_{ρ}} |h(x)|.  (2.14)

∗) An IBM Personal Computer equipped with the 8087–math–coprocessor, and the Microsoft C compiler 3.0 running on top of MS–DOS 3.1.

Furthermore, we define A_{ρ}^{+} = A \ {0}.

Note that the functions in A_{ρ}, as well as those in E_{ρ}, are uniquely determined by their Taylor series about the origin. Before defining the Borel transform, let us list first some simple bounds on Taylor coefficients. Consider two functions g ∈ A_{σ} and h_{B} ∈ E_{σ} for some σ > 0, and suppose that

g(x) = ∑_{n=0}^{∞} c_{n} x^{n},  |x| < σ,  (2.15)

h_{B}(t) = ∑_{n=0}^{∞} b_{n} t^{n},  t ∈ ^{|}C.  (2.16)

Then for all positive τ < σ and for all k ∈ ^{|}N we have

|c_{k}| ≤ σ^{−k}∥g∥_{σ} ,  ∥g∥_{τ} ≤ ∑_{n=0}^{∞} |c_{n}| τ^{n} ,  (2.17)

|b_{k}| ≤ \frac{κ\sqrt{k}}{σ^{k} k!}∥h_{B}∥_{σ} ,  ∥h_{B}∥_{τ} ≤ κ ∑_{n=0}^{∞} |b_{n}| \frac{τ^{n} n!}{\sqrt{n}} ,  (2.18)

where κ is some constant, independent of g and h_{B}. The first two estimates are trivial. The third bound follows by using Cauchy’s formula with contour |t| = nσ, and then Stirling’s formula. The last inequality is obtained by computing the norm of monomials t ↦ t^{n}.

We define now our Borel transform B, which differs slightly from the standard one. (see e.g. [6] for the standard definition). To make the difference more transparent, we define B in terms of its action on power series. An integral representation will be given in Proposition 2.8.

Definition 2.7. ^{∗)} For fixed σ > 0 consider the spaces

e_{σ} = ⋂_{ρ<σ} E_{ρ} ,  a_{σ} = ⋂_{ρ<σ} A_{ρ} .  (2.19)

Assume that h ∈ a_{σ} has the Taylor series

h(x) = ∑_{n=0}^{∞} c_{n} x^{n} .  (2.20)

Then we define its Borel transform Bh ∈ e_{σ} by

(Bh)(t) = ∑_{n=0}^{∞} c_{n} \frac{(−\frac{1}{2})!}{(n − \frac{1}{2})!} t^{n} ,  t ∈ ^{|}C .  (2.21)

∗) We prefer the notation x! instead of Γ(x+1)=∫_{0}^{∞}t^{x}e^{−t} dt for the Euler gamma-function.

Note that from (2.17), (2.18) it follows that B is a one–to–one map from a_{σ} to e_{σ}. If a function h_{B} ∈ e_{σ} is written in the form (2.16), then its inverse Borel transform is given by (2.15), with

c_{n} = \frac{(n − \frac{1}{2})!}{(−\frac{1}{2})!} b_{n} .  (2.22)

Proposition 2.8. Assume that 0 < ρ < σ. Then there is a constant κ such that for g ∈ A_{ρ} and h_{B} ∈ E_{σ},

(Bg)(t) = g(0) + t ∫_{0}^{∞} \frac{dx}{x^{1/2} (1 + x)^{3/2}} \{\frac{1}{2πi} ∮_{|z|=ρ} \frac{dz}{z} exp (\frac{t}{(1 + x)z} ) \frac{g(z) − g(0)}{z}\} , (2.23)

∥Bg∥_{ρ} ≤ κ ∥g∥_{ρ} ,  (2.24)
(B^{−1}h_{B})(x) = \frac{1}{\sqrt{π}} ∫_{−∞}^{∞} dt exp (−t^{2}) h_{B}(xt^{2}) ,  (2.25)

∥B^{−1}h_{B}∥_{ρ} ≤ κ ∥h_{B}∥_{σ}.  (2.26)

Proof. Assume that the above conditions are satisfied. Then it follows from (2.17) and (2.18) that for ρ < τ < σ

∥B^{−1}h_{B}∥_{ρ} ≤ const. ∥h_{B}∥_{τ} ≤ const. ∥h_{B}∥_{σ}.  (2.27)

This proves (2.26). To show (2.25) we use the fact that in E_{τ} the function h_{B} can be approximated by polynomials. Thus it is sufficient to consider h_{B}(t) = t^{n}. Using the identity \sqrt{π} = (−1/2)! , we obtain

\frac{1}{\sqrt{π}} ∫_{−∞}^{∞} dt e^{−t^{2}}(xt^{2})^{n} = \frac{1}{\sqrt{π}} ∫_{0}^{∞}dt e^{−t} t^{n− \frac{1}{2}} x^{n} = \frac{(n − \frac{1}{2})!}{(−\frac{1}{2})!} x^{n},  (2.28)

which proves (2.25). Assume now that 0 < τ < ρ. Then by (2.17), (2.18), it follows that

∥Bg∥_{τ} ≤ const. ∥g∥_{ρ} .  (2.29)

Since polynomials are dense in A_{ρ}, it suffices to show (2.23) for g(x) = x^{n}. Assuming that n > 0 (the case n = 0 is trivial), the contour integral in (2.23) can be evaluated as follows.

\begin{aligned}\frac{1}{2πi} ∮_{|z|=ρ} \frac{dz}{z} exp (\frac{t}{(1 + x)z} ) z^{n−1} & = (1 + x)^{1−n} \frac{1}{2πi} ∮_{|z|=ρ} \frac{dz}{z} exp (\frac{t}{z} ) z^{n−1} \\ & = (1 + x)^{1−n} ∑_{k=0}^{∞} \frac{t^{k}}{k!} · \frac{1}{2πi} ∮_{|z|=ρ} \frac{dz}{z} z^{n−k−1} \\ & = (1 + x)^{1−n} \frac{t^{n−1}}{(n − 1)!} . \end{aligned}  (2.30)

Using this together with a standard formula [37] for the beta–function, the r.h.s. of (2.23) becomes

\frac{t^{n}}{(n − 1)!} ∫_{0}^{∞} \frac{dx}{x^{1/2} (1 + x)^{n+1/2}} = \frac{(−\frac{1}{2})!}{(n − \frac{1}{2})!} t^{n}.  (2.31)

There remains (2.24) to be shown. But this estimate is now an immediate consequence of (2.23), since the exponential factor in the contour integral is bounded by exp(|t|/ρ).

This completes our discussion of the domain questions related to our version of the Borel transformation, and we start now the analysis of the renormalization transformation T , as defined in (2.13). From the above discussion it follows that the map T is related to the map T_{B} by a “change of coordinates” in function space, namely

T = B^{−1} T_{B} B .  (2.32)

Proposition 2.9. Let ρ = 7/2. Then the transformation T is a C^{∞} map from A_{ρ}^{+} to A_{ρ}^{+}, and

(T h)(x) = h̃(x) = \frac{1}{c} · \frac{1}{\sqrt{1 + \frac{x}{33}}} (B^{−1}( [Bh]^{8} ))( \frac{\frac{3}{11} + \frac{x}{22}}{1 + \frac{x}{33}} )  (2.33)

for h in A_{ρ}^{+}. Here, c = c(h) is defined such that h̃(0) = 1.

Proof. The regularity property of T could be proved directly by using (2.33). However we will not do this, since things already follow from Proposition 2.2 and Proposition 2.8. To show (2.33) we start with formula (2.1). Using (2.25) we get

h̃(x) ≡ (B^{−1}h̃_{B})(x) = \frac{1}{c} · \frac{1}{\sqrt{π}} ∫_{−∞}^{∞} dt exp (−t^{2}) exp ((λ − \frac{L^{2}λ}{1 + λℓ}) xt^{2})

· ∫_{−∞}^{∞} ds exp (−(1 + λℓ)s^{2}) [h_{B}((\frac{α}{1 + λℓ}\sqrt{x} t + s)^{2}]^{ℓ} .

(2.34)

Next we change variables from (t, s) to (t, τ), setting

s = s(t, τ) ≡ q(x) τ − \frac{α}{1 + λℓ} \sqrt{x} t ,  (2.35)

where

q(x) = (\frac{1 + (α^{2} − λ)x}{1 + λℓ + λ((α^{2} − λ)ℓ − 1)x})^{1/2} .  (2.36)

Then the integral with respect to dt in (2.34) can be computed explicitly, and we obtain

h̃(x) = \frac{1}{c} · \frac{1}{\sqrt{1 + λℓ + λ((α^{2} − λ)ℓ − 1)x}} · (B^{−1} ( [Bh]^{ℓ} ))(q(x)^{2}).  (2.37)

The equation (2.33) follows by inserting the numerical values given in (2.2) and (2.3).

Theorem 2.4 is now a corollary of the following theorem for T .

Theorem 2.10. For ρ = 7/2 there exists a fixed point h^{∗} of T in A_{ρ}^{+}. The function h_{B}^{∗} = Bh^{∗} is non–Gaussian.

The proof of this theorem is outlined in the next section. The details are given in Section 5.

3. Proof

We prove Theorem 2.10 by reducing it to a large number of simple inequalities. We start by rewriting the fixed point problem for the RG transformation T as a fixed point problem for a contraction M. M is then disassembled into a product of maps which can be bounded individually. This procedure leads to a very large number of inequalities which need to be checked, and we use at this point a computer to perform the bounds, and to assemble them properly.

3.1. The Contraction M

In order to simplify some estimates, let us introduce the norm

∥h∥ = ∑_{n=0}^{∞} \frac{1}{n!} |((\frac{d}{dz})^{n} h) (0)|.  (3.1)

Furthermore, we denote by A the Banach space of functions h in A_{1} for which the norm (3.1) is finite. Instead of T , we will now analyze the transformation

R = J_{ρ}^{−1} T J_{ρ},  ρ = 7/2,  (3.2)

where J_{r} is defined for r > 0 by

(J_{r}h)(z) = h(\frac{z}{r}).  (3.3)

Since A contains A_{τ} for every τ > 1, it follows from Proposition 2.2 and Proposition 2.8 that R is a C^{∞} map on A \ {0}.

We will now show how the fixed point problem for R can be reduced to a fixed point problem for a contraction. More precisely, we first define a map M, which has a “good chance” of being a contraction, and whose fixed point (if it exists) is related to the fixed point of R. The necessary properties of M are proved afterwards. A numerical analysis (whose details are irrelevant here) indicates that R has a nontrivial hyperbolic fixed point, with a one–dimensional local unstable manifold which is approximately parallel to the lines

L_{h} = \{h + s · e| s ∈ IR, e(z) = z + \frac{3}{5}z^{2}\}.  (3.4)

From these findings we can expect that, given a function h close to the fixed point of R, there are points on R(L_{h}) which are even closer. This motivates the following definition of M. To simplify the notation, let us identify functions σ in A with sequences (σ_{2}, σ_{3}, …). The identification is made by means of the power series

σ(z) = ∑_{n=0}^{∞} σ_{n+2} z^{n},  |z| ≤ 1.  (3.5)

Consider the family s ↦ U_{s} of “unpack” operators on A, given by

(U_{s} σ)(z) = 1 + az + (W_{s} σ)(z),  (3.6)

(W_{s} σ)(z) = s · e(z) + σ_{2} z^{2} + \frac{1}{3}σ_{3} z^{3} + \frac{1}{2}σ_{4} z^{4} + ∑_{n=5}^{∞} σ_{n}z^{n},  (3.7)

where a = .6989740 … (see Section 5) and e(z) = z + (3/5)z^{2} are fixed. Then we define M by the equation

U_{s}M σ = RU_{s}σ ,  s = s(σ).  (3.8)

Here, s = s(σ) is implicitly defined by the requirement that the r.h.s. of (3.8) lies in the range of U_{s} (see Section 3.2 below). Note that if σ^{∗} is a fixed point of M in A, then

h^{∗} = J_{ρ}^{−1}U_{s} σ^{∗},  s = s(σ^{∗}),  (3.9)

is a fixed point of T in A_{ρ}.

The factors 3 and 2 in (3.7) have been chosen with the intention to minimize the norm of the tangent map of M near σ^{∗}. Our next goal is to verify the conditions for the contraction mapping principle. This amounts to show that there is a function σ_{0} in A such that

a) M is C^{1} in a neighborhood U_{β} = {σ ∈ A | ∥σ − σ_{0}∥ < β } of σ_{0} ,

b) The tangent map DM_{σ} for σ ∈ U_{β} is bounded by ∥DM_{σ}∥ ≤ θ < 1 ,

c) σ_{0} is an approximate fixed point of M, i.e. ∥Mσ_{0} − σ_{0}∥ ≤ ε < β(1 − θ) .

As a consequence of a), …, c) there is a unique fixed point σ^{∗} of M in U_{β}, and Theorem 2.10 follows.

3.2. Formulas for M and DM

We start the proof of a), …, c) by replacing a) by a stronger but more convenient condition. Consider the function s( .) defined by (3.8). For given σ in A, s(σ) is obtained as the fixed point of the map

x ↦ s̃(x, σ) ≡ (RU_{x} σ)′(0) − a.  (3.10)

To find this fixed point we use Newtons’s method, i.e. we iterate the map

x ↦ N_{σ}(x) = [ \frac{∂s̃}{∂x}(x, σ) − 1 ]^{−1}(s̃(x, σ) − x),  (3.11)

starting with x in some given interval I. To guarantee convergence, it is sufficient to verify the conditions

\frac{∂s̃}{∂x}(x, σ) = (DR_{U_{x}σ} e)′(0) > 2  and  N_{σ}(x) ⊂ I,  (3.12)

for all pairs (x, σ) in I × U_{β}. Since s̃ is C^{∞} on I × U_{β}, it follows that s( .), and thus also M, are C^{∞} on U_{β}.

In the next subsection we will outline how the inequalities b), c) and (3.12) are proved on a computer. As a preparatory step we decompose M and DM into products of more elementary maps. First we have

Mσ = U_{s}^{−1}RU_{s} σ,  (3.13)

where s = s(σ) is the fixed point of (3.11). Note that all three factors in (3.13) are nonlinear. The nonlinear parts of R are given by

(Eh)(z) = [h(z)]^{8},  (3.14)

and

(Sh)(z) = [h(0)]^{−1} h(z).  (3.15)

In order to be able to work with the spaces e_{ρ} and A only, we combine E with a scaling J_{r} with r = 8.956 > 8. This leads to the following decomposition of R.

R = SK,  K = QV B^{−1}J_{r}EB .  (3.16)

Here, B is the Borel transform (2.21), and the remaining factors are defined by

(V h)(z) = h(\frac{1}{ρ · r} · \frac{\frac{3}{11} + \frac{ρ}{22}z}{1 + \frac{ρ}{33}z} ),  (3.17)

and

(Qh)(z) = [1 + \frac{ρ}{33}z]^{− \frac{1}{2}} h(z).  (3.18)

Using the definitions of R and U_{s}, we get the following expression for the tangent map DM of M.

DM_{σ} δσ = W_{δs}^{−1} DR_{U_{s}σ}W_{δs} δσ,  (3.19)

where s = s(σ), and where δs is given by

δs = [1 − \frac{∂s̃}{∂x}]^{−1} \frac{∂s̃}{∂σ}|_{(s(σ),σ)} δσ.  (3.20)

The tangent map DR of R, as obtained from (3.14), …, (3.18), is

DR_{h} = DS_{Kh} QV B^{−1}J_{r} DE_{Bh} B,  (3.21)
(DE_{h} δh)(z) = 8 [h(z)]^{7} δh(z),  (3.22)
(DS_{h} δh)(z) = [h(0)]^{−2} (h(0) · δh(z) − h(z) · δh(0) ).  (3.23)

3.3. Bounds used on the computer

We first give a definition of what we call a bound in the context of computer assisted proofs.

Let Σ, Σ′ be sets, and define P(Σ) and P(Σ′) to be the set of all subsets of Σ and Σ′ respectively. Let furthermore F and G be maps from D_{F} ⊂ P(Σ) and D_{G} ⊂ P(Σ) to P(Σ′).

Definition 3.1. G is called a bound on F if

a) D_{F} ⊇ D_{G},

b) F (K) ⊆ G(K) , ∀K ∈ D_{G}. “ G is a bound on F ” will also be written as G ≥ F.

In this language, our estimate on the tangent map DM of M is a bound G on the map

F: K ↦ {∥DM_{σ}∥ | σ ∈ K},  K ⊂ A.  (3.24)

Bounds of the type just defined have some general properties which make it possible to “mechanize” estimates. One of them is the fact that if G = G_{1} ∘ G_{2} is well defined, with G_{1} ≥ F_{1} and G_{2} ≥ F_{2}, then G is a bound on F_{1} ∘ F_{2}. In order to ensure that bounds can be properly composed, we adapt the following procedure.

• Given Σ and Σ′, we define a set of \underline{standard sets},

std(Σ) ⊂ P(Σ), std(Σ′) ⊂ P(Σ′).  (3.25)

• Given a map

F: P(Σ) ⊇ D_{F} → P(Σ′) ,  (3.26)

we then construct a map

G: std(Σ) ⊇ D_{G} → std(Σ′) ,  (3.27)

such that G ≥ F.

Standard sets will be defined below for Σ = IR, A, and e_{ρ}. For products of such spaces we define

std(Σ_{1} × Σ_{2}) = std(Σ_{1}) × std(Σ_{2}) .  (3.28)

Let us start with the bounds for binary operations on real numbers, such as “sum”, “product”, “equal”, . . .. Each of these operations is regarded as a map from P(IR×IR) to P(IR). For example

product(K) = {x · y| (x, y) ∈ K} ,  K ∈ std(IR × IR) ,  (3.29)
std(IR) = {[x^{<}, x^{>}] | x^{<}, x^{>} ∈ ℜ; x^{<} ≤ x^{>}} ,  (3.30)

where ℜ is a fixed finite set of rational numbers, containing zero and one (see e.g. [1, 2, 24] for a discussion of such a set). Then for each of the maps F listed above, there is a bound

G: std(IR × IR) ⊇ D_{G} → std(IR)  (3.31)

with nonempty domain, and the graph of G is a finite set. It is clear that in general G is not unique. We have chosen ℜ and G in such a way that G(K) can be evaluated on a computer. The details can be found in Section 5. Following the tradition of [1, 2], the elements of ℜ and of std(IR) are referred to as \underline{reps} and \underline{scalars}, respectively. The bounds on “sum”, “product”, …, are now regarded as given, and they will be denoted by “s sum”, “s product”, …. In order to make formulas more readable, we will also use notations like

{x · y} ≡ s product((x, y)) .  (3.32)

Having chosen all the necessary scalar bounds, we proceed by considering maps of the type (3.26), where Σ is a function space. A simple example of such a map would be

norm(K) = {∥h∥ | h ∈ K} ,  K ⊂ A .  (3.33)

Following our standard procedure, we describe now first the standard sets for A. We fix, once and for all, an integer N > 0. Then we define \underline{vectors} as (N + 3)–tuples v = (v_{0}, v_{1}, …, v_{N}, v^{(G)}, v^{(H)}) of scalars. To every vector v we associate a set

\begin{aligned}v̂ = \{h ∈ A | & h(z) = ∑_{i=0}^{N} c_{i} z^{i} + z h^{(G)}(z) + z^{N+1} h^{(H)}(z) ; c_{i} ∈ v_{i}; \\ & h^{(G)} ∈ A, ∥h^{(G)}∥ ∈ v^{(G)}; h^{(H)} ∈ A, ∥h^{(H)}∥ ∈ v^{(H)}\} , \end{aligned}  (3.34)

and we choose std(A) to be the collection of all these sets v̂. In order to simplify the notation, we shall only discuss here vectors v whose general components v^{(G)} and higher order components v^{(H)} are intervals which have zero as their lower boundary. Then the correspondence v ↦ v̂ is one–to–one, and we can identify vectors and standard sets.

We are now in the position to give a bound on the norm (3.33). For example, if s abs is a bound on the absolute value function, then the map

v ↦s norm(v) = s abs( [s abs( […
… s abs( [s abs( [v_{0} + v_{1}] ) + v_{2}] ) + …] ) + v^{(H)}] )

(3.35)

clearly satisfies all the necessary requirements. Similarly we can construct vector–valued bounds for elementary operations on A such as “sum”, “product”, . . .. For a description of these bounds we refer to Section 5. A detailed discussion can also be found in [1–5, 7].

The bounds considered so far are the basic building blocks used to prove Theorem 2.10. Such blocks can be assembled to provide bounds for the map

F: K ↦ {∥Mσ − σ∥ | σ ∈ K} ,  K ⊂ A ,  (3.36)

and for the map given in (3.24). Due to the large number of steps involved in this process, we have written this assembly plan in the form of a computer program. The main part of this program is the equivalent of Section 3.2, with the original maps replaced by their corresponding bounds. A large portion of the other parts of the program contain the exact definitions of the general purpose bounds such as s abs, v add, . . .. Finally, the program contains yet another part which is new and specific to our problem. Namely, we bound maps from and to the spaces e_{ρ}. These bounds will be discussed in the next subsection.

To complete our discussion on bounds we define now the standard sets for the spaces e_{ρ} (see (2.19) for the definition of e_{ρ}). We define a \underline{Borel–function} to be a list b = (b_{0}, b_{1}, …, b_{N}, b^{(G)}, b^{(H)}, b^{(E)}), whose first N + 3 components are scalars, and whose last component b^{(E)} is a rep (i.e. an element of ℜ). To every Borel–function b with b^{(E)} = ρ, we associate a set

\begin{aligned}b̂ = \{h_{B} ∈ e_{ρ} | & h_{B}(z) = ∑_{i=0}^{N} d_{i} z^{i} + z h_{B}^{(G)}(z) + z^{N+1} h_{B}^{(H)}(z) ; d_{i} ∈ b_{i}; \\ & h_{B}^{(G)} ∈ E_{ρ} , ∥h_{B}^{(G)}∥_{ρ} ∈ b^{(G)} ; h_{B}^{(H)} ∈ E_{ρ} , ∥h_{B}^{(H)}∥_{ρ} ∈ b^{(H)}\} , \end{aligned}  (3.37)

and we choose std(e_{ρ}) to be the collection of all these sets b̂. In order to simplify the notation, we shall only discuss Borel–functions b whose general components b^{(G)} and higher order components b^{(H)} are intervals which have zero as their lower boundary. Then the correspondence b ↦ b̂ is one–to–one, and we can identify Borel–functions and standard sets.

3.4. Bounds on Borel–Functions

We construct here bounds for the following three maps.

a) The Borel transform

\begin{matrix}F: P(A) ⊇ D_{F} & → P(e_{1}) \\ v & ↦ \{Bh | h ∈ v\} \end{matrix} (3.38)

b) The inverse Borel transform

\begin{matrix}F: P(e_{ρ}) ⊇ D_{F} & → P(A) \\ b & ↦ \{(BJ_{c})^{−1}h_{B} | h_{B} ∈ b\} \end{matrix} (3.39)

c) The product of Borel–functions

\begin{matrix}F: P(e_{ρ_{1}} × e_{ρ_{2}}) ⊇ D_{F} & → P(e_{ρ_{3}}) \\ K & ↦ \{g_{B,1} · g_{B,2} | g_{B,1} × g_{B,2} ∈ K\} \end{matrix} (3.40)

In the construction of these bounds we will use a map c ↦ {c} which associates to every real number c, |c| sufficiently small, a scalar {c} which contains c. This allows to define a product of real numbers with scalars, namely (c, s) ↦ {c · s} ≡ {{c} · s}.

The Borel Transform In order to bound the Borel transform (3.38), it suffices to construct a map

G: std(A) ⊇ D_{G} → std(e_{1}) ,  (3.41)

with the property that for all vectors v ∈ D_{G},

{Bh | h ∈ v} ⊆ G(v).  (3.42)

Let now v = (v_{0}, v_{1}, …, v_{N}, v^{(G)}, v^{(H)}) be a vector. Then we define a Borel–function b = (b_{0}, b_{1}, …, b_{N}, b^{(G)}, b^{(H)}, 1) in std(e_{1}) by

b_{i} = \{\frac{(−\frac{1}{2})!}{(i − \frac{1}{2})!} · v_{i}\} ,  (3.43)

b^{(G)} = [0, b_{G}^{>}] , b_{G}^{>} = \{\frac{(−\frac{1}{2})!}{(\frac{1}{2})!} · v^{(G)}\}^{>} ,  (3.44)

b^{(H)} = [0, b_{H}^{>}] , b_{H}^{>} = \{\frac{(−\frac{1}{2})!}{(N + \frac{1}{2})!} · v^{(H)}\}^{>} ,  (3.45)

where { }^{>} denotes the upper boundary of the scalar { }. We claim that the map G: v ↦ b satisfies (3.42). It is clear that this is the case for polynomials, i.e. when v^{(G)} and v^{(H)} are zero. It is also easy to see, that it then sufficies to restrict the analysis to functions

h(z) = z^{k} · l(z) ,  l ∈ A ,  (3.46)

for k = 1 or k = N + 1. We will show that for t ∈ ^{|}C

|t^{−k} (Bh)(t)| ≤ \frac{(−\frac{1}{2})!}{(k − \frac{1}{2})!} exp(|t|) ,  (3.47)

from which (3.42) follows. To prove (3.47), we use the integral representation (2.23) for the Borel transform B. We have

h_{B}(t) = (Bh)(t)

= t · ∫_{0}^{∞} \frac{dx}{(1 + x)^{3/2} \sqrt{x}} [\frac{1}{2π i} ∮_{|z|=1} \frac{dz}{z} exp (\frac{t}{z (1 + x)}) \frac{h(z)}{z}]

= t · ∫_{0}^{∞} \frac{dx}{(1 + x)^{3/2} \sqrt{x}} [\frac{1}{2π i} ∮_{|z|=1} \frac{dz}{z} exp (\frac{t}{z (1 + x)}) z^{k−1} l(z)]

= t^{k} · ∫_{0}^{∞} \frac{dx}{(1 + x)^{k+1/2} \sqrt{x}} [\frac{1}{2π i} ∮_{|z|=1} \frac{dz}{z} · \frac{exp (\frac{t}{z(1+x)}) − ∑_{ν=0}^{k−2} \frac{1}{ν!} (\frac{t}{z(1+x)})^{ν}}{(\frac{t}{z(1+x)})k−1} l(z)] .

(3.48)

The contour integral is bounded by using

\begin{aligned}|\frac{e^{ξ} − ∑_{ν=0}^{k−2} \frac{ξ^{ν}}{ν!}}{ξ^{ν−1}} | & = | ∑_{ν=k−1}^{∞} \frac{ξ^{ν−k+1}}{ν!} | = |∑_{ν=0}^{∞} \frac{ξ^{ν}}{(ν + k − 1)!}| \\ & = \frac{1}{(k − 1)!} |∑_{ν=0}^{∞} \frac{(k − 1)! ν!}{(ν + k − 1)!} · \frac{ξ^{ν}}{ν!} | ≤ \frac{1}{(k − 1)!} exp(|ξ|) , \end{aligned}  (3.49)

and therefore, using a standard integral representation for the beta–function [37], we get

\begin{aligned}|t^{−k} (Bh)(t)| & ≤ ∫_{0}^{∞} \frac{dx}{(1 + x)^{k+1/2} \sqrt{x}} · \frac{1}{2π} ∫_{0}^{2π} \frac{dφ}{(k − 1)!} exp (\frac{|t|}{1 + x}) ∥l∥_{1} \\ & ≤ \frac{(−\frac{1}{2})!}{(k − \frac{1}{2})!} ∥l∥_{1} exp(|t|) . \end{aligned}  (3.50)

This proves (3.47).

The Inverse Borel Transform We bound the map (3.39) for c > ρ and ρ ∈ ℜ by constructing a map

G: std(e_{ρ}) ⊇ D_{G} → std(A) ,  (3.51)

which satisfies

{(BJ_{c})^{−1}h_{B} | h_{B} ∈ b} ⊆ G(b) ,  b ∈ D_{G} .  (3.52)

Given a Borel–function b = (b_{0}, b_{1}, …, b_{N}, b^{(G)}, b^{(H)}, ρ), we define a vector v = (v_{0}, v_{1}, …, v_{N}, v^{(G)}, v^{(H)}) by choosing

v_{i} = \{\frac{(i − \frac{1}{2})!}{(−\frac{1}{2})!} · b_{i} · c^{i}\} ,  (3.53)

v^{(G)} = [0, v_{G}^{>}] , v_{G}^{>} = \{(b^{(G)})^{>} \frac{(\frac{1}{2})!}{(−\frac{1}{2})!} · \frac{1}{(1 − τ/ρ)^{3/2}} · \frac{1}{(1 − c/τ)}\}^{>},  (3.54)

v^{(H)} = [0, v_{H}^{>}] , v_{H}^{>} = \{(b^{(H)})^{>} \frac{(N − \frac{1}{2})!}{(−\frac{1}{2})!} · \frac{c^{N+1}}{(1 − τ/ρ)^{N+3/2}} · \frac{1}{(1 − c/τ)}\}^{>}, (3.55)

where τ = (2/5)ρ + (3/5)c. We claim that the map G: b ↦ v satisfies (3.52). As above, the “polynomial part” (3.53) is trivial, so we may assume that b only contains functions of the form

h_{B}(t) = t^{k} l_{B}(t) ,  l_{B} ∈ E_{ρ} ,  (3.56)

with k = 1 or k = N + 1. The inverse Borel transform of such a function h_{B} is given by

\begin{aligned}h(z) = (B^{−1}h_{B})(z) & = \frac{1}{\sqrt{π}} ∫_{−∞}^{∞} dt exp (−t^{2}) h_{B} (z t^{2}) \\ & = \frac{z^{k}}{\sqrt{π}} ∫_{−∞}^{∞} dt exp (−t^{2}) t^{2k} l_{B} (z t^{2}) . \end{aligned}  (3.57)

Therefore

|h(z)| ≤ \frac{|z|^{k}}{\sqrt{π}} ∫_{−∞}^{∞} dt exp (−t^{2}) t^{2k} ∥l_{B}∥_{ρ} exp (\frac{|z| t^{2}}{ρ} ) ,  (3.58)

and

sup_{|z|≤1} |J_{τ}^{−1} h(z)| = sup_{|z|≤1} |h(τ z)|

≤ \frac{τ^{k}}{\sqrt{π}} ( ∫_{−∞}^{∞} t^{2k} exp (−t^{2} (1 − τ/ρ)) dt) ∥l_{B}∥_{ρ}

= \frac{(k − \frac{1}{2})!}{(−\frac{1}{2})!} · \frac{τ^{k}}{(1 − τ/ρ)^{k+1/2}} · ∥l_{B}∥_{ρ} .  (3.59)

To bound the norm of J_{τ}^{−1}h as an element of A (see (3.1) for the definition of A) we use Cauchy’s formula with contour |z| = τ/ρ, and obtain

\begin{aligned}∥(BJ_{c})^{−1}h_{B}∥ & = ∥J_{c}^{−1}h∥ ≤ \frac{(c/τ)^{k}}{1 − (c/τ)} sup_{|z|=1} |h(τz)| \\ & ≤ \frac{(k − \frac{1}{2})!}{(−\frac{1}{2})!} · \frac{c^{k}}{(1 − τ/ρ)^{k+1/2}} · \frac{1}{1 − c/τ} · ∥l_{B}∥_{ρ} . \end{aligned}  (3.60)

This establishes (3.52) for the bound G, defined by (3.53), …, (3.55).

Remark. The particular choice τ = (2/5)ρ + (3/5)c has been made in order to minimize the r.h.s. of (3.60) for k = 1. For large k, (3.60) yields a very poor bound even for optimal choices of τ. We follow therefore the strategy to transform “higher order bounds” into “lower order bounds” whenever this is possible without negative consequences for subsequent estimates. More details can be found below in the discussion of the product of Borel–functions.

The Product of Borel–Functions Given ρ_{1}, ρ_{2}, ρ_{3} ∈ ℜ, satisfying ρ_{3} > ρ_{1} + ρ_{2}, we construct a bound

G: std (e_{ρ_{1}} × e_{ρ_{2}}) ⊇ D_{G} → std (e_{ρ_{3}})  (3.61)

on the product (3.40), satisfying

{h_{B,1} · h_{B,2} | h_{B,1} ∈ b_{1}, h_{B,2} ∈ b_{2}} ⊆ G ((b_{1}, b_{2})) .  (3.62)

Given two Borel–functions b_{1} and b_{2} in e_{ρ_{1}} and e_{ρ_{2}} respectively, we define G((b_{1}, b_{2}))= b_{3}, where

b_{3,i} = { ∑_{k=0}^{i} b_{1,k} b_{2,i−k}} , i = 0, …, N ,  (3.63)

b_{3}^{(G)} = [0, (b_{3}^{(G)})^{>}] ,  (3.64)
b_{3}^{(H)} = [0, 0] ,  (3.65)

with (b_{3}^{(G)})^{>} to be specified below. The sum in (3.63) is defined by repeated application of the scalar bound “s sum”. For the precise order of summation see Section 5. Since, as noted above, the inverse Borel transform (3.53), …, (3.55) gives poor estimates on higher order terms, we have chosen here to add all of these terms to the general bound b_{3}^{(G)}. This can be done by using the inequality

x^{k} ≤ (\frac{k}{e} )^{k}· exp(x) ,  k ∈ ^{|}N, x ∈ IR .  (3.66)

We define, for ε = {ρ_{3} − (ρ_{1} + ρ_{2})}^{>}

\begin{aligned}(b_{3}^{(G)})_{>} & = \{(\frac{1}{εe}) b_{1}^{(G)} b_{2}^{(G)} + (\frac{2N + 1}{εe})^{2N+1} b_{1}^{(H)} b_{2}^{(H)} \\ & + ∑_{i=N+1}^{2N} | ∑_{k=i−N}^{N} b_{1,k} b_{2,i−k}| (\frac{i − 1}{e (ρ_{1} + ρ_{2} + ε)})^{i−1} \\ & + ∑_{i=0}^{N} |b_{1,i}| b_{2}^{(G)} (\frac{i}{(ρ_{1} + ε) e})^{i} + ∑_{i=0}^{N} |b_{2,i}| b_{1}^{(G)} (\frac{i}{(ρ_{2} + ε) e})^{i} \\ & + ∑_{i=0}^{N} |b_{1,i}| b_{2}^{(H)} (\frac{N + i}{(ρ_{1} + ε) e})^{N+i} + ∑_{i=0}^{N} |b_{2,i}| b_{1}^{(H)} (\frac{N + i}{(ρ_{2} + ε) e})^{N+i} \}^{>}. \end{aligned} (3.67)

Again, the sums and products in (3.67) are to be replaced by their corresponding scalar bounds. Similarly s ↦ |s| is a shorthand notation for the bound on the absolute value function. It is now clear that the map G: (b_{1} , b_{2}) ↦ b_{3} satisfies (3.62), and is therefore a bound on the product (3.40).

Remark. The domain D_{G} of G is defined in terms of the domains of the maps “s sum”, “s product”, …. Namely, D_{G} is restricted to pairs (b_{1}, b_{2}), for which the scalar bounds in (3.63) and (3.67) can be evaluated. Similar restrictions apply to the other bounds discussed in this subsection. Their exact definitions can be found in the computer program in Section 5. Such restrictions are implemented there in the form of “conditional statements”, which result in error messages in the case of a domain violation.

4. Infinite Volume Limit and Correlations

In this section we construct the infinite volume measure corresponding to the function h_{B}^{∗} described in Theorem 2.4, and discuss its two point function.

Denote by f^{∗} the function defined by the equation

f^{∗}(t) = const. exp (−\frac{1}{3} t)h_{B}^{∗}(t) ,  (4.1)

where h_{B}^{∗} is the fixed point of T_{B}. Since h_{B}^{∗} is a solution of equation (2.1) for λ = 1/3, f^{∗} solves equation (1.19), for λ = 0, and for some choice of the constant in (4.1). From Theorem 2.4 it follows that f^{∗} satisfies the bound

|f^{∗}(t)| ≤ const. exp (−\frac{1}{3} Re t + \frac{2}{7} |t|) ,  (4.2)

and therefore f^{∗} is exponentially decreasing along the positive real axis. A repeated application of the identity (1.19) for f^{∗} leads to a bound

\frac{1}{k(δ)} exp (−(\frac{3}{8} + δ) t^{2}) < f^{∗}(t^{2}) < k(δ) exp (−(\frac{3}{8} − δ) t^{2}),  t ∈ IR,  (4.3)

for every δ > 0, 0 < k(δ) < ∞.

We consider now finite volume measures dμ^{(n)} associated with a sequence of cubes Λ_{1}, Λ_{2}, …

Λ_{n} = {−L^{n}, …, L^{n} − 1}^{3} .  (4.4)

If ϕ is a spin configuration on Λ_{n}, then we define dμ^{(n)} by the equation

dμ^{(n)}(ϕ) = dμ_{C_{n}}(ϕ) F_{n}(ϕ) ,  (4.5)

where

F_{n}(ϕ) = \frac{1}{Z_{n}} · ∏_{j∈Λ_{n}} f^{∗}(((A^{∗}Aϕ)_{j})^{2}) ,  (4.6)

and where

C_{n} = \frac{ℓ}{2}(1 + L^{−2} ∑_{k=2}^{n} L^{2k}(A^{∗})^{k}A^{k}) .  (4.7)

The normalization constants Z_{n} are chosen such that the measures dμ^{(n)} are probability measures. C_{n} is the hierarchical covariance introduced in (1.10), restricted to the cube Λ_{n}.

Proposition 4.1. For n = 2, 3, … the covariances C_{n} and the functions F_{n} satisfy the following recursion relations

C_{n} = α^{2} B^{∗} C_{n−1} B + Γ ,  (4.8)
F_{n−1}(ϕ) = F̃_{n}(ϕ) ≡ ∫ dμ_{Γ}(ψ) F_{n}(α B^{∗} ϕ + ψ) ,  (4.9)

with B^{∗} = L^{d/2}(A^{∗})^{2}A and Γ = ℓ/2 · Id., as defined in Section 1. Furthermore the normalization constants Z_{n} satisfy Z_{n} = Z_{1} for all n.

Proof. The recursion relation (4.8) is an immediate consequence of the definition (4.7). In order to prove (4.9) we compute F̃_{n},

F̃_{n}(ϕ) = \frac{1}{Z_{n}} ∫_{−∞}^{∞}( ∏_{j∈Λ_{n}} \frac{dψ_{j}}{\sqrt{πℓ}})exp(−\frac{1}{2} (ψ, \frac{2}{ℓ}ψ)) ∏_{j∈Λ_{n}} f^{∗}(((A^{∗}A(α B^{∗} ϕ + ψ))_{j})^{2}) . (4.10)

Since the integrand only depends on the block part s_{[L^{−1}j]} = (A^{∗}Aψ)_{j} of ψ_{j}, the fluctuations (1 − A^{∗}A)ψ can be integrated out, and therefore

F̃_{n}(ϕ) = \frac{1}{Z_{n}} ∏_{i∈Λ_{n−1}} \frac{1}{\sqrt{π}} ∫_{−∞}^{∞} ds_{i} exp (−s_{i}^{2}) ∏_{j:[L^{−1}j]=i} f^{∗}(((L(A^{∗})^{2}A ϕ)_{j} + s_{i})^{2})

= \frac{1}{Z_{n}} ∏_{i∈Λ_{n−1}} \frac{1}{\sqrt{π}} ∫_{−∞}^{∞} ds_{i} exp (−s_{i}^{2}) [f^{∗}((α (A^{∗}A ϕ)_{i} + s_{i})^{2})]^{ℓ}

= \frac{1}{Z_{n}} ∏_{i∈Λ_{n−1}} f^{∗}(((A^{∗}A ϕ)_{i})^{2}) = \frac{Z_{n−1}}{Z_{n}} F_{n−1}(ϕ) .

(4.11)

There remains to be shown that Z_{n} = Z_{n−1}. Using (4.8), we get

\begin{aligned}1 & = ∫ dμ_{C_{n}}(ϕ) F_{n}(ϕ) = ∫ dμ_{C_{n−1}}(ϕ) ∫ dμ_{Γ}(ψ) F_{n}(α B^{∗} ϕ + ψ) \\ & = ∫ dμ_{C_{n−1}}(ϕ) \frac{Z_{n−1}}{Z_{n}} F_{n−1}(ϕ) = \frac{Z_{n−1}}{Z_{n}} . \end{aligned}  (4.12)

This completes the proof of Proposition 4.1.

In the following we discuss the thermodynamic limit of the measures dμ^{(n)}. It is convenient to do this by considering the corresponding characteristic functionals S_{n}. The characteristic functional, or Fourier transform, of the measure dμ^{(n)}, is defined by the equation

S_{n}{g} = ∫ dμ^{(n)}(ϕ) exp(i < ϕ, g >) ,  (4.13)

where < , > denotes the inner product in ℓ_{2}( ^{/ /} Z^{3}).

The limit n → ∞ will be performed by using the following properties.

Proposition 4.2. Let g be a complex valued function on Λ_{m} and let β > |g_{j}| for all j ∈ Λ_{m}. Then for n ≥ m

|S_{n}{g}| ≤ exp (s(g) + κ · 8^{m−n}) ,  (4.14)

where κ is some constant independent of m,n, and g, and where

s(g) = κ · ∑_{j∈Λ_{n}} [ (β^{4/3} + β^{−2/3}) |g_{j}|^{2/3} + |g_{j}|^{2} ] .  (4.15)

Furthermore the characteristic functionals S_{n} are entire analytic in all of their arguments g_{j}, and they satisfy S_{n}{0} = 1.

Proof. The characteristic functional S_{n} can be written as

\begin{aligned}S_{n}\{g\} = \frac{1}{Z_{1}} ∫ dμ_{C_{n}}(ϕ) & exp(i < (1 − A^{∗}A) ϕ, g >) \\ & · ∏_{j∈Λ_{n}} f^{∗}(((A^{∗}Aϕ)_{j}^{2}) exp (i · g_{j} (A^{∗}Aϕ)_{j}) . \end{aligned}  (4.16)

As in (4.10) the integral over the fluctuations r = (1 − A^{∗}A)ϕ factorizes. It can be computed explicitly, namely

|∫ dμ_{C_{n}}(r) exp(i < r, (1 − A^{∗}A)g >)| = |exp (−\frac{1}{2} < (1 − A^{∗}A)g, C_{n} (1 − A^{∗}A)g >)|

≤ exp (2 < g, g >) .

(4.17)

The remaining terms will be bounded by using the exponential decay of the function f^{∗}. From (4.3) it follows that there are constants k and κ, such that for 0 < p < 1

f(t^{2}) exp(|σ t|) ≤ (f(t^{2}))^{1−p} (k exp (−t^{2}/4))^{p} exp(|σ t|)
≤ (f(t^{2}))^{1−p} exp (4 · |σ/p|^{2} + p · ln k)
≤ (f(t^{2}))^{1−p} exp (κ (β^{4/3} + β^{−2/3}) σ^{2/3}) .

(4.18)

The last inequality is obtained by taking p = (σ/β)^{2/3} < 1. This bound can be applied to each factor in (4.16) separately, yielding

|S_{n}\{g\}| ≤ \frac{1}{Z_{1}} exp(s(g)) ∫ dμ_{C_{n}}(ϕ) ∏_{j∈Λ_{n}} [f(((A^{∗}Aϕ)_{j})^{2})]^{1−p_{j}} ,  (4.19)

where s(g) is given by (4.15), and where

p_{j} = (|g_{j}| / β)^{2/3} < 1 .  (4.20)

Denote by F the product in (4.19). Using the relation (4.8) the integral in (4.19) can be written as the integral of dμ_{C_{n−1}} F̃, with

F̃(ϕ) = ∫ dμ_{Γ}(ψ) F (α B^{∗} ϕ + ψ) .  (4.21)

The r.h.s. of (4.21) factorizes into one–dimensional integrals, which can be bounded by using the fact that f^{∗} is a solution of the equation (1.19) for λ = 0. Applying Hölders inequality we get

\begin{aligned}F̃(ϕ) & = ∏_{i∈Λ_{n−1}} \frac{1}{\sqrt{π}} ∫_{−∞}^{∞} ds_{i} exp (−s_{i}^{2}) [f((α(A^{∗}Aϕ)_{i} + s_{i})^{2})]^{8(1−_{p_{i}})} \\ & ≤ ∏_{i∈Λ_{n−1}} [f(((A^{∗}Aϕ)_{i})^{2})]^{1−_{p_{i}}} , \end{aligned}  (4.22)

where

p̃_{i} = \frac{1}{8} ∑_{j:[L^{−1}j]=i} p_{j} .  (4.23)

Integrating this bound on F̃, we get

|S_{n}\{g\}| ≤ \frac{1}{Z_{1}} exp(s(g)) ∫ dμ_{C_{n−1}}(ϕ) ∏_{i∈Λ_{n−1}} [f(((A^{∗}Aϕ)_{i})^{2})]^{1−_{p_{i}}} .  (4.24)

The steps (4.19) to (4.24) can now be iterated n − 1 times. The integral which we obtain that way is bounded by Z_{1}^{1−a_{n}}, where a_{n} is the average value of the function j ↦ p_{j} on the cube Λ_{n}. This proves the bound (4.14). The analyticity of the functionals S_{n} follows from the absolute convergence of the integral (4.13). This completes the proof of Proposition 4.2.

Corollary 4.3. There exists a thermodynamic limit measure dμ for the sequence of finite volume measures dμ^{(n)}. The Fourier transform S,

S{g} = ∫ dμ(ϕ) exp(i < ϕ, g >) ,  (4.25)

is analytic in all g_{j}, and it satisfies the bound

|S_{n}{g}| ≤ exp (s(g)) ,  (4.26)

with s(g) given by (4.15).

To be more specific, S is defined on the space D of functions g on ^{/ /} Z^{3} with compact support, and is obtained as the limit of a subsequence of (S_{1}, S_{2}, …). The limit is constructed by using Montel’s theorem. It then follows (see e.g.[21]) that S is the Fourier transform of a probability measure dμ on the (algebraic) dual D^{∗} of D.

In the remaining part of this section we analyze the second moment of the infinite volume measure dμ. Under an assumption on the tangent map of the RG transformation T_{B}, we will show that the long distance behavior of the two point correlation function is canonical, i.e. that the critical index η, as defined through equation (1.1), is zero in our model.

The moments of dμ are obtained by differentiating the characteristic functional (4.25) with respect to the parameters g_{j}. Note that by Cauchy’s theorem, the moments of the finite volume measures dμ^{(n)} converge to the moments of dμ, along the same sequence Ω of indices as used in the construction of S. Since the fluctuations (1 − A^{∗}A)g contribute only a trivial factor to (4.25), we shall assume that (1 − A^{∗}A)g = 0. Then S{g} can be written as a partition function

Z(f) = lim_{n∈Ω} \frac{1}{Z_{n}} ∫ dμ_{C_{n}} (ϕ) ∏_{j∈Λ_{n}} f_{j}((A^{∗}Aϕ)_{j}) ,  (4.27)

with a single spin distribution f: j ↦ f_{j} given by

f_{j}(t) = f^{∗}(t^{2}) exp(i · t g_{j}) .  (4.28)

Following the RG philosopy, we study the long distance behavior of the measure dμ by successively integrating out small scale degrees of freedom in (4.27). From the identity (1.6), we obtain

Z(f) = Z( f̂ ) ,  (4.29)

f̂_{i}(t) = \frac{1}{\sqrt{π}} ∫_{−∞}^{∞} ds exp (−s^{2}) ∏_{j:[L^{−1}j]=i} f_{j}(α t + s) .  (4.30)

Note that the moments of dμ can also be written in the form (4.27), if we allow some of the functions f_{j} to be replaced by the corresponding g_{j}−derivatives. We shall only consider here the two point functions

S_{2}(k, m; h) = lim_{n∈Ω} \frac{1}{Z_{n}} ∫ dμ_{C_{n}} (ϕ) ∏_{j∈Λ_{n}\\{k,m\}} f^{∗}(((A^{∗}Aϕ)_{j})^{2}) ∏_{j∈\{k,m\}} h((A^{∗}Aϕ)_{j}) . (4.31)

In this particular case, and for |k − m| > 3, the identity (4.29) reduces to

S_{2}(k, m; h) = S_{2} ([L^{−1}k] , [L^{−1}m]; \frac{1}{ℓ}T h) ,  (4.32)

with T defined by the equation

(T h)(t) = ℓ · \frac{1}{\sqrt{π}} ∫_{−∞}^{∞} ds exp (−s^{2}) [f^{∗}((α t + s)^{2})]^{ℓ−1} h(α t + s) .  (4.33)

We choose now, once and for all, the function h to be

h(t) = t · f^{∗}(t^{2}) .  (4.34)

For this choice of h the two point function (4.31) becomes

S_{2}(k, m; h) = ∫ dμ(ϕ) (A^{∗}Aϕ)_{k} (A^{∗}Aϕ)_{m} ,  (4.35)

and we can apply the identity (4.32) to estimate its long distance behavior. The functions T h, T^{2}h, … can be computed explicitly. The result is

(T^{n}h)(t) = (α ℓ)^{n} h^{(n)}(t) ≡ (α ℓ)^{n} · \frac{ℓ}{L^{2} − 1} · [h_{2}(t) − L^{−2n}h_{1}(t)] ,  (4.36)

where

h_{1}(t) = t · (f^{∗})′(t^{2}) ,  h_{2}(t) = \frac{L^{2} − 1}{ℓ} · t · f^{∗}(t^{2}) + h_{1}(t) .  (4.37)

Remark. The map T is related to an extension of the RG transformation T_{B}. This extension has two additional eigenvalues of modulus > 1, namely α ℓ and α ℓ/L^{2}. The functions h_{1} and h_{2} are the corresponding eigenvectors for T.

We now iterate the relation (4.32) as many times as possible. The maximum number of iterations depends on the points k, m ∈ ^{/ /} Z^{3}, and is given by

n_{0} = max {n | [L^{−n}k] ≠ [L^{−n}m]}

= \frac{log |k − m|}{log L} + O (\frac{1}{|k − m|}) . (4.38)

This leads to the following bound on the two point function (4.35)

\begin{aligned}S_{2}(k, m; h) & = S_{2} ([L^{−n_{0}}k] , [L^{−n_{0}}m]; (\frac{1}{ℓ}T )n_{0} h) \\ & = α^{2n_{0}} S_{2} ([L^{−n_{0}}k] , [L^{−n_{0}}m]; h^{(n_{0})}) \\ & = α^{2n_{0}} (\frac{ℓ}{L^{2} − 1} · S_{2} ([L^{−n_{0}}k] , [L^{−n_{0}}m]; h_{2}) − O (L^{−2n_{0}})) \\ & = \frac{Z}{|k − m|^{d−2}} + O (\frac{1}{|k − m|^{d−1}} ) . \end{aligned}  (4.39)

where Z is some nonnegative constant. Here, we have used that there are three constants X, Y, Z, such that

S_{2} ([L^{−n_{0}}k] , [L^{−n_{0}}m]; εh_{1} + h_{2}) = Xε^{2} + Yε + Z ,  (4.40)

for every ε and for every choice of (k, m, n_{0}) satisfying (4.38). This can be seen by iterating the RG transformation defined in (4.29). For the first step, the integrals (4.30) are given by

f̂_{i}(t) = \frac{1}{\sqrt{π}} ∫_{−∞}^{∞} ds exp (−s^{2}) [f^{∗}((α t + s)^{2})]^{ℓ−p} [(ε h_{1} + h_{2})(α t + s)]^{p} ,  (4.41)

with p = 2 if i = [L^{−n_{0}−1}k] = [L^{−n_{0}−1}m], and p = 0 in all other cases. This shows that the r.h.s. of (4.40) is independent of k, m, and n_{0}. To bound the constants X, Y, and Z, we can follow the proof of Proposition 4.2.

Remark. This does not prove that the critical index η as defined in (1.1) is zero, since the field strength Z in (4.39) might vanish. To eliminate this possibility, we would have to analyze the map T ( or the tangent map of T_{B} ) in more details. A sufficient result would be that the funtion g, defined by

g(t) = [f^{∗}(t^{2})]^{−1} [h_{2}(t)]^{2} ,  (4.42)

has a nonzero component in the spectral subspace associated with the eigenvalue 8 of T.