# A geometric perspective on the scaling limits
of critical Ising and $\varphi^{4}_{d}$ models

Michael Aizenman  Departments of Physics and Mathematics, Princeton
University, Princeton NJ, USA<EMAIL_ADDRESS>
(Date: December 5, 2021)

###### Abstract.

The lecture delivered at the _Current Developments in Mathematics_ conference
(Harvard-MIT, Jan. 2021) focused on the recent proof of the Gaussian structure
of the scaling limits of the critical Ising and $\varphi^{4}$ fields in the
marginal case of four dimensions (joint work with Hugo Duminil-Copin). These
notes expand on the background of the question addressed by this result,
approaching it from two partly overlapping perspectives: one concerning
critical phenomena in statistical mechanics and the other functional integrals
over Euclidean spaces which could serve as a springboard to quantum field
theory. We start by recalling some basic results concerning the models’
critical behavior in different dimensions. The analysis is framed in the
models’ stochastic geometric random current representation. It yields
intuitive explanations as well as tools for proving a range of dimension
dependent results, including: the emergence in $2D$ of Fermionic degrees of
freedom, the non-gaussianity of the scaling limits in two dimensions, and
conversely the emergence of Gaussian behavior in four and higher dimensions.
To cover the marginal case of $4D$ the tree diagram bound which has sufficed
for higher dimensions needed to be supplemented by a singular correction. Its
presence was established through multi-scale analysis in the recent work with
HDC.

###### Contents

  1. 1 Introduction
  2. 2 Constructive quantum field theory  and functional integration
    1. 2.1 An outline
    2. 2.2 Lattice regularization
  3. 3 The statistical mechanical perspective
  4. 4 Preliminaries: the ferromagnetic models’ phase transition
    1. 4.1 Sharpness of the phase transition and general critical exponent bounds
    2. 4.2 Bounds on the two point function at $\beta_{c}$
  5. 5 Ising model’s random current representation
    1. 5.1 Definition and the switching lemma
    2. 5.2 Spin correlations as connectivity amplitudes
    3. 5.3 The $4$-point truncated correlation and clusters’ intersection probability
  6. 6 Emergent fermionic structures in $d=2$ dimensions
    1. 6.1 Pfaffian correlations for boundary spins in planar models
    2. 6.2 Emergent planarity in $2D$ finite range models
    3. 6.3 Non-gaussian structure of the scaling limit of $2D$ bulk correlations
  7. 7 The $U_{4}$ gaussianity criterion
  8. 8 Gaussianity of the scaling limits in $d>4$ dimensions
    1. 8.1 Sketch of the proof of Theorem 2.2 for the Ising case
    2. 8.2 Sketch of the proof of Theorem 2.2 for $\varphi^{4}$ variables
  9. 9 A decomposition of $\varphi^{4}$ variables into Ising constituents
    1. 9.1 The Griffiths-Simon class of scalar variables
    2. 9.2 Dimension balancing through point splitting
  10. 10 Gaussianity at the marginal dimension $d=4$
    1. 10.1 A suggestive dimensional analysis
    2. 10.2 The challenge
    3. 10.3 A suggestive dichotomy
    4. 10.4 Logarithmically improved bounds
  11. 11 Some open questions

##  1\. Introduction

Scaling limits of critical systems have been of obvious interest for the
theory of critical phenomena in statistical mechanics, and beyond that also
for the constructive program of quantum field theory. A recent result in this
area is the long awaited proof of the “marginal triviality” of the scaling
limits of critical $4D$ Ising and $\varphi^{4}_{4}$ models. It was derived
jointly with Hugo Duminil-Copin, and its details are presented in [6]. As the
manuscript is directly available, the goal of these notes is to present some
of the background, and place the discussion of four dimensions in the broader
context of the system’s different behavior in other dimensions.

We start with the presentation of the field theoretic perspective on the
question addressed in [6], and its relation to the natural statistical
mechanics question concerning the critical model’s fluctuating magnetization
field.

The discussion then pivots to a class of ferromagnetic one component spin
systems, which includes both the Ising model and the lattice $\varphi^{4}$
fields. Their critical behavior, which is dimension dependent, can be traced
to the stochastic geometry of corresponding systems of _random currents_ whose
definition is recorded below. Their intersection and percolation properties
are in direct relation with the structure of the correlation functions.

To demonstrate the dependence on dimension, and explain the transition which
occurs at $4D$ we list here some of the model’s dimension dependent features.
Specifically (postponing the references, that are provided below):

  * $\partial$(2D):

Along the boundaries of planar Ising models, the multi-spin correlations are
given by Pfaffians of the boundary-to-boundary two point function. This is
indicative of the model’s integrability through _emergent_ fermionic degrees
of freedom.

  * 2D:

In general, planar Ising models are integrable, in the sense that the model’s
free energy can be expressed as the a sum of the contribution of non-
interacting degrees of freedom. Unlike the boundary case, in the bulk these
are related to order-disorder fermionic variables which propagate with complex
phases (expressed in Kac-Ward path amplitudes, Kaufman spinors, Kadanoff
order-disorder variables, Smirnov parafermions). However, the the model’s
original spin variables scale into an interactive field theory.

  * 3D:

In three dimension the correlations are expected to have as their scaling
limit an interactive $\varphi^{4}$ field theory. The challenge of elucidating
the non-trivial critical behavior in $3D$, which is of obvious interest for
statistical mechanics, continues to attract the attention of both
mathematicians and theoretical physicists.

  * D$\geq$4:

In dimensions $d\geq 4$ the critical models’ scaling limits are Gaussian, and
as such they correspond to non-interactive field theories.

While the author’s CMD 2020(/21) lecture was focused on the last result, which
was derived in the recent joint work with Hugo Duminil-Copin [6], the purpose
of these notes is to expand on its background and natural context.

The presence of stochastic geometric scaffoldings behind the spin correlations
is not a new observation, but even so its range of applicability continues to
spread. Such representations yield insights into duality relations and tools
for rigorous results on the critical behavior in models which are out of reach
of explicit solutions. Given the focus of these notes we shall not delve here
into the full range of such methods, and focus on the _random current
representation_ (RCR). It provides a useful tool for the analysis of Ising
models, and other systems with similar spin-flip symmetry. It allows to cast
in intuitive stochastic geometric terms the results of some of the delicate
cancellations which affect the structure of the model’s correlation functions.

The study of correlation functions of critical models on $\mathbb{Z}^{d}$ and
the statistical field theories which emerge in their scaling limits, has also
attracted attention as a possible springboard towards quantum field theory.
Let us start by presenting this relation, borrowing heavily from the
introduction of [6].

##  2\. Constructive quantum field theory
and functional integration

###  2.1. An outline

Quantum field theories with local interaction play an important role in the
physics discourse, where they appear in subfields ranging from high energy to
condensed matter physics. The mathematical challenge of proper formulation of
this concept led to constructive quantum field theory programs (CQFT). A path
to the goal was charted through the proposal to define quantum fields as
operator valued distributions whose essential properties are formulated as the
Wightman axioms [85]. Wightman’s reconstruction theorem allows one to recover
this structure from the collection of the corresponding correlation functions,
defined over the Minkowski space-time. An essential feature of quantum physics
is the unitarity of the time evolution operator, generated by a self adjoint
operator which, preferably, is bounded below. A potential link with
statistical mechanics is suggested by the formal step of analytic continuation
of the time variable, under which unitary evolution in time, $e^{itH}$ acting
in the suitable Hilbert space of states, is related to the semigroup
$e^{-\beta H}$ which is of independent interest as the (unnormalized) density
operator of the thermal state at temperature $1/\beta$. The relation was
placed on firm footing by the Osterwalder-Schrader theorem [68, 69] which lays
down a program in which correlation functions satisfying the Wightman axioms
may potentially be obtained through analytic continuation from those of random
distributions defined over the corresponding Euclidean space that meet a
number of conditions: suitable analyticity, permutation symmetry, Euclidean
covariance, and reflection-positivity.

More explicitly, seeking natural candidates for scalar _Euclidean fields_ ,
one ends up with the task of constructing probability averages over random
distributions $\varphi(x)$, for which the expectation value of functionals
$F(\varphi)$ would have properties fitting the formal expression

(2.1) $\langle F\rangle=\frac{1}{\rm norm}\int
F(\varphi)\exp[-\mathcal{A}(\varphi)]\,d\varphi,$

where $d\varphi$ is an integral over (generalized) functions and
$\mathcal{A}(\varphi)$ is a local ‘action’. In this context, one is tempted to
consider expressions of the form

(2.2)
$\mathcal{A}(\varphi)=(\varphi,K\varphi)+\int_{\mathbb{R}^{d}}P(\varphi(x))\,dx$

with $(\varphi,K\varphi)$ a positive definite and reflection-positive
quadratic form, and $P(\varphi)$ a polynomial (or a more general function)
whose terms of order $\varphi^{2k}$ are interpreted heuristically as
representing $k$-particle interactions. An example of a quadratic form with
the above properties (at $J,b>0$) and also rotation invariance is

(2.3)
$(\varphi,K\varphi)=\int_{\mathbb{R}^{d}}\left(J\|\nabla\varphi(x)\|^{2}+b|\varphi(x)|^{2}\right)\,dx.$

The functionals $F(\varphi)$ to which (2.1) is intended to apply include the
smeared averages

(2.4) $\Phi_{f}(\varphi):=\int_{\mathbb{R}^{d}}f(x)\varphi(x)dx$

associated with continuous functions of compact support $f\in
C_{0}(\mathbb{R}^{d})$. By linearity, the expectation values of products of
such variables take the form

(2.5)
$\boxed{\langle\prod_{j=1}^{n}\Phi_{f_{j}}\rangle:=\int_{(\mathbb{R}^{d})^{n}}dx_{1}\dots
dx_{n}\,S_{n}(x_{1},\dots,x_{n})\,\prod_{j=1}^{n}f(x_{j})}$

with $S_{n}(x_{1},\dots,x_{n})$ characterizing the probability measure on the
space of distribution which corresponds to the expectation value
$\langle\,\cdot\,\rangle$. This is summarized by saying that in a
distributional sense

(2.6)
$\langle\prod_{j=1}^{n}\varphi(x_{j})\rangle=S_{n}(x_{1},\dots,x_{n})\,,$

with $S_{n}$ referred to as the Schwinger functions of the corresponding
euclidean field theory.

The joint distribution of the fields can also be described in terms of the
Ursell functions $U_{n}$ which bear the following relation to the moment
generating functional

(2.7)
$\langle\exp\left\\{\Phi_{f}\right\\}\rangle=\exp\left\\{\sum_{n=1}^{\infty}\frac{1}{n!}\int_{(\mathbb{R}^{d})^{n}}dx_{1}\dots
dx_{n}\,U_{n}(x_{1},\dots,x_{n})\,\prod_{j=1}^{n}f(x_{j}),\right\\}$

for all test functions $f\in C_{0}(\mathbb{R}^{d})$.

A relatively simple class of Euclidean fields are the Gaussian fields, which
result from $H$ with only quadratic terms. More generally, Gaussian fields of
flip-symmetric distribution (with $S_{2n+1}(x_{1},...,x_{2n+1})\equiv 0$) are
characterized by:

(2.8) $U_{2n}(x_{1},...,x_{2n})=0\quad\mbox{$\forall n\geq 2$\,, \quad and
\quad}U_{2}(x_{1},x_{2})=S_{2}(x_{1},x_{2})\,$

or, equivalently, the Wick’s law:

(2.9)
$\boxed{S_{2n}(x_{1},\dots,x_{2n})=\sum_{\pi}\prod_{j=1}^{n}S_{2}(x_{\pi(2j-1)},x_{\pi(2j)})\,=:\,\mathcal{G}_{n}[S_{2}](x_{1},\dots,x_{2n})}$

where $\pi$ ranges over pairing permutations of $\\{1,\dots,2n\\}$.

The presence in the continuum theory of particle interactions is expressed in
the deviation from Wick’s law. In situation in which $S_{2}(x,y)$ has the
interpretation of a propagation amplitude, $U_{2n}$ can be viewed as
representing effective $n$-particle interactions. Gaussian fields, for which
such multi-particle interactions are absent, have on occasions been referred
to as trivial.111Triviality is a misnomer when applied to a scaling limit,
since even in such cases the determination of the limit’s two point function
$S_{2}$ would require some non-trivial analysis.

However, even in the simple looking case of the Gaussian _free field_ , with
$H$ consisting of just the quadratic term (2.3), one finds that (2.1) requires
some care. It cannot be interpreted literally since ipso-facto the resulting
measure is supported on non-differentiable functions for which the integral in
the exponential, evaluated through the function’s Fourier transform, is almost
surely divergent. Furthermore, in dimensions $d>1$, the function $\varphi$
itself fails to be locally square integrable. Generally known methods allow to
tackle these difficulties. However, beyond one dimension (in which case one
gets the Wiener measure) the result is not a random function but a random
distribution. This in turn complicates the interpretation of the next
polynomial term in (2.2).

The heuristic “renormalization group” analysis by Wilson [86] indicates that
in low enough dimensions, specifically $d<4$ for $\varphi^{4}$, the
difficulties can be tackled through cutoff-dependent counter-terms. In
particular, to produce what would be regarded as a $\varphi^{4}$ field theory
the local interaction term in (2.2) is taken to be of the form

(2.10) $P(\varphi)=\lambda\varphi^{4}\,-b\varphi^{2}\,,$

allowing for $\lambda$ and $b$ to be adjusted as the _ultraviolet_ (finite
volume) and _infrared_ (short distance) cutoffs cutoffs are removed (the first
$R\to\infty$ and the second $a\to 0$). The goal of this process is to produce
convergence of the Schwinger functions $S_{n}(x_{1},..,x_{n})$ to a non-
trivial limit.

###  2.2. Lattice regularization

Partially successful attempts to carry rigorously a constructive project along
the above lines have been the focus of a substantial body of works. The means
employed have included various versions of rigorous renormalization group
analysis, with counter-terms, which are allowed to depend on regularizing
cutoffs, scale decomposition, renormalization group flows, and/or regularity
structures.

In our discussion we focus on the attempts to construct a $P(\varphi)$
expectation value functional (2.1) as the limit of its discretized versions

(2.11) $\boxed{\langle F(\varphi)\rangle=\frac{1}{\rm norm}\int
F(\varphi)\exp{[-H(\varphi)]}\prod_{x\in
V_{a,R}}e^{-\lambda\varphi_{x}^{4}+b\varphi_{x}^{2}}d\varphi_{x}.}$

Here $\\{\varphi_{x}\\}_{x\in V_{a,R}}$ is a collection of variables
associated with the sites of the finite graph

(2.12)
$\mathcal{V}_{a,R}=(a\mathbb{Z})^{d}\cap[-\mathcal{L},\mathcal{L}]^{d}\,$

with an a-priori distribution of the form
$\rho(d\varphi_{x})=e^{-\lambda\varphi_{x}^{4}+b\varphi_{x}^{2}}d\varphi_{x}$
and $H$ is the short range interaction

(2.13)
$H(\varphi)=-\frac{1}{2}\sum_{\\{x,y\\}\subset\mathcal{V}_{a,\mathcal{L}}}J\,(\varphi_{x}-\varphi_{y})^{2}\,.$

Quantities whose joint distribution we track in the scaling limit are based on
the collections of random variables of the form

(2.14) $\boxed{T_{f}(\varphi):=\frac{1}{\sqrt{\langle\big{[}\sum_{x\in
a\mathbb{Z}^{d}\cap[-1,1]^{d}}\varphi_{x}\big{]}^{2}\rangle}}\sum_{x\in\mathcal{V}_{a,\mathcal{L}}}f(x)\,\varphi_{x}}$

where $f$ ranges over $C_{0}(\mathbb{R}^{d})$ \- the collection of compactly
supported continuous functions.

######  Definition 2.1.

Given a sequence of joint values of $(a,\mathcal{L},J,\lambda,b)$ for which
$a\to 0$ and $\mathcal{L}\to\infty$, we say that model scaling limit exists if
for any finite collection of test functions $f\in C_{0}(\mathbb{R}^{d})$ the
joint distributions of the random variables $\\{T_{f}(\varphi)\\}$ has a
proper limit (a probability distribution with respect to which the variables
are almost surely finite).

Through a standard probabilistic construction, the limit can be presented as a
random field $\varphi$. 222By the Kolmogorov extension theorem, one may start
by selecting sequences of the parameter values so as to establish convergence
in distribution for a countable collection of test functions $f$, which is
dense in $C_{0}(\mathbb{R}^{d})$, and then use the uniform local integrability
of the rescaled correlation function and of the limiting Schwinger functions,
to extend the statement by continuity arguments to all $f\in
C_{0}(\mathbb{R}^{d})$. One may then recast the limiting variables as
associated with a single random $\varphi$, as in (2.4).

A regularity condition which it is natural to add here is that
$S_{2}(x,y)<\infty$ for non-coincidental pairs ($x\neq y$) and

(2.15) $\lim_{\|x-y\|\to\infty}S_{2}(x,y)=0$

A point of fundamental importance here is that the correlations of the
resulting field are observed on a scale ($L$) with is much larger than $a$ –
the range of the interactions due to which they emerge. For that to occur, the
system’s parameters $(J,\lambda,b)$ need to be very close to the critical
manifold along which the lattice system’s correlation length $\xi$ diverges
(on the scale set by $a$.)

Thus, the scaling limits discussed here are constructed the continuum limits
of the distribution of the variables $T_{f,L}$ induced by (2.1) in different
limits, for which the order of the different scales is

(2.16) $a\ll L\ll\xi(\beta)\leq\mathcal{L}\,.$

where, to summarize: $a$ is the lattice scale, $L$ the continuum limit scale,
i.e. the scale which is prodded through the variables $T_{f,L}(\sigma)$, $\xi$
is the correlation length, $\mathcal{L}$ is the finite systems cutoff, and
$A\ll B$ means $A/B\to 0$.

The constructive field theory program has yielded non-trivial scalar field
theories $\varphi^{4}_{2}$ and $\varphi^{4}_{2}$ (where the subscript is the
space dimension and the upper script the degree of the polynomial
$P(\varphi)$) [21, 42, 48, 69]. The progression of constructive results was
halted when it was proved that for $d>4$ the scaling limits of the field’s
lattice versions at $\beta\leq\beta_{c}$ yield only Gaussian fields [1, 36].
(Gauge field theories are not discussed here, cf. [56]).

In the $80$’s various partial results have indicated that the same may hold
true for the critical dimension $d=4$ (cf. [11, 14, 18, 39, 51]), however a
proof which covers also the strong coupled $\varphi^{4}$ (of which Ising is an
example) was reached only in the recent work [6]. Its main result is the
completion of the following statement which now covers also the marginal (and
thus the hardest) case $d=4$.

######  Theorem 2.2 (Gaussianity of $\varphi^{4}_{d}$, $d\geq 4$).

For any dimension $d\geq 4$, any random field reachable by the above
construction as a scaling limit of measures of the form (2.1) in limits
meeting the condition (2.16) is a generalized Gaussian process.

The previous results concerning $4D$ include proofs of convergence to Gaussian
field if one starts from small enough $\lambda$, i.e. sufficiently small
perturbations of the Gaussian free field[18, 31, 39, 51]. The rigorous
renormalization techniques presented in these works yield also more detailed
asymptotic, for that case. In comparison, the result proven in [6] covers
arbitrarily “hard” $\varphi^{4}$ fields (i.e. large $\lambda$). However, it
does not currently provide comparable information on of the exact scale of the
logarithmic corrections, and the exact expression for the covariance of the
limiting Gaussian field.

To keep the presentation simple, we first discuss the standard Ising model,
and then present the way the analysis extends to ferromagnetic system of
lattice $\varphi^{4}$ variables.

##  3\. The statistical mechanical perspective

What from the perspective of constructive field theory may be taken as a
disappointment is a positive and constructive result from the perspective of
statistical mechanics. The theoreticians’ goal there is to understand the
critical behavior in models which lie beyond the reach of exact solutions. The
proven gaussianity of the critical model’s scaling limits in $d\geq 4$
dimensions has therefore also a positive aspect.

Among the simply stated examples of the mean value functional (2.11) are the
scaled versions of the Gibbs equilibrium measures of the Ising model on
$\mathbb{Z}^{d}$.

A configuration of Ising spin variables on a finite graph
$\mathbb{G}=(\mathcal{V},\mathcal{E})$ (written in terms of its vertex set and
edge set) is a collection of binary variables
$\\{\sigma_{x}\\}_{x\in\mathcal{V}}$ which define a function
$\sigma:\mathcal{V}\mapsto\\{-1,1\\}$. The standard presentation of its
Hamiltonian is

(3.1)
$\boxed{{H}_{\mathbb{G}}(\sigma):=-\sum_{\\{x,y\\}\in\mathcal{E}}J_{x,y}\sigma_{x}\sigma_{y}-h\sum_{x\in\mathcal{V}}\sigma_{x}}$

with $J:=\\{J_{x,y}\\}_{\\{x,y\\}\in\mathcal{E}}$ the coupling coefficients,
and $h$ is the magnetic field parameter.

The system’s Gibbs equilibrium states at inverse temperature $\beta$
correspond to expectation value averages with respect to probability measures
on $\\{-1,1\\}^{\mathcal{V}}$ for which

(3.2) $\boxed{\mu_{\mathcal{V},\beta}(\\{\sigma\\})=e^{-\beta
H(\sigma)}/Z_{\mathcal{V}}(\beta)}\,.$

The normalizing factor is the partition function

(3.3) $Z_{\mathcal{V}}(\beta)\ :=\
\sum_{\sigma\in\\{-1,+1\\}^{\mathcal{V}}}\exp(-\beta{H}_{G}(\sigma))$

(An attentive reader may note the small discrepancy between (3.1) and (2.13).
The difference amounts to purely quadratic terms which even when not constant
can be absorbed in the definition of the a-priori measure.)
The Gibbs measure for the infinite graph $\mathbb{Z}^{d}$ is defined as the
$\mathcal{L}\to\infty$ limit of the above in
$\mathcal{V}=[-\mathcal{L},\mathcal{L}]^{d}$.

For generic values of the temperature, and/or the interaction parameters
$J_{x,y}$ the spin variables’ covariance decays exponentially in the distance,
with a finite correlation length $\xi(\beta)$. However, at $h=0$ the model
undergoes a phase transition which is continuous in the sense that
$\lim_{\beta\searrow\beta_{c}}\xi(\beta)=\infty$. At the critical point the
system’s spin fluctuations are meaningfully correlated at all scales. One way
to described the correlation’s structure is through the field theory which
emerges as a scaling limit of the variables

(3.4)
$\boxed{T_{f,L}(\sigma):=\frac{1}{\sqrt{\langle\big{[}\sum_{x\in\mathbb{Z}^{d}\cap[-L,L]^{d}}\sigma_{x}\big{]}^{2}\rangle}}\sum_{x\in\mathbb{Z}^{d}}f(x/L)\,\sigma_{x}}$

where $f$ ranges over $C_{0}(\mathbb{R}^{d})$ \- the collection of compactly
supported continuous functions. Looking up from the perspective of the lattice
the relation of the different scales is as presented in (2.16), with $a=1$.

For the Ising model at $h=0$, in which case the only adjustable thermodynamic
parameter is $\beta$, it is expected that the scaling limits will be the same
for all choices of $L,\mathcal{L}\to\infty$ and $\beta\searrow\beta_{c}$
meeting the condition (2.16).

Ising and $\varphi^{4}$ lattice systems can each be presented as the limit of
the other. The direction “$\varphi\Rightarrow$ Ising” is rather obvious. The
somewhat more surprising converse relation is through the Griffiths-Simon
representation of real variables with the $\varphi^{4}$ distribution as a
limit (in distribution) of the empirical average of ferromagnetically coupled
elemental Ising spins [44, 78]. The collection of such variables forms what we
call the Griffiths-Simon class.

The more explicit definition of the Griffiths-Simon class of variables is
postponed here to section 9, even though references will be made to it earlier
in the text. The most relevant point about it are summarized above.

The analysis of Ising models is facilitated by the random current
representation (RCR). Its power derives from a combinatorial identity (the
“switching lemma”) which made its appearance in the Griffiths-Hurst-Sherman
work on their eponymous correlation inequality [45]. It was further developed
in [1] as a tool for expressing the spin covariances in stochastic geometric
terms.

Combined with simple geometric considerations, RCR allows to present the
Ursell function $U_{4}$ of (2.7) as a multi-site connection amplitude for
random currents linking prescribed sources. From this follow (omitting here
the credits, which are given in the statements’ more detailed presentations in
the corresponding sections, below):

  * •

A simple explanation of the Pfaffian and fermionic relations which appear in
the planar case, and a positive elementary lower bound on $|U_{4}|/S_{4}$ in
$d=2$ dimensions.

  * •

An intuitive explanation, alas not yet a proof, of the non gaussian nature of
the scaling limits in $d=3$ dimension.

  * •

A relation between truncated correlations which quantifies the statement that
the model’s scaling limit is Gaussian if and only if Wick’s law holds that the
level of the $4$-point function.

  * •

A diagrammatic upper bound on $U_{4}$ in terms of the model’s two point
function. When combined with the “infrared bound” on the latter, it yields a
simple proof of the triviality of the scaling limits in $d>4$ dimensions.

  * •

A stepping stone for the extension of the above to $d=4$.

For the latter, an essential use was made of the fact that the random current
representation produces not only a diagrammatic upper bound on $U_{4}$ but
actually an identity from which one may extract also a lower bound. Both play
a role in the multi scale analysis which was needed for the marginal
dimension.

The random current representation was also instrumental in establishing what
may be viewed as a more preliminary statement about the model’s phase
structure, which are recalled below.

##  4\. Preliminaries: the ferromagnetic models’ phase transition

Statistical Mechanics, has had as its initial success the explanation of the
roots of the observed thermodynamic behavior: the meaning of entropy
(Boltzmann), the roots of irreversibility and the phenomena of phase
transition. Its impact has however extended quite further: it played an
essential role in the emergence of quantum physics (Plank), information theory
(Shannon), financial market’s analysis (spin glass), virology (contact
process), complexity in classical and quantum computations, and current
attempts to conceptualize the fabric of space-time (SYK).

Ising spin systems, through various generalizations of the initial model, play
a similarly fecund role within statistical mechanics. They offer an arena for
the formulation and testing of a range of concepts and analytical tools, only
few of which will be mentioned here.

The microscopic structure of real matter is obviously more complex than that
of the Ising model. And so is the phase structure of water and of materials
exhibiting ferromagnetic transition. Yet it is an observed fact that at
continuous phase transitions the critical behavior in many different systems
agree to an astounding degree (upon smooth adjustments of the thermodynamic
parameters). A common explanation is that since the scale on which the systems
are observed is vastly larger than that of its microscopic constituents, the
scale invariant critical correlation functions may be governed by continuum
theories which emerge from local interactions, and these come with a limited
collection of relevant parameters.

This observation gives impetus to the study the phase transitions in models of
simplified microscopic structure, such as the model which Ising was invited to
study by his research advisor Lenz. Its basic setup is described in (3.1).

We shall focus here on the models formulated on the $d$-dimensional graphs
$\mathbb{Z}^{d}$, with the translation invariant nearest neighbor interaction.
However it may be of interest to note that some of the basic results on the
Ising models’ phase structure are valid in the greater generality of
transitive (i.e. invariant under a transitive group of automorphism) and
ameanable graphs, with homogeneous interaction (by which is meant here
invariance under a transitive group of graph homomorphisms).

###  4.1. Sharpness of the phase transition and general critical exponent
bounds

A natural starting point is the following pair of general statements on the
phase structure of spin models in the class described above.

######  Theorem 4.1.

For any Ising model on a transitive and amenable graph, with a homogeneous
interaction (3.1) with $\|J\|_{1}:=\sum_{x}J_{x,x_{0}}<\infty$,

  1. 1)

(Existence of the limit [43, 35])
The limit defining the Gibbs state (formulated here with the free boundary
conditions) exists for all $(\beta,h)\in[0,\infty)\times\mathbb{R}$

  2. 2)

(Lee-Yang analyticity [63])
For any $\beta<\infty$ the state is real analytic in $h$ over
$\mathbb{R}\setminus\\{0\\}$

  3. 3)

(Analyticity at high temperatures [24])
For $\beta<1/\left[\|J_{0,x}\|_{1}\right]$ the state is a real analytic in
$(\beta,h)$, in the sense that so are all the expectation values $\langle
F\rangle_{\beta,h}$ of local measurable functions of the spins.

A relevant implication of (2) is that phase transitions in these models can
occur only along the line $h=0$. In dimensions $d>1$ at low enough
temperatures the state is in fact discontinuous across that line, i.e the
system exhibits a _first order_ phase transition. The proof of this fact is a
celebrated results of Peierls [70], in which he has elevated the model’s
status and confirmed the hopes of its creator.

The discontinuity is manifested in a jump in the magnetization
$M(\beta,h):=\langle\varphi_{0}\rangle_{\beta,h}$, or equivalently in the
strict positivity of the _residual_ (or _spontaneous_) magnetization

(4.1) $M^{*}(\beta)=\lim_{h\searrow 0}M(\beta,h)\,.$

Griffiths’ correlation inequalities imply that $M^{*}(\beta)$ is monotone
increasing in $\beta$ and thus the first order phase transition occurs along
the line segment $\beta\in(\beta_{c},\infty]$.333As to the situation at
$\beta_{c}$: with one notable exception [5, 82, 16] it is expected that
$M(\beta_{c},0)=0$. Proven cases include the nearest neighbor models on
$\mathbb{Z}^{d}$ in all dimensions [10, 7] ($3D$ being the last to be proven).

While the model’s high and low temperature regimes can be prodded by
convergent expansions, its behavior close to $(\beta_{c},0)$ can be accessed
only through non-perturbative means. No exact solution was found beyond the
two-dimensional nearest neightbor case. Nevertheless, the following could be
established rigorously (joint works with Barsky and Fernandez [4]).

######  Theorem 4.2 ([4]).

For any ferromagnetic Ising model on $\mathbb{Z}^{d}$ with a shift invariant
pair interaction, of exponential decay in $|x-y|$, there exists
$\beta_{c}\leq\infty$ such that:

$\displaystyle\forall\beta<\beta_{c}:$
$\displaystyle\quad\,\,\langle\varphi_{x}\varphi_{y}\rangle_{\beta,0}\leq
A(\beta)\,e^{-\|x-y\|/\xi(\beta)}$ with $A(\beta),\xi(\beta)<\infty$
$\displaystyle\forall\beta>\beta_{c}:$
$\displaystyle\\!\\!\\!\\!\\!\\!\langle\varphi_{x}\varphi_{y}\rangle_{\beta,0}\geq
M^{*}(\beta)^{2}$ $\displaystyle\quad\mbox{with $M^{*}(\beta)>0$}\,.$

And, approaching the critical point $(\beta_{c},0)$ from three different
directions:

(4.3) $\boxed{M(\beta,0)\geq\rm{C}|\beta-\beta_{c}|_{+}^{1/2}\,,\quad
M(\beta_{c},h)\geq\rm{C}\,h^{1/3}\,,\quad\frac{\partial M}{\partial
h}(\beta,0)\geq\frac{\rm{C}}{|\beta_{c}-\beta|_{+}^{1}}}$

with constants $\rm{C}$ which may vary from one equation to another.

Though stated initially for $\mathbb{Z}^{d}$, the given proofs readily
generalize to Ising spin systems on amenable transitive graphs with homogenous
ferromagnetic interaction of exponential decay. Furthermore, for the above
results - with the exception of the exponential decay at $\beta<\beta_{c}$,
the assumption of exponential decay of $J_{x,y}$ can be relaxed to just
$\|J\|_{1}<\infty$.

In essence, this general result can be seen as conveying two statements. The
first is that in the above described Ising models the phase transition occurs
directly from the regime of exponential decay of correlations to that of long
range order. That has been referred to as sharpness of the phase transition.

Its second part implies that the model’s critical exponents444Allowing for the
presence of logarithmic corrections we define the critical exponents
cautiously as the limits of the corresponding log ratios, e.g.
$\delta^{-1}=\lim_{h\searrow 0}\,\,\log M(\beta_{c},h)/\log h$.  are bounded
by what turns to be their values in the model’s mean-field version (for which
they are easily calculable)

(4.4) $\hat{\beta}\leq 1/2\,,\qquad\delta\geq 3\,,\qquad\gamma\geq 1\,\,.\\\ $

The proof of Theorem 4.2 proceeds through partial differential inequalities
(PDI), listed in section 11, which were derived for the order parameter
$M(\beta,h)$. These build on relations which are derived through the model’s
stochastic geometric representation. Partly overlapping results were developed
for the independent percolation models, and the comparison of the two has been
quite instructive.

Let us add that another path to the sharpness of the phase transition was more
recently presented in [28]. It applies more generally to $Q$-state FK random
cluster models, a class which includes both Ising ($Q=2$) and independent
percolation ($Q=1$).

As is also explained below, for the Ising model with the nearest neighbor
interaction the critical exponent bounds (4.4) are proven to be saturated in
dimensions $d\geq 4$ [10]. The existence of an upper critical dimension, above
which the critical behavior simplifies is a phenomenon shared also by some
other statistical mechanical models. In particular, for the independent
percolation the corresponding critical exponent bounds are saturated in
dimensions $d\geq 6$ (through the combinations of the results of [13, 17,
52]).

###  4.2. Bounds on the two point function at $\beta_{c}$

In the analysis of the critical behavior in high dimensions an important role
is played by the following bounds on the decay rate of the spin-spin
correlations at the critical point.

######  Theorem 4.3.

For the Ising model on $\mathbb{Z}^{d}$, $d>2$, with the nearest neighbor
interaction ($J_{u,v}=J\delta_{\|u-v\|,1}$), at the critical point

(4.5) $\boxed{\frac{\rm{C_{1}}}{\|x\|^{d-1}}\leq\beta
J\,\langle\sigma_{x}\sigma_{y}\rangle_{\beta_{c},0}\leq\frac{\rm{C_{2}}}{\|x-y\|^{d-2}}}$

The very important upper bound is based on the model’s reflection positivity,
through either the transfer matrix analysis of Glimm and Jaffe [42] or (with
better constant) the _Gaussian domination_ bound of Fröhlich, Simon and
Spencer [38]. For the spacial form of the upper bound in (4.5) the original’s
Frourier-transform is combined with the Messager Miracle-Sole monotonicity
[65], as was done by Sokal [79].

The lower bound in (4.5) is due to Simon [77], through a generally applicable
strategy with roots in the work of Dobrushin [24] and, going further back,
Hammersley [50] (in the context of percolation models).

Combined with (LABEL:eq:offcriticality) these bounds demonstrate the point
made above: power law behavior of
$\langle\varphi_{0}\varphi_{x}\rangle_{\beta,0}$, which is essential for
scaling limits, can be found in the vicinity the model’s critical point and
only there. The extension of this statement to the truncated correlations at
the high $\beta$ phase as was recently presented in [27].

In the analysis of the marginal case of $4D$ we added to (4.5) the observation
that $\langle\varphi_{0}\varphi_{x}\rangle_{\beta_{c},0}\cdot\|x\|^{d-2}$ is
monotone decreasing in the scales of $\|x\|$ [6]. (This played a useful role
in combining the contributions of different scales to the effective coupling
in $4D$, resulting in a logarithmic-type correction to the previous upper
bound by $C/L^{d-4}$.)

Let us add that for $d>4$ the lace expansion method is getting ever closer to
a proof that the upper bound in (4.5) is assymptotically saturated (cf. [73]
and references therein).

##  5\. Ising model’s random current representation

###  5.1. Definition and the switching lemma

In the random current representation (RCR), the Ising model’s defining degrees
of freedom $\\{\sigma_{x}\\}_{x\in\mathcal{V}}$ are summed over and replaced
by integer valued variables $\\{n(x,y)\\}_{(x,y)\in\mathcal{E}}$ associated
with the graphs edges. The transition casts the partition function as that of
a system of loops. The spin variables reappear as source insertion operators
in a systems or random currents with source constraints.

Due to the positivity of the RCR measure, and its convenient combinatorial
properties, this representation uncovers a stochastic geometric underpinning
of the spin-spin correlation functions, and sheds lights on the structure of
the model’s correlation functions, and by implication of the related field of
spin fluctuations.

In formulating the RCR we focus on the spin-flip symmetric Ising Hamiltonian
(5.1) with $h=0$, i.e. at zero magnetic field. (The extension to $h\geq 0$ is
not difficult, and is also useful [4]) 555A natural extension of RCR to $h\geq
0$ is obtained through the addition of a “ghost site” – a device which was
first employed by Griffiths in his derivation of correlation inequalities for
odd numbers of spins [43].

######  Definition 5.1.

For an Ising model on a graph $\mathbb{G}=\\{\mathcal{V},\mathcal{E}\\}$, with
the Hamiltonian

(5.1)
${H}_{\mathbb{G}}(\sigma):=-\sum_{\\{x,y\\}\in\mathcal{E}(\mathbb{G})}J_{x,y}\sigma_{x}\sigma_{y}\,.$

a current configuration $\mathbf{n}$ is an integer-valued function
$\mathbf{n}:\mathcal{E}\to\mathbb{Z}_{+}$. Referring to the edges as $(x,y)$,
the current’s set of sources is the set of sites

(5.2)
$\boxed{\partial\mathbf{n}\,:=\,\\{x\in\mathcal{V}:\,(-1)^{\sum_{(x,y)\in\mathcal{E}}\mathbf{n}(x,y)}=-1\\}}$

At given $\beta\geq 0$, we associate to the current configurations the weight
function

(5.3)
$\boxed{w(\mathbf{n}):=\prod_{\\{x,y\\}\subset\mathcal{E}}\frac{\displaystyle(\beta
J_{x,y})^{\mathbf{n}(x,y)}}{\mathbf{n}(x,y)!}}$

######  Lemma 5.2.

For any $\beta\geq 0$ the Ising model’s partition function (at $h=0$) can be
expressed as the following sum over sourceless random currents

(5.4)
$Z(\Lambda,\beta):=\sum_{\sigma:\mathcal{V}\to\\{-1,1\\}}\prod_{(x,y)\in\mathcal{E}}\exp(\beta
J_{x,y}\sigma_{x}\sigma_{y})=\boxed{2^{|\Lambda|}\sum_{\begin{subarray}{c}\mathbf{n}:\,\mathcal{E}\to\mathbb{Z}_{+}\\\
\partial\mathbf{n}=\emptyset\end{subarray}}w(\mathbf{n})}\,.$

Furthermore, the Gibbs state expectation value of even products of spins can
be presented as the effect on the sum of the insertion of sources at the
corresponding sites:

(5.5) $\boxed{\langle\prod_{x\in
A}\sigma_{x}\rangle_{\Lambda,\beta}=\frac{\displaystyle\sum_{\mathbf{n}:\,\partial\mathbf{n}=A}w(\mathbf{n})}{\displaystyle\sum_{\mathbf{n}:\partial\mathbf{n}=\emptyset}w(\mathbf{n})}\,}$

(the odd correlations vanish by the model’s symmetry).

The proof is elementary, starting from Taylor’s expansion

(5.6) $\exp(\beta J_{x,y}\sigma_{x}\sigma_{y})=\sum_{\mathbf{n}(x,y)\geq
0}\frac{(\beta
J_{x,y}\sigma_{x}\sigma_{y})^{\mathbf{n}(x,y)}}{\mathbf{n}(x,y)!}\,.$

It should be noted that any sourceless configuration, i.e. one with
$\partial\mathbf{n}=\emptyset$, can be viewed as the edge count of a
multigraph which is decomposable into a union of loops. Configuration with
$\partial\mathbf{n}=A$, such as the one appearing in the numerator of (5.5),
can be viewed as describing the edge count of a multigraph which is
decomposable into a collection of loops and of paths connecting pairwise the
sources, i.e. the sites of $A$.

For any $\beta$ the insertion of sources at $A=\\{x,y\\}$ produces a locally
detectable change in the constraints on $\mathbf{n}$. In addition, it requires
the admissible multigraphs to link the two sites. Since at small $\beta$
typical graphs are sparse, the latter condition should be expected to produce
an exponentially decaying factor. In contrast, at high $\beta$, where one may
expect robust percolation, the existence of a long connection would not in
itself cause a detectable effect.

More explicit statements can be reached through combinatorial identities, such
as the one presented by Griffiths-Hurst-Sherman [45] in their derivation of
the GHS inequality. The following formulation is particularly suited for our
purpose 666A simple way to process the combinatorics underlying Lemma 5.3 is
through the multigraph representation of the combination of the two random
currents, cf. [45, 1]. The switching argument can, and has been, deployed in a
number of slightly varying setups,.

######  Lemma 5.3 (Switching lemma).

Let $\mathbb{G}_{1}\supset\mathbb{G}_{2}$ be a pair of graphs, one included in
the other, and $J:\mathcal{E}_{1}\to\mathbb{Z}_{+}$ a set of coupling
constants which for the two graphs coincide on shared edges. Then for any pair
of vertex sets $A\subset\mathcal{V}_{1}$ and $B\subset\mathcal{V}_{2}$

(5.7) $\sum_{\begin{subarray}{c}\mathbf{n}_{1}:\partial\mathbf{n}_{1}=A\\\
\mathbf{n}_{2}:\partial\mathbf{n}_{2}=B\end{subarray}}F(\mathbf{n}_{1}+\mathbf{n}_{2})w(\mathbf{n}_{1})w(\mathbf{n}_{2})=\\\
\sum_{\begin{subarray}{c}\mathbf{n}_{1}:\partial\mathbf{n}_{1}=A\Delta B\\\
\mathbf{n}_{2}:\partial\mathbf{n}_{2}=\emptyset\end{subarray}}F(\mathbf{n}_{1}+\mathbf{n}_{2})w(\mathbf{n}_{1})w(\mathbf{n}_{2})\bm{1}_{{\mathbf{n}_{1}+\mathbf{n}_{2}}\in\mathcal{F}_{B}}.$

where $\mathbf{n}_{j}$ are summed over the random current configurations on
the corresponding graphs $\mathbb{G}_{j}$, $\mathcal{F}_{B}$ is the set of
currents $\mathbf{m}$ on $\mathbb{G}_{1}$ for which there exists a sub-current
$\mathbf{n}^{\prime}\leq\mathbf{m}$ supported on $\mathcal{E}_{2}$ with
$\partial\mathbf{n}^{\prime}=B$, and $A\Delta B$ denotes the symmetric
difference of sets ($:=(A\setminus B)\cup(B\setminus A)$).

The representation (5.5) and related combinatorial identities were employed in
Griffiths-Hurst-Sherman’s proof of the concavity of the magnetization
$M(\beta,h)$ as a function of $h$ over $h\geq 0$. The term _random currents_
was coined in [1] in the process of developing a stochastic geometric
perspective on the structure of the spin-spin correlations between distant
sites.

###  5.2. Spin correlations as connectivity amplitudes

The switching lemma allows to establish some exact cancellations though which
emerges a picture of the stochastic geometry underlying the structure of the
model’s spin-spin correlations. Following are three such relations, each
derived by an elementary application of Lemma 5.3.

For this purpose we denote :

${\bf P}^{A_{1},...,A_{k}}_{\Lambda,\beta}[\dots]$ – the probability average
over independent $k$-tuples
of currents $\\{\mathbf{n}_{1},...,\mathbf{n}_{k}\\}$ with source sets
$\partial\mathbf{n}_{j}=A_{j}$
${\bf E}^{A_{1},...,A_{k}}_{\Lambda,\beta}[\dots]$ – the corresponding
expectation value
$\bm{1}[x\stackrel{{\scriptstyle\mathbf{m}}}{{\leftrightarrow}}y]$ – the
indicator function of the event that $x$ and $y$
are connected by a path along edges of non-vanishing $\mathbf{m}$

i. The onset of long range order as a percolation transition

(5.8)
$\boxed{\left[\langle\sigma_{x}\sigma_{y}\rangle_{\Lambda,\beta}\right]^{2}=\frac{\displaystyle\sum_{\begin{subarray}{c}\mathbf{n}_{1}:\partial\mathbf{n}_{1}=\emptyset\\\
\mathbf{n}_{2}:\partial\mathbf{n}_{2}=\emptyset\end{subarray}}w(\mathbf{n}_{1})w(\mathbf{n}_{2})\bm{1}[x\stackrel{{\scriptstyle\mathbf{n}_{1}+\mathbf{n}_{2}}}{{\longleftrightarrow}}y]}{\displaystyle\sum_{\begin{subarray}{c}\mathbf{n}_{1}:\partial\mathbf{n}_{1}=\emptyset\\\
\mathbf{n}_{2}:\partial\mathbf{n}_{2}=\emptyset\end{subarray}}w(\mathbf{n})}\,={\bf
P}^{\emptyset,\emptyset}_{\Lambda,\beta}[x\stackrel{{\scriptstyle\mathbf{n}_{1}+\mathbf{n}_{2}}}{{\longleftrightarrow}}y]}$

By implication, the presence of long range order corresponds to percolation,
i.e. the appearance of an infinite cluster, in a duplicated system of
sourceless random currents [1]. To be precise, the natural definition of these
terms refers to infinite system, while (5.8) refer to finite ones, albeit
arbitrarily large. However, with some experience in the theory of percolation,
the correspondence is easy to establish.

A more explicit discussion of the extension of the RCR formalism to infinite
systems can be found in [7]. There, RCR percolation arguments (uniqueness of
the infinite cluster) are combined with bounds based on reflection positivity
for a proof of the continuity of the spontaneous magnetization in the n.n.
models. (Unlike the previous proofs of such statements, that argument covers
also the $3D$ case).

ii. Cluster based hitting probability

The following can be viewed as an RCR random cluster analog of the random
walk’s _hitting probability_ (for paths conditioned at two ends).

$\displaystyle\frac{\langle\sigma_{x}\sigma_{y}\rangle_{\Lambda,\beta}\langle\sigma_{y}\sigma_{z}\rangle_{\Lambda,\beta}}{\langle\sigma_{x}\sigma_{z}\rangle_{\Lambda,\beta}}$
$\displaystyle=$
$\displaystyle\frac{\displaystyle\sum_{\begin{subarray}{c}\mathbf{n}_{1}:\partial\mathbf{n}_{1}=\\{x,z\\}\\\
\mathbf{n}_{2}:\partial\mathbf{n}_{2}=\emptyset\quad\end{subarray}}w(\mathbf{n}_{1})w(\mathbf{n}_{2})\bm{1}[x\stackrel{{\scriptstyle\mathbf{n}_{1}+\mathbf{n}_{2}}}{{\longleftrightarrow}}y]}{\displaystyle\sum_{\begin{subarray}{c}\mathbf{n}_{1}:\partial\mathbf{n}_{1}=\\{x,z\\}\\\
\mathbf{n}_{2}:\partial\mathbf{n}_{2}=\emptyset\quad\end{subarray}}w(\mathbf{n})}\,$
$\displaystyle:=$ $\displaystyle\boxed{{\bf
P}^{\\{x,z\\},\emptyset}_{\Lambda,\beta}[y\in{\mathbf{C}}_{\mathbf{n}_{1}+\mathbf{n}_{2}}(x)]}\,.$

In the last expression the probability refers to a pair of random currents
with $\partial(\mathbf{n}_{1}+\mathbf{n}_{2})=\\{x,z\\}$. The corresponding
multigraph includes a path linking the two points decorated by a collection of
loops, which in the absence of percolation are all finite.

The equality expresses the probability that the random cluster
${\mathbf{C}}_{\mathbf{n}_{1}+\mathbf{n}_{2}}$, which by the source
constraints includes a path linking $x$ with $z$, includes also the point $y$.
While clusters are of a more complicated geometry than simple random paths, it
is natural to expect the path analogy to be useful, and in particular in high
dimensions.

iii. Covariance and the probability of avoided intersections
Partial derivatives of correlations are expressed in truncated correlation
functions, e.g.

(5.10) $\displaystyle\mbox{
}\quad\qquad\frac{1}{\beta}\frac{\partial}{\partial
J_{u,v}}\langle\sigma_{x}\sigma_{y}\rangle=\langle\sigma_{x}\sigma_{y}\sigma_{u}\sigma_{v}\rangle-\langle\sigma_{x}\sigma_{y}\rangle\,\langle\sigma_{u}\sigma_{v}\rangle=:\langle\sigma_{x}\sigma_{y}\,;\,\sigma_{u}\sigma_{v}\rangle$

(omitting the common subscript $(\Lambda,\beta)$). Such expressions appear in
the derivative in $\beta$ of the energy density
$\langle\sigma_{x}\sigma_{y}\rangle$ with $\\{x,y\\}$ a pair of nearest
neighbors, and of the magnetic susceptibility
$\chi(\beta)=\sum_{y}\langle\sigma_{0}\sigma_{y}\rangle\,.$

The random current expression for such truncated correlations is

(5.11)
$\displaystyle\langle\sigma_{x}\sigma_{y}\,;\,\sigma_{u}\sigma_{v}\rangle$
$\displaystyle=\langle\sigma_{x}\sigma_{u}\rangle\,\langle\sigma_{y}\sigma_{v}\rangle{\bf
P}^{\\{x,u\\},\\{y,v\\}}[u\stackrel{{\scriptstyle\mathbf{n}_{1}+\mathbf{n}_{2}}}{{\not\longleftrightarrow}}v]$
$\displaystyle\,\,+\,\,\langle\sigma_{x}\sigma_{v}\rangle\,\langle\sigma_{y}\sigma_{u}\rangle{\bf
P}^{\\{x,v\\},\\{y,u\\}}[u\stackrel{{\scriptstyle\mathbf{n}_{1}+\mathbf{n}_{2}}}{{\not\longleftrightarrow}}v]\,.$

In case $u$ and $v$ are neighbors and $x$ and $y$ are far away, the
probabilities of no intersection refer to a pair of random clusters in which
is imbedded a pair of long paths that start next to each other.

For a hint of what one may expect, let us note that for the simple random
walks such probabilities are uniformly bounded away from $0$ only in
dimensions $d>4$. This simple process can be viewed as describing critically
damped random paths. It is then natural to ask whether the same, or a close
analog of that, is true for the connected clusters associated with the random
current system at $\beta_{c}$ (which as we saw is its percolation threshold).

As it turns out, a version of the $d>4$ part of the last assertion is
provable, and it plays a role in the proofs that the critical exponent bounds
(4.4) are saturated in those dimensions. [1, 10].

###  5.3. The $4$-point truncated correlation and clusters’ intersection
probability

Since the main objective of these notes is the structure of the multi-spin
correlations we now fast forward to the Ursell function $U_{4}$, which plays a
key role in our discussion.

Upon power series expansion, or equivalently differentiation, of (2.7) one
finds that for sign-flip symmetric fields

(5.12)
$U_{4}(x_{1},...,x_{4}):=\langle\sigma_{x_{1}}\sigma_{x_{2}}\sigma_{x_{3}}\sigma_{x_{4}}\rangle-\\\
-\left[\langle\sigma_{x_{1}}\sigma_{x_{2}}\rangle\langle\sigma_{x_{3}}\sigma_{x_{4}}\rangle+\langle\sigma_{x_{1}}\sigma_{x_{3}}\rangle\langle\sigma_{x_{2}}\sigma_{x_{4}}\rangle+\langle\sigma_{x_{1}}\sigma_{x_{4}}\rangle\langle\sigma_{x_{2}}\sigma_{x_{3}}\rangle\right]$

As a direct expression of the deviation of the $4$-point function’s from
Gaussian behavior, $|U_{4}(x_{1},...,x_{4})|$ should be measured against
$\langle\sigma_{x_{1}}\sigma_{x_{2}}\sigma_{x_{3}}\sigma_{x_{4}}\rangle$, or
equivalently the largest of the product of the two pair correlation functions.
For that RCR yields the following pair of expressions.

######  Theorem 5.4 ([1]).

For any ferromagnetic system of Ising spins, at $h=0$,

(5.13) $U_{4}(x_{1},...,x_{4})=\\\ =\
-2\langle\sigma_{x_{1}}\sigma_{x_{2}}\sigma_{x_{3}}\sigma_{x_{4}}\rangle\,\,{\bf
P}^{\\{x_{1},x_{2},x_{3},x_{4}\\},\emptyset}[\mbox{all $x_{j}$ are
$(\mathbf{n}_{1}+\mathbf{n}_{2})$ interconnected}]\,\\\
=\boxed{-2\langle\sigma_{x_{1}}\sigma_{x_{2}}\rangle\,\langle\sigma_{x_{3}}\sigma_{x_{4}}\rangle\,\,{\bf
P}^{\\{x_{1},x_{2}\\},\\{x_{3},x_{4}\\}}[\\{x_{1},x_{2}\\}\stackrel{{\scriptstyle\mathbf{n}_{1}+\mathbf{n}_{2}}}{{\longleftrightarrow}}\\{x_{3},x_{4}\\}]}\hskip
48.36958pt$

This important pair of identities are a simple consequence of Lemma 5.3:
switching all sources to $\mathbf{n}_{1}$, one finds that the resulting terms
produce perfect cancellation if the
$(\mathbf{n}_{1}+\mathbf{n}_{2})$-connected cluster of $x_{j}$ form two
distinct clusters. The alternative is that all the sources are connected, and
in that case the overall factors is $1-3=-2$. From this observation follows
the first line in (5.13). The transition from the first expression to the
second is by yet another application of the switching lemma.

Let us add that the equality of the two expressions in (5.13) is somewhat
remarkable since the symmetry under permutations is obvious in one case but
not the other. In this respect, the second relation is somewhat reminiscent of
the (assumed) _crossing symmetry_ of the conformal field theory, though here
the relation is not limited to critical models.

We next turn to the implication of this relation for the model in different
dimensions: $d=2$, $d>4$, and then the marginal case of $d=4$. (One may note
that we omit the case $d=3$, and not by chance.)

##  6\. Emergent fermionic structures in $d=2$ dimensions

###  6.1. Pfaffian correlations for boundary spins in planar models

Since the breakthrough results of Onsager [67], and numerous works which
followed, among the Ising models the planar ones form a special class, with
strong links to other integrable systems. Their partition functions at $h=0$
admits a determinant expression, in terms of $\\{\beta J_{x,y}\\}$-dependent
matrices of size $2|\mathcal{V}|\times 2|\mathcal{V}|$. For translation
invariant interactions on homogeneous graphs of degree $\nu$ the matrices can
be presented as block diagonal with $4\times 4$ blocks, indexed by the
momentum (the Fourier transform variable), and through that a calculable
expression follows.

Furthermore, a rich collection of methods exist for the lattice correlation
functions (cf. [75, 57, 58, 22, 15] and references therein). There is also a
(partly separate) set of tools for the description of the conformally
invariant field theory which is expected to emerge in the models’ scaling
limit (cf. [71, 23, 22]).

We shall not discuss here the special methods which have been devised for the
analysis of the exactly solvable case of $2D$. However, since in our
discussion of the higher dimensions we shall invoke the random current
representation, let us note here some of its implications for the planar case.
A striking example is the effectiveness of (5.13) in explaining the following
curiosity, which was noted and derived by other means by Groeneveld-Boel-
Kasteleyn [47].

######  Theorem 6.1 (Pfaffian structure of boundary spin correlations [47]).

For any Ising model on a planar graph $\mathbb{G}$, with only nearest-neighbor
couplings $\\{J_{x,y}\\}$ (which need not be uniform), at $h=0$ and any
$\beta$ the boundary spins’ correlation functions have the Pfaffian structure.

Figure 1. A configuration of sites along the boundary of a planar domain. The
signature of each pairings’ contributions to the Pfaffian equals the parity of
the number of crossings of the pairing paths ($(-1)^{2}$ in the depicted
example).

More explicitly: for any $2n$-tuple $(x_{1},\dots,x_{2n})$ of different sites
located along the outer boundary, or the boundary of a fixed face of
$\mathbb{G}$,

(6.1)
$\boxed{\langle\sigma_{x_{1}}\cdots\sigma_{x_{2n}}\rangle_{\mathbb{G},\beta}\
=\
\sum_{\pi\in\Pi_{2n}}\operatorname{sgn}(\pi)\prod_{j=1}^{n}\langle\sigma_{x_{j}}\sigma_{x_{\pi(j)}}\rangle_{\mathbb{G},\beta}\,}$

where the sum is over pairing $\\{j\leftrightarrow\pi(j)\\}$ 777Pairings
satisfy $\pi(j)\neq j$ and $\pi(\pi(j))=j$. The signature is easily computed
as the parity of the number of crossings in a generic collection of paths
linking the corresponding pairs, if drawn on the same side of the graph’s
boundary, as depicted in Figure 1.  and $\operatorname{sgn}(\pi)$ is the
parity of the number of pair crossings in any continuous deformation, along
the boundary line, of the labeled configuration under which $\\{x_{j}\\}$
emerge relabeled as $\\{x_{\pi(j)}\\}$.

Equivalently, (6.1) can be expressed by saying that for the sites labeled in a
cyclic order the correlation is given by a Pfaffian of the upper triangular
array $[A_{i,j}]$ whose entries are given by the two point correlation
function:888Pfaffians appeared very early in the exact solution of the free
energy of two-dimensional Ising models [32, 55, 34], and are linked to
determinants through the relation ${\bf Pf}_{n}(A)^{2}=\text{det}(A)$, where
$A$ is the antisymmetric matrix whose entries for $j<k$ are given by
$A_{j,k}$.

(6.2)
$\langle\sigma_{x_{1}}\cdots\sigma_{x_{2n}}\rangle_{\mathbb{G},\beta}={\bf
Pf}_{n}\left(\left[\langle\sigma_{x_{i}}\sigma_{x_{j}}\rangle\right]_{1\leq
i<j\leq 2n}\right)\,.$

Pfaffian structure of correlation functions is a fermionic counterpart to the
Wick rule of the (bosonic) correlations of Gaussian fields. In both cases all
the $n$-point functions are simply expressible in terms of the two-point
function. Each evokes integrability in the sense of the existence of a non-
interactive structure. However, in contrast to the Gaussian correlations,
Pfaffians indicate the presence of fermonic degrees of freedom. It is of
interest to note that our RCR expression for $U_{4}$ sheds light on both
cases.

Let $\\{x_{1},...,x_{4}\\}$ be a cyclicly labeled configuration of points
along the boundary of a finite planar graph $\mathbb{G}$. In that situation
any pair of paths within the domain with intertwined end points
($x_{1}\leftrightarrow x_{3}$ and $x_{2}\leftrightarrow x_{4}$), on its
boundary have to intersect, and thus

(6.3) ${\bf
P}^{\\{x_{1},x_{3}\\},\\{x_{2},x_{4}\\}}[\\{x_{1},x_{3}\\}\stackrel{{\scriptstyle\mathbf{n}_{1}+\mathbf{n}_{2}}}{{\longleftrightarrow}}\\{x_{2},x_{4}\\}]=1\,.$

Combined with the last expression in (5.13) this implies

(6.4) $\displaystyle
U_{4}(x_{1},...,x_{4})=-2\langle\sigma_{x_{1}}\sigma_{x_{3}}\rangle\,\langle\sigma_{x_{2}}\sigma_{x_{4}}\rangle\,,$

which by the definition of $U_{4}$ means

(6.5)
$\langle\sigma_{x_{1}}\sigma_{x_{2}}\sigma_{x_{3}}\sigma_{x_{4}}\rangle=\langle\sigma_{x_{1}}\sigma_{x_{2}}\rangle-\langle\sigma_{x_{1}}\sigma_{x_{3}}\rangle\,\langle\sigma_{x_{2}}\sigma_{x_{4}}\rangle+\langle\sigma_{x_{1}}\sigma_{x_{4}}\rangle\,\langle\sigma_{x_{2}}\sigma_{x_{3}}\rangle\,.$

This proves (6.1) for the $4$-point function. The argument’s extension to
higher correlations can be found in [9].

###  6.2. Emergent planarity in $2D$ finite range models

The above strict relations do not carry over into the extensions of the
ferromagnetic model to planar models with more general variables in the
Griffiths-Simon class. The reason for that is also clear from the above
argument: Pfaffian correlations for systems built from Ising spins require
probability $1$ for the actual intersection of paths whose planar projections
cross. This feature is lost in the construction depicted in Figure 2.

Nevertheless, taking into account the expected fractal nature of the relevant
RCR clusters at $\beta_{c}$, it is natural to conjecture that the Pfaffian
structure of correlations emerges at the models’ critical points. A related
situation shows up in $2D$ models on $\mathbb{Z}^{2}$ with finite range
interactions which are not planar (and thus also no longer solvable). For this
case the following result was proven using the RCR and percolation type
techniques.

######  Theorem 6.2.

[[8]] Let $J$ be a set of coupling constants for an Ising model over the
restriction of $\mathbb{Z}^{2}$ to the upper half-plane $\mathbb{H}$, which
are:

  * (i)

ferromagnetic, $J_{x,y}\geq 0$ for every
$x,y\in\mathbb{\mathbb{Z}}^{2}\cap\mathbb{H}$,

  * (ii)

finite-range and such that the induced graph is connected,

  * (iii)

translation invariant, i.e. that $J_{x,y}=J(x-y)$, and

  * (iv)

reflection invariant: $J_{0,(a,b)}=J_{0,(-a,b)}=J_{0,(a,-b)}=J_{0,(b,a)}$ for
every $a,b\in\mathbb{Z}$.

Then, for any collection of boundary points
$x_{1}=(k_{1},0),\dots,x_{2n}=(k_{2n},0)$ satisfying
$k_{1}<k_{2}<\dots<k_{2n}$

(6.6) $\boxed{\langle\sigma_{x_{1}}\cdots\sigma_{x_{2n}}\rangle_{\beta_{c}}\
=\ {\bf
Pf}_{n}\big{(}\big{[}\langle\sigma_{x_{i}}\sigma_{x_{j}}\rangle_{\beta_{c}}\big{]}_{1\leq
i<j\leq 2n}\big{)}\big{[}1+o(1)\big{]}\,}$

where $\beta_{c}$ is the critical inverse-temperature of the model, and $o(1)$
is a function of the points $x_{1},\dots,x_{2n}$ which tends to zero for
configuration sequences with $\min_{1\leq i<j\leq
2n}\\{|x_{i}-x_{j}|\\}\to\infty$.

An intuitive explanation of the mechanism at work here is the fractality of
the connecting paths, and of the connected clusters, at the critical point.
Even if, due to the finite range of hopping the connections over a two
dimensional graph are not strictly planar, when the projections of two fractal
paths cross, there would typically be multiple switchbacks, in which case non-
intersection would require a large number of avoided linkages. The probability
that all such opportunities to connect would be missed may be expected to
vanish exponentially in the number of such encounters. By scaling, that number
should typically grow at least logarithmicaly in the distance (with a related
estimate for the probability of exceptional behavior).

As this outline suggest, the proof of Theorem 6.2 is seeped in percolation
type arguments, spelled in [8]. The stochastic geometric picture suggests that
also the following may be true.

######  Conjecture 1.

The conclusion of Theorem 6.2 holds also for similar systems with the Ising
spins replaced by spin variables in the Simon-Griffiths class (at the
correspondingly adjusted $\beta_{c}$).

With only some coincidental exceptions [15], Ising spin systems which are not
strictly planar are beyond the reach of exact solutions. However there has
been recent progress on rigorous perturbative methods which allow to prove a
degree of universality in certain aspects of the critical behavior, of models
close enough to the solvable case [41, 40]. In comparison, Theorem 6.2
provides less detailed information, but it is is not limited to weak
couplings.

###  6.3. Non-gaussian structure of the scaling limit of $2D$ bulk
correlations

The RCR explanation of the Pfaffian structure of the boundary-to-boundary spin
correlations makes it also clear that this behavior does not extend to the
multi-spin correlations in the bulk.

It was also noted in [1] that by elementary geometric considerations in the
symmetric case of the vertices of a square allied with the symmetry axes, for
two random currents whose sources at the pairs of opposite vertices, the
probability of paths intersection is at least $1/2$, and thus for any $\beta$:

(6.7) $\boxed{|U_{4}(x_{1},...,x_{4})|\ \geq\
\langle\sigma_{x_{1}}\sigma_{x_{3}}\rangle\,\langle\sigma_{x_{2}}\sigma_{x_{4}}\rangle}$

Related considerations allow to deduce that the $2D$ model’s scaling limit is
definitely not gaussian. However, since this model is solvable by other means,
we should not delve into this point here.

##  7\. The $U_{4}$ gaussianity criterion

The non-vanishing in the scaling limit of the properly rescaled $4$-point
function $U_{4}$ clearly implies that the limit is not Gaussian. The converse
statement is not obvious, but for the models considered here is also true.
This fact was first noted and proven by C. Newman [66](Theorem 8) through the
Lee-Yang property.

######  Proposition 7.1 ([66]).

If, for an Ising model, $U_{2m}(x_{1},...,x_{2n})\equiv 0$ for some
$m\in\mathbb{N}$, then the model is Gaussian.

More quantitative estimates can be derived from the expressions provided in
[66] for the Ursel functions in terms of the Lee-Yang zeros. Alternatively,
the statement can also be deduced from the following bound on the $N$-point
functions’ deviation from the corresponding Week rule, which was derived in
[1] through the random currents’ switching lemma.

######  Proposition 7.2 ([1](Theorem 12.1)).

In ferromagnetic Ising models, for any $n\in\mathbb{N}$

(7.1) $0\leq\
\mathcal{G}_{2n}[S_{2}](x_{1},\dots,x_{4})-S_{2n}(x_{1},...,x_{2n})\\\ \leq\
\boxed{\,\frac{-3}{2}\sum_{1\leq j<k<l<m\leq
2n}U_{4}(x_{j},x_{k},x_{l},x_{m})\cdot\mathcal{G}_{2n-4}[S_{2}](...,\cancel{x_{j}},...,\cancel{x_{k}},...,\cancel{x_{l}},...,\cancel{x_{m}},...)}$

where $\mathcal{G}_{2n}$ is the Wick functional defined in (2.9)

One should note that (7.1) is homogeneous of degree $1$ in each of the spin
variables, and hence the relation passes onto the scaling limit. It follows
that the limit is Gaussian if and only if the ratio of
$|U_{4}(x_{1},...,x_{4})|$ to $S_{4}(x_{1},...,x_{4})$ tends to zero, i.e. the
Wick law holds at the level of the $4$-point function.

It follows that a quantifier of the deviation from gaussianity can be found in
the dimensionless ratio

$\frac{U_{4}(x_{1},...,x_{4})}{\langle\sigma_{x_{1}}\sigma_{x_{2}}\sigma_{x_{3}}\sigma_{x_{4}}\rangle}$

for quadruples of sites at pairwise distances of order $L$. For concreteness
sake, one may represent this through the scale dependent ratio

(7.2)
$\boxed{R_{L}(\beta):=\frac{\displaystyle\sum_{x_{1},...,x_{4}\in\Lambda(L)}|U_{4}(x_{1},...,x_{4})|}{\Big{[}\displaystyle\sum_{x_{1},x_{2}\in\Lambda(L)}S_{2}(x_{1},x_{2})\Big{]}^{2}}}\left.\begin{cases}\leq\\\
\geq\frac{1}{3}\end{cases}\\!\\!\\!\\!\\!\\!\right\\}\frac{\displaystyle\sum_{x_{1},...,x_{4}\in\Lambda(L)}|U_{4}(x_{1},...,x_{4})|}{\displaystyle\sum_{x_{1},...,x_{4}\in\Lambda(L)}S_{4}(x_{1},...,x_{4})}$

where the inequalities are stated just to convey the essential equivalence of
the two slightly different ratios.

With the above choice, $R_{L}$ can be presented as minus the Ursel function

(7.3) $R_{L}(\beta)=-\left[\langle T_{L}^{4}\rangle_{\beta}-3\langle
T_{L}^{2}\rangle_{\beta}^{2}\right]\,.$

associated with the block-average of $\sigma_{x}$:

(7.4)
$T_{L}(\sigma):=\frac{1}{\sqrt{\Sigma_{L}}}\sum_{x\in\Lambda(L)}\,\sigma_{x}\,$

which by its normalization is of unit variance: $\langle
T_{L}^{2}\rangle_{\beta}=1$.

Proposition 7.2 has the following notable implication.

######  Corollary 7.3.

In any sequential limit in which $\beta\nearrow\beta_{c}$, $L\to\infty$ and

(7.5) $R_{rL}(\beta)\to 0\,\quad\mbox{for each $r<\infty$}\,$

the collection of variables $\\{T_{f,L}:f\in C_{0}(\mathbb{R}^{d})\\}$ of
(3.4) converges in distribution, and their joint limit describes a Gaussian
field.

###### Proof.

By a standard general argument, to establish the joint convergence of a
collection of variables to a Gaussian limit it sufficed to prove convergence
to a Gaussian limit for each of the variables’ linear combinations. We
establish that by focusing on the moment generating functions of $T_{f,L}$,
with in the limit $L\to\infty$ carried at fixed $f$.

For convenience, we present the argument first for $T_{L}$ of (7.4). (The
corresponding $f$, being an indicator function is not in
$C_{0}((\mathbb{R}^{4})$, but continuity is not essential for the argument.)

i) Applying (7.1) to $\langle T_{L}^{2n}\rangle$ we learn, through elementary
combinatorics, that for any $n\geq 2$

(7.6) $0\ \leq\ \frac{(2n)!}{2^{n}n!}\langle T_{L}^{2}\rangle^{n}-\langle
T_{L}^{2n}\rangle\ \leq\ \frac{3}{2}{2n\choose 4}\ R_{L}\ \langle
T^{2n-4}\rangle$

Upon summation over $n\geq 2$ with weights $z^{2n}/(2n)!$ (and noting that
also the middle term vanishes for $n=0,1$) this yields

(7.7) $0\ \leq\ \exp\big{\\{}\frac{z^{2}}{2}\langle
T_{L}^{2}\rangle\big{\\}}-\langle\exp\big{\\{}zT_{L}\big{\\}}\rangle\ \leq\
\frac{z^{4}}{2^{4}}\ R_{L}\ \langle\exp\big{\\{}zT_{L}\big{\\}}\rangle\,,$

Recalling that $\langle T_{L}^{2}\rangle=1$ one gets, under the stated
assumption,

(7.8)
$\boxed{\Big{|}\langle\exp\big{\\{}zT_{L}\big{\\}}\rangle-\exp\\{z^{2}/2\\}\Big{|}\
\leq\ 2^{-4}\ z^{4}\exp\\{z^{2}/2\\}\cdot\ R_{L}\ \to 0}$

from which the first claim follow.

ii) For the more general variables $T_{f,L}$, of (3.4), the moment analysis
yields

(7.9)
$\boxed{\Big{|}\langle\exp\big{\\{}zT_{f,L}\big{\\}}\rangle-\exp\big{\\{}\frac{z^{2}}{2}\langle
T_{f,L}^{2}\rangle\big{\\}}\Big{|}\ \leq\ 2^{-4}\
z^{4}\exp\big{\\{}\frac{z^{2}}{2}\langle T_{|f|,L}^{2}\rangle\big{\\}}\cdot\
\widetilde{R}_{|f|,L}\ }$

with

(7.10)
$\widetilde{R}_{f,L}(\beta):=\frac{\sum_{x_{1},...,x_{4}\in\Lambda(L)}|U_{4}^{(\beta)}(x_{1},...,x_{4})|\prod_{j=1}^{4}f(x_{j}/L)}{\left[\sum_{x_{1},x_{2}\in\Lambda(L)}S_{2}^{(\beta)}(x_{1},x_{2})\right]^{2}}\leq
r^{d}\|f\|_{\infty}^{4}R_{rL}(\beta)$

and

(7.11) $\langle
T_{f,L}^{2}\rangle=\frac{\sum_{x,y\in\Lambda(rL)}\langle\sigma_{x}\sigma_{y}\rangle\,f(x/L)\,f(y/L)}{\sum_{x,y\in\Lambda(L)}\langle\sigma_{x}\sigma_{y}\rangle}\leq
r^{d}\|f\|_{\infty}^{2}\,.$

The inequalities in the last two expressions are based on the Messager
Miracle-Solé monotonicity of the two point function [65], as explained in [6].
Taken together they imply convergence of the moment distribution function to a
finite limit which coincides with that of a Gaussian random variable of
variance

(7.12) $\lim_{L\to\infty}\langle
T_{f,L}^{2}\rangle=\frac{\int_{\Lambda(rL)^{2}}S_{2}(x,y)\,f(x)\,f(y)\,dx\,dy}{\int_{\Lambda(rL)^{2}}S_{2}(x,y)\,dx\,dy}\,.$

The stated distributional convergence for $T_{f,L}$ follows, as it did for
$T_{L}$. ∎

##  8\. Gaussianity of the scaling limits in $d>4$ dimensions

###  8.1. Sketch of the proof of Theorem 2.2 for the Ising case

The exact expression in (5.13) leads to the following pair of upper bounds [1]
.

######  Lemma 8.1.

For the Ising model on any finite graph, at $h=0$ and $\beta\geq 0$ (which are
omitted below)

(8.1) $|U_{4}(x_{1},...,x_{4})|\leq 2\
\langle\sigma_{x_{1}}\sigma_{x_{2}}\rangle\langle\sigma_{x_{3}}\sigma_{x_{4}}\rangle\
\times\\\ \times{\bf
P}^{\\{x_{1},x_{2}\\},\\{x_{3},x_{4}\\},\emptyset,\emptyset}[{\mathbf{C}}_{\mathbf{n}_{1}+\mathbf{n}_{3}}(x_{1})\cap{\mathbf{C}}_{\mathbf{n}_{2}+\mathbf{n}_{4}}(x_{3})\neq\emptyset]$

where the probability refers to four independently distributed random currents
with the prescribed sources:

$\partial(\mathbf{n}_{1})=\\{x_{1},x_{2}\\}\quad,\partial(\mathbf{n}_{2})=\\{x_{3},x_{4}\\},\quad\partial(\mathbf{n}_{3})=\partial(\mathbf{n}_{4})=\emptyset\,.$

And [consequently] the following diagrammatic bound holds

(8.2)
$\boxed{|U_{4}(x_{1},...,x_{4})|\,\leq\,2\,\sum_{u}\langle\sigma_{u}\sigma_{x_{1}}\rangle\langle\sigma_{u}\sigma_{x_{2}}\rangle\langle\sigma_{u}\sigma_{x_{3}}\rangle\langle\sigma_{u}\sigma_{x_{4}}\rangle}$

Inequality (8.1) is established through analysis which we omit here (cf. [1]).
However the transition to (8.2) is then elementary. It amounts to a bound on
the probability of intersection by the expectation value of its size (number
of points),

(8.3) ${\bf
P}^{\\{x_{1},x_{2}\\},\\{x_{3},x_{4}\\},\emptyset,\emptyset}[{\mathbf{C}}_{\mathbf{n}_{1}+\mathbf{n}_{3}}(x_{1})\cap{\mathbf{C}}_{\mathbf{n}_{2}+\mathbf{n}_{4}}(x_{3})\neq\emptyset]\
\leq\ \\\ \leq\,{\bf
E}^{\\{x_{1},x_{2}\\},\\{x_{3},x_{4}\\},\emptyset,\emptyset}[\,\,|{\mathbf{C}}_{\mathbf{n}_{1}+\mathbf{n}_{3}}(x_{1})\cap{\mathbf{C}}_{\mathbf{n}_{2}+\mathbf{n}_{4}}(x_{3})|\,\,]\,\
=\ \\\ =\sum_{u}{\bf
P}^{\\{x_{1},x_{2}\\},\\{x_{3},x_{4}\\},\emptyset,\emptyset}[u\in{\mathbf{C}}_{\mathbf{n}_{1}+\mathbf{n}_{3}}(x_{1})\cap{\mathbf{C}}_{\mathbf{n}_{2}+\mathbf{n}_{4}}(x_{3})]\hskip
34.14322pt\mbox{ }$

Combined with (5.2), which provides an exact expression for the last term, one
obtains the _digrammatic_ upper bound stated in (8.2).

The gaussianity of the Ising model’s scaling limits for $d>4$ was established
through the combination of the tree diagram bound (8.2) of [1] and the
_infrared_ upper bound (for the nearest neighbor interaction of constant
strength $J$)

(8.4) $\boxed{\forall\beta\leq\beta_{c}:\quad\beta
J\langle\sigma_{x}\sigma_{y}\rangle_{\beta}\leq\frac{C}{|x-y|^{d-2}}\,}$

which was presented above in (4.5). Omitting some details, which are covered
in the original references, the combination of the two bounds yields for the
nearest neighbor Ising model the estimate

(8.5) $\boxed{R_{L}(\beta)\leq\
C\frac{L^{d}(\Sigma_{L})^{4}}{(\Sigma_{L})^{2}}\leq\frac{C}{L^{d-4}}}$

which holds with a uniform constant for all $\beta<\beta_{c}$ and $L<\infty$.
The claimed results then follows by a direct application of the criterion
stated in Corollarly 7.3.

###  8.2. Sketch of the proof of Theorem 2.2 for $\varphi^{4}$ variables

The proof of the $\varphi^{4}$ version of Theorem 2.2 in [1] proceeded through
an extension of the above analysis to a broader class of single spin
distributions. This is enabled by the observation, of Griffiths and Simon
[78]) that the $\varphi^{4}$ single spin distribution can be presented as the
limiting distribution of weighted averages of ferromagnetically coupled Ising
spins. The explicit definition of the GS class of variables is given next, but
is not essential for the following summary.

By the linearity of the relation between the Ising spins and their aggregates
$\varphi$, statements that were derived for the Ising spin model through
relations which are homogeneous of degree $1$ in the spin variables apply also
to systems based on $\varphi^{4}$ variables (all that is needed is the
substitution $\sigma_{x}\Rightarrow\varphi_{x}$). That applies in particular
to (7.1) of Proposition 7.2 Hence the entire discussion of Section 7 applies
mutatis mutants also to systems based on $\varphi^{4}$ variables.

At first sight, the above rule does not apply to the tree diagram bound (8.2),
where the left side is homogeneous of degree $4$ in $\sigma$, while the right
is of degree $6$. This was not an issue for Ising spins since there
$\sigma^{2}\equiv 1$. However one may proceed by first transforming the
relation into a dimensionally balanced one. From this perspective,
$\sigma^{2}$ is not dimensionless but $\beta J_{u,v}\sigma_{u}\sigma_{v}$ is.

Dimensional deficiency in Ising relations can in many cases be corrected
through “point splitting”. A case in point is the following dimensionally
balanced version of the diagrammatic inequality (8.2).

######  Proposition 8.2 ([11]).

For any ferromagnetic system of Ising spins with the Hamiltonian (9.4) at
$h=0$, at each $\beta\geq 0$ (whose presence as a subscript on $U$ and $S$ is
omitted below)

$\displaystyle\boxed{|U_{4}(x_{1},...,x_{4})|\leq
2\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}y\notin\\{x_{1},...,x_{4}\\}\\\
u,v\end{subarray}}\\!\\!\\!\\!S_{2}(x_{1},y)\,S_{2}(x_{2},y)\,[\beta
J_{y,u}]S_{2}(u,x_{3})\,\,[\beta J_{y,v}]S_{2}(v,x_{4})}$ (8.6)
$\displaystyle\boxed{+2\left[\sum_{u\neq
x_{1}}S_{2}(x_{1},x_{2})S_{2}(x_{1},x_{3})[\beta
J_{x_{1},u}]S_{2}(u,x_{4})+\mbox{$3$ permutations}\right]}$

where $S_{2}(x,y)=\langle\sigma_{x}\sigma_{y}\rangle_{\beta}$.

This version of the tree diagram bound extends verbatim to systems with
variables in the Griffiths Simon class, and in particular the lattice field
variables with the $\varphi^{4}$-type a priori measure. The only change is to
switch the notation to
$S_{2}(x,y)=\langle\varphi_{x}\varphi_{y}\rangle_{\beta}$, and make the
similar adjustment in $U_{4}$. (This will be an obvious consequence of the
deconstruction of the $\varphi$ variables into Ising spins which we postponed
to section 9.)

The other essential ingredient for the Ising analysis was the _Infrared Bound_
(8.5). However already the original derivation (in its different forms [42,
38, 80]) has been directly valid for all single site distributions of sub-
exponential decay.

By the above means, the proof of the gaussianity of the Ising model’s scaling
limits extends to the generality in which it is stated in Theorem 2.2.

##  9\. A decomposition of $\varphi^{4}$ variables into Ising constituents

###  9.1. The Griffiths-Simon class of scalar variables

Through some minor adjustments, the results presented above for the
ferromagnetic Ising model extend to similar lattice systems with various other
single variable distributions, including the $\varphi^{4}$ fields. A device
which was designed for such a purpose is the Simon-Griffiths representation of
$\varphi^{4}$ variables as the distributional limit of block averages of
ferromagnetically coupled Ising “elementary” constituents.

This technique was used by Griffiths [44] for an extension of the Lee-Yang
theorem, as well as his eponymous inequalities, to a broader range of spin
distributions. Among the well recognized examples in that class are the
“classical” bounded spins, with uniform distribution in $[-S,S]$, and their
“quantized” analogs in which the spins take values in $[-S,S]\cap\mathbb{Z}$
with equal a-priori weights. The construction was further extended by B. Simon
and R. Griffiths to cover also the $\varphi^{4}$ distribution [78] which is of
interest here. Following is its more explicit formulation.

Figure 2. The decorated graph, in which the sites $x$ of a graph of interest
are replaced by “blocks” $\mathcal{B}_{x}$ of sites indexed as $(x,n)$. The
Ising “constituent spins” $\sigma_{x,n}$ are coupled pairwise through intra-
block couplings $\delta_{x,y}K_{n,m}$ and inter-block couplings $J_{x,y}$. The
lines indicate a possible realization of the corresponding random current.

######  Definition 9.1.

A probability measure on $\rho$ on $\mathbb{R}$ is said to belong to the
Griffiths-Simon (GS) class if either of the following conditions is satisfied
$1)$ the expectation values with respect to $\rho$ can be presented as

(9.1) $\int
F(\varphi)\,\rho(d\varphi)=\frac{1}{\rm{norm}}\sum_{\underline{\sigma}\in\\{-1,1\\}^{N}}F(\sum_{j=1}^{N}\alpha_{j}\sigma_{j})\,\,e^{\sum_{i,j=1}^{N}K_{i,j}\alpha_{i}\alpha_{j}\sigma_{i}\sigma_{j}}$

with some $\\{b_{j}\\}\subset\mathbb{R}$, and $K_{i,j}\geq 0$.
$2)$ $\rho$ can be presented as a (weak) limit of probability measures of the
above type, and is of sub-gaussian growth:

(9.2) $\int e^{|\varphi|^{\alpha}}\rho(d\varphi)<\infty\quad\mbox{for some
$\alpha>2$.}$

A random variable is said to be of Griffiths-Simon type if its probability
distribution is in the GS class.

The Griffiths condition (9.1) can be equivalently expressed by saying that
$\phi$ can be presented as

(9.3) $\phi=\sum_{j}\alpha_{j}\sigma_{j}\,.$

The ferromagnetic system of such variables $\\{\varphi_{x}\\}$ with the
Hamiltonian

(9.4)
${H}(\varphi):=-\sum_{\\{x,y\\}\subset\mathbb{Z}^{d}}J_{x,y}\varphi_{x}\varphi_{y}-h\sum_{x\in\mathbb{Z}^{d}}\varphi_{x}$

can be presented as an Ising spin system, with the interaction

(9.5) $\displaystyle\mbox{ }\hskip
36.98866pt\hfill\boxed{H+K=-\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}\\{x,y\\}\in\mathcal{E}\\\
i,j\in\\{1,...,N\\}\end{subarray}}\alpha_{i}\alpha_{j}J_{x,y}\sigma_{x,i}\sigma_{y,j}-h\sum_{x,j}\alpha_{j}\sigma_{x,j}+K(\sigma)}\,.$

where
$K(\sigma)=-\sum_{x}\sum_{i,j}K_{i,j}\alpha_{i}\alpha_{j}\sigma_{i}\sigma_{j}$.

The infinite volume Gibbs equilibrium states of such systems are then
constructible as a double limit, in which one first takes $N\to\infty$, to
produce the desired convergence of the distribution of the $\varphi$
variables, followed by the infinite volume limit $\mathcal{L}\to\infty$.

Of particular relevance to us is the observation of [78] that a variable
$\\{\varphi_{x}\\}$ with the probability distribution

$\rho_{\lambda,b}(d\varphi)=e^{-\lambda\varphi^{4}+b\varphi^{2}}d\varphi/\text{norm}$

can be produced as the $N\to\infty$ distributional limit of the collection of
the block averages of elemental Ising spins $\\{\sigma_{x,j}\\}$ (the dots in
Fig. 2)

(9.6) $\varphi_{x}^{(N)}=\frac{\alpha(N)}{\sqrt{N}}\sum_{j=1}^{N}\sigma_{x,j}$

which interacted through the “ultra-local” on-site coupling

(9.7) $K(\sigma)=-\frac{g(N)}{N}\left[\sum_{j=1}^{N}\sigma_{x,j}\right]^{2}$

at suitably adjusted values for $\\{\alpha(N),g(N)\\}$. The explicit values of
the required adjustments in the parameters of $K$ are easy to determine, but
are not relevant for the present discussion, beyond the fact of feasibility of
this construction for any $\lambda\geq 0$ and $b$ (of arbitrary sign).

Underlying the last assertion is the observation that in this case $K$ is a
mean-field interaction of $N$ spins. In the limit $N\to\infty$ it undergoes a
phase transition at $g=1$, which is explicitly analyzable. Expanding the well
known expression for the entropy as function of
$\frac{1}{N}\sum_{j}\sigma_{x,j}$, to the fourth order (around its maximum),
one finds that at the mean-field critical point the leading $\varphi^{2}$ term
vanishes, and $\varphi^{4}$ emerges. With suitable fine-tuning of
$\\{\alpha(N),g(N)\\}$ the distribution of $\varphi_{x}^{(N)}$ can be made to
converge to any $\rho_{\lambda,b}$ at $\lambda>0$.

To summarize: through the above construction, any system of $\varphi^{4}$
variables associated with the sites of a finite graph $\mathbb{G}$, and
coupled through the graph’s edges, is presentable as the limit ($N\to\infty$)
of a system of constituent Ising spins associated withe the decorated graph
produced by replacing each vertex of $\mathbb{G}$ by the the complete graph of
$N$ vertices. In it, the “microscopic sites” are indexed by $\\{x,j\\}$ and
two sites are neighbors if they either lie in a common block (i.e. same $x$)
or in a pair on neighboring blocks (as depicted in Fig. 2).

###  9.2. Dimension balancing through point splitting

The linearity of the decomposition (9.3) allows to extend to systems with
variables in the GS class many of the results that were originally derived for
ferromagnetic Ising spin models. In particular, a suitably weighted version of
(8.1) holds, allowing to assign a probabilistic interpretation to the ratio

$U_{4}(x_{1},...,x_{4})/S_{4}(x_{1},...,x_{4})$

also for the field variables. However the intersection event needs to be
interpreted now in terms of the connected clusters of the elemental
constituents of the $\varphi$ variables (the small dots on Figure 2). One may
quickly spot two problems, which are actually related:

  1. i.

bounding the probability that two clusters intersect by the size of their
intersection on the enhanced graph may lead to a more severe over-estimate of
$U_{4}$ than the one which was discussed above for the Ising model,

  2. ii.

without the restriction to $\sigma^{2}=1$, the diagrammatic bound (8.2) is
unbalanced and hence cannot be valid as stated for the more general spins

To reduce the over-counting, one may bound the probability of intersection
through the number of the sites of the graph $\mathcal{V}$, i.e. the boxes,
rather than the “internal dots” in the decorated graph.

Furthermore, for the above purpose, the probability that a current with
specified sources reaches a box can be bounded above through the mean flux
into the box through edges connected to the sources. This results in the
following bound (Proposition 7.1 in [11]). As an added benefit, this provides
a dimensionally balanced substitute for the site hitting relation (5.2):

######  Lemma 9.2 (Box hitting probability [11]).

Under the assumptions of Proposition 8.2, for any subset
$\mathcal{B}\subset\mathcal{V}$, and pair of sites $\\{x,y\\}$ in
$\mathcal{B}^{c}:=\mathcal{V}\setminus\mathcal{B}$

(9.8) $\displaystyle\boxed{\langle\sigma_{x}\sigma_{z}\rangle\,{\bf
P}^{\\{x,z\\},\emptyset}[{\mathbf{C}}_{\mathbf{n}_{1}+\mathbf{n}_{2}}(x)\cap\mathcal{B}\neq\emptyset]\
\leq\\!\\!\sum_{u\in\mathcal{B},\,\,v\in\mathcal{B}^{c}}\\!\\!\\!\langle\sigma_{x}\sigma_{u}\rangle\
[\beta J_{u,v}]\,\langle\sigma_{v}\sigma_{y}\rangle}$

In this inequality the right hand side is an estimate on the double current’s
flux into $B$ through edges which in $B$’s complement are
$\mathbf{n}_{1}+\mathbf{n}_{2}$ connected to $x$. 999To avoid confusion let us
recall that random current’s flux is directionless, and void of a meaningful
sign. The current’s “conservation law” involves only the flux’s parity. The
proof of this bound is an exercise in geometric combinatorics based on the
switching lemma. The argument, which can be found in the provided reference,
can be simplified a bit for the $x\Leftrightarrow y$ symmetrized version of
(9.8), which for practical purposes is as useful..

With (9.8) applied in place of (5.2) in the derivation of the diagrammatic
bound (8.2), one gets the dimensionally balanced inequality (8.2). As
explained above, this allows to extend the proof of the gaussianity of the
Ising model’s scaling limits in $d>4$ dimensions to variables in the GS class,
including the $\varphi^{4}$ case.

##  10\. Gaussianity at the marginal dimension $d=4$

###  10.1. A suggestive dimensional analysis

The above sketched proof of the scaling limits’ gaussianity in $d>4$
dimensions may be recapitulated in a suggestive dimensional argument, which
helps to frame the next question.

The proof’s key input consists of the tree diagram bound (8.2) and the
_infrared upper bound_ (4.5) To focus on the essence, one may initially
proceed under the (reasonable) assumption that: i) at $\beta_{c}$ the two-
point function is of comparable values for pairs of sites at similar
distances, ii) on the scales under consideration $S_{2}(x,y)$ decays by a
power law.101010The behavior assumed in (10.1) does not hold for
$\beta\neq\beta_{c}$ at distances much larger than the correlation length, but
is a common feature of critical models with power law decay of correlations.
In the standard notation that would be presented as

(10.1)
$\beta_{c}J\langle\sigma_{x}\sigma_{y}\rangle_{\beta_{c}}\approx\frac{C}{|x-y|^{d-2+\eta}}\,,$

for which (8.4) implies $\boxed{\eta\geq 0}$.

Then, for quadruples of points at mutual distances of order $L$, in lattice
units (!), the sum in the tree diagram bound (8.2) contributes a factor
$L^{d}$ while the summand has two extra correlation function factors
$S_{2}(\cdot,\cdot)$, in comparison to $S_{4}(x,y,z,t)$. each $S_{2}$ factor
being of order $1/L^{d-2+\eta}$.

This suggests that for any collection of four point which are at mutual
distances of order $L$

(10.2)
$\boxed{\frac{|U_{4}^{(\beta)}(x_{1},...,x_{4})|}{S_{4}^{(\beta)}(x_{1},...,x_{4})}\leq
C\,\frac{L^{d}}{L^{2(d-2+\eta)}}=\frac{C}{L^{d-4+2\eta}}}$

which for $d>4$ vanishes in the limit $L\to\infty$.

Up to technical details, the above captures the essence of the argument
underlying the gaussianity of the Ising model’s scaling limit in [1]. A
similar bound for $\varphi^{4}$ models was derived also in [36], by other
means.

The above dimensional argument may appear similar to Wilson’s heuristic
renormalization group analysis [86]. However it should be stressed that:

  1. i.

the bounds on $U_{4}$ of [1, 36] are expressed in terms of relations among the
“dressed” quantities, not just their initial values near a gaussian starting
point,

  2. ii.

in general, the two point function’s critical behavior is corrected by the
critical exponent $\eta$, and hence the dimensional analysis relies on the
non-perturbative fact (implied by a reflection positivity argument) that in
this model $\eta\geq 0$.

###  10.2. The challenge

The initial proofs of gaussianity in high dimensions ($d>4$) [1, 36], as well
as the heuristic outline given above, do not extend to $d=4$ (unless $\eta>0$
which however does not seem to be the case). More explicitly, if at this
marginal dimension $\eta=0$ (as expected), then the tree diagram bound on
$R_{L}$ no longer decays to $0$ for $L\to\infty$.

Still, one may note that for $d=4$ the simple diagrammatic upper bound on
$R_{L}$ at least does diverge in the limit $L\to\infty$. At fist sight this
may not appear as adding much information since from (5.13) one learns that in
any dimension

(10.3) $0\leq R_{L}(\beta)\leq 2$

(which is similar to Glimm and Jaffe’s universal upper bound of the
renormalized coupling constant [42]). Nevertheless, upon closer inspection,
the non-divergence of the upper bound actually provides a meaningful hint. To
explain the point, let us look closer at what it represents.

In terms of the random current representation, the all important quotient is
bounded by the intersection probability,

(10.4)
$\frac{|U_{4}(x_{1},...,x_{4})|}{\langle\sigma_{x_{1}}\sigma_{x_{2}}\rangle\langle\sigma_{x_{3}}\sigma_{x_{4}}\rangle}\
\leq\ 2\,\,{\bf
P}^{\\{x_{1},x_{2}\\},\\{x_{3},x_{4}\\},\emptyset,\emptyset}[Q\neq\emptyset]$

where, to abbreviate, we denote
$Q:={\mathbf{C}}_{\mathbf{n}_{1}+\mathbf{n}_{3}}(x_{1})\cap{\mathbf{C}}_{\mathbf{n}_{2}+\mathbf{n}_{4}}(x_{3})$.

In the diagrammatic bound (8.1) the probability that $Q$ is non-empty is
estimated by its expected size, i.e.,

(10.5) ${\bf
P}^{\\{x_{1},x_{2}\\},\\{x_{3},x_{4}\\},\emptyset,\emptyset}[Q\neq\emptyset]\
\leq\ {\bf E}^{\\{x_{1},x_{2}\\},\\{x_{3},x_{4}\\},\emptyset,\emptyset}[|Q|]$

with $|Q|=\sum_{y}\bm{1}[y\in Q]$. However the exact relation between the two
quantities (given that $|Q|$ is an integer) is

(10.6) ${\bf P}[Q\neq\emptyset]\ =\ \frac{{\bf E}\big{[}|Q|\big{]}}{{\bf
E}\big{[}|Q|\,\big{|}|Q|\geq 1\big{]}}\,.$

In the present situation the numerator is easy to express exactly in terms of
the two point function, and it is that upper bound to which the scaling
estimate (10.2) applies.

Hence, at the marginal dimension gaussianity of the scaling limit would follow
if

(10.7) $\lim_{L\to\infty}{\bf E}_{L}\big{[}|Q|\,\big{|}|Q|\geq
1\big{]}=\infty$

where the subscript on ${\bf E}_{L}$ indicates that the limit refers to
configurations $x_{1},x_{2},x_{3},x_{4}$ in which all distances are of sizes
comparable to $L$.

###  10.3. A suggestive dichotomy

Continuing from the last point: at the intuitive level one could expect that
if $\eta=0$ (with no relevant logarithmic corrections) then the conditional
expectation in (10.7) may diverge due to the fact that $|Q|$ is the sum of
contributions from different scales which should be approximately independent
and each bounded below. (If, to the contrary $\eta>0$, then the bound (10.2)
suffices to cover also the case $d=4$.)

A bit more explicitly: the switching lemma suggests that the expected value of
the size of the intersection within $[-\ell,\ell]^{2}$ of a pair of RCR
cluster which share one of their two sources, and have the other at distances
of order $L\gg\ell$ from the common one is of the order of

(10.8) $B_{\ell}:=\sum_{\begin{subarray}{c}x\in\mathbb{Z}^{4}\\\
\|x\|<\ell\end{subarray}}\langle\sigma_{0}\sigma_{x}\rangle_{\beta_{c}}^{2}$

to which we refer as the _bubble diagram_ on scale $\ell$:

This in turn suggests that if at the critical point (for simplicity considered
first) the value of $S_{2}(0,x)\|x\|^{2}$ shows consistent behavior across
scales, i.e. in the scaling regime of interest the following limit exists

(10.9) $\kappa=:\lim_{L\to\infty}S_{2}(0,x)\|x\|^{2}$

then the ratio $U_{4}/S_{4}$ in (10.2) vanishes for one of two complementary
reasons:

  * •

If $\kappa=0$ then the original tree diagram bound on $U_{4}/S_{4}$ for
roughly equidistant points tends to zero

  * •

If $\kappa>0$ then the bubble diagram diverges, (10.7) holds, and the scaling
limit is gaussian due to the divergence of the correcting factor in the
denominator of (10.6).

###  10.4. Logarithmically improved bounds

There are some gaps in the argument outlined above. Among those:

  1. 1)

what if the behavior of $S_{2}(0,x)\|x\|^{2}$ shows variation in scales, being
small on many scales and then non-negligible on other, higher scales, thus
avoiding each of the above arguments

  2. 2)

establish the suggested relation of $B_{L}$ with ${\bf
E}_{L}\big{[}|Q|\,\big{|}|Q|\geq 1\big{]}$ for $Q$ the intersection size
defined below (10.4)

  3. 3)

formulate the analysis so it applies to limits in which also $\beta$ is
allowed to vary, as long as $L/\xi(\beta)\to 0$.

The first question was handled by proving a relative monotonicity (i.e., up to
a correction by a constant factor) in scales of $S_{2}(0,x)\|x\|^{d-2}$
(Theorem 5.6 in [6]) on which a remark was made below (4.5). The other points,
for which more substantial analysis was required, were handled in [6] through
estimates proving abundance of scale on which the two point function satisfies
a regularity condition, analysis of the conditional expectation, and proof of
sufficient independence in the contribution of different regular scales to
${\bf E}_{L}\big{[}|Q|\,\big{|}|Q|\geq 1\big{]}$.

Through such means the following improvement of the tree diagram bound (8.2)
was established.

######  Theorem 10.1 (Improved tree diagram bound [6]).

For the n.n.f. Ising model in dimension $d=4$, there exist $c,C>0$ such that
for every $\beta\leq\beta_{c}$, every $L\leq\xi(\beta)$ and every
$x,y,z,t\in\mathbb{Z}^{d}$ at a distance larger than $L$ of each other,

(10.10)
$|U_{4}^{\beta}(x,y,z,t)|\leq\frac{C}{B_{L}(\beta)^{c}}\sum_{u\in\mathbb{Z}^{4}}\langle\sigma_{u}\sigma_{x}\rangle_{\beta}\langle\sigma_{u}\sigma_{y}\rangle_{\beta}\langle\sigma_{u}\sigma_{z}\rangle_{\beta}\langle\sigma_{u}\sigma_{t}\rangle_{\beta},$

From this, arguing through the dichotomy described in section 10.3 one gets
the following improved version of (7.9)

######  Proposition 10.2.

There exist $c,C>0$ such that for the n.n. ferromagnetic Ising model on
$\mathbb{Z}^{4}$, every $\beta\leq\beta_{c}$, every $L\leq\xi(\beta)$, and
test function $f\in C_{0}(\mathbb{R}^{4})$,

(10.11)
$\boxed{\Big{|}\langle\exp\big{\\{}zT_{f,L}\big{\\}}\rangle-\exp\big{\\{}\frac{z^{2}}{2}\langle
T_{f,L}^{2}\rangle\big{\\}}\Big{|}\ \leq\ 2^{-4}\
z^{4}\exp\big{\\{}\frac{z^{2}}{2}\langle T_{|f|,L}^{2}\rangle\big{\\}}\
\frac{\widetilde{R}_{|f|,L}\ }{(\log L)^{c}}}$

with $\widetilde{R}_{f,L}(\beta)\leq r^{d}\|f\|_{\infty}^{4}R_{rL}(\beta)$

Let us recall that the variance of $T_{f,L}$ is bounded above uniformly in $L$

(10.12) $\langle
T_{f,L}^{2}\rangle=\frac{\sum_{x,y\in\Lambda(rL)}\langle\sigma_{x}\sigma_{y}\rangle\,f(x/L)\,f(y/L)}{\sum_{x,y\in\Lambda(L)}\langle\sigma_{x}\sigma_{y}\rangle}\
\leq\ r^{d}\|f\|_{\infty}^{2}\,.$

If the corresponding scaling limit exists then

(10.13) $\lim_{L\to\infty}\langle
T_{f,L}^{2}\rangle=\frac{\int_{\mathbb{R}^{4}\times\mathbb{R}^{4}}f(x)f(y)S_{2}(x,y)dx\,dy}{\int_{[-1,1]^{4}\times[-1,1]^{4}}S_{2}(x,y)dx\,dy}$

The claimed gaussianity of any such scaling limit follows, since for any test
function $f\in C_{0}(\mathbb{R}^{4})$ the right hand side of (10.11) tends to
zero as $L\to\infty$, while the variance of $T_{f,L}$ converges to a finite
limit, and for all $z\in\mathbb{C}$

(10.14)
$\lim_{L\to\infty}\langle\exp\big{\\{}zT_{f,L}\big{\\}}\rangle=\exp\big{\\{}\frac{z^{2}}{2}\lim_{L\to\infty}\langle
T_{f,L}^{2}\rangle\big{\\}}\,.$

Through the adjustments discussed in section 8.2 the above bounds, and the
conclusion, extend also to limits of ferromagnetic lattice $\varphi^{4}$
systems.

##  11\. Some open questions

1) Scaling limits in $d=3$ dimensions

In closing the above summary let us note the strange coincidence that at
present $3D$, which is of obvious relevance for statistical mechanics, is the
hardest dimension to reach by rigorous methods. The effectiveness of the
random current representation in shedding light on the model’s behavior in
both $d\geq 4$ and $d=2$ dimensions, leads one to ask whether the stochastic
geometric perspective could be of help also in furthering the understanding of
the model’s non-trivial behavior in $3D$.

Among the approaches to $3D$ which have been explored in physics literature is
the expansion in $\varepsilon=4-d$, starting from the upper critical dimension
$d=4$ [87]. From the opposite direction, more recently the $3D$ Ising model
has also been explored through conformal bootstrap [29], which is an extension
of an approach that is particularly effective in $2D$.

The constructive field theory program has yielded a non-trivial $\varphi^{4}$
theory over $\mathbb{R}^{3}$ (c.f.[21] and references therein). The
construction shows that, focusing on the ultraviolet regime, for the weakly
coupled $\varphi^{4}_{3}$ fields the state of zero renormalized-interaction is
unstable. It would be of interest to supplement this by two sets of results:
i) boost the results beyond the perturbative regime to the field’s critical
manifold, ii) relate the scaling limits of the critical $\varphi^{4}$ theories
to that of the Ising model.

This challenge beckons for a merger of the non-perturbative methods deployed
in the analysis of the Ising / $\varphi^{4}$ renormalized coupling constants
with some guidance from the perturbative studies of soft
$\lambda\varphi^{4}_{3}$ fields.

2) Logarithmic corrections

As we saw, the existence of logarithmic-type corrections in $4D$ plays a role
in the gaussianity of scaling limit. It will be of interest to obtain sharper
control of such corrections, the like of which are also expected to show up in
the power law scaling relations in $4D$.

The critical exponent bounds listed in Theorem 4.2 were derived through
partial differential inequalities (PDI) on the order parameter
$M(\beta,\hat{h})$ and its derivative $\chi(\beta,\hat{h})=\frac{\partial
M(\beta,\hat{h})}{\partial\hat{h}}$ (with $\hat{h}:=\beta h$). Leading
examples are:

$\displaystyle\frac{d}{d\beta}\chi(\beta)$
$\displaystyle\,\leq\,\|J\|\chi(\beta)^{2}$ (11.1)
$\displaystyle\beta\frac{\partial M}{\partial\beta}$ $\displaystyle\,\leq\
\beta\|J\|\,\,M\cdot\frac{\partial M}{\partial\widehat{h}}$

which hold for both percolation and Ising models and

(11.2) $M\leq\widehat{h}\frac{\partial M}{\partial\widehat{h}}\ +\
M^{1+\alpha}\beta\frac{\partial}{\partial\beta}M+M^{2+\alpha}\quad\rm{at}\,\,\alpha=\begin{cases}1&\rm{Ising}\\\
0&\rm{percolation}\end{cases}$

It was also shown that under certain conditions mildly modified relations hold
also in the other direction. The conditions allowing such reversal are the
convergence of the bubble diagram $\mathcal{B}(\beta_{c})<\infty$ for Ising
type models, and the triangle diagram $\mathcal{T}(\beta_{c})<\infty$ for
percolation. These are defined by:

$\displaystyle\mathcal{B}$
$\displaystyle=(\beta\|J|)^{2}\sum_{x}S_{2}(0,x)S_{2}(x,0)=(\beta\|J|)^{2}\int_{p\in[-\pi,\pi]^{d}}\widehat{S}_{2}(p)^{2}\,dp$
$\displaystyle\mathcal{T}$
$\displaystyle=\sum_{x,y}\tau(0,x)\tau(x,y)\tau(y,0)=\int_{p\in[-\pi,\pi]^{d}}\widehat{\tau}(p)^{3}\,dp\,$

where $\tau(x,y)$ is the probability of connection in a percolation model and
$\widehat{F}$ is the Fourier transform of $F$.

By such means it was proven that some of these models’ critical exponents
assume simple (mean field) values in dimensions $d>d_{c}$ with

(11.4) $\boxed{d_{c}=\begin{cases}4&\rm{Ising}\\\
6&\rm{percolation}\end{cases}}\,$

Can the relation expressed by PDI be sharpened enough to yield the logarithmic
corrections at the marginal dimensions?
(If experience is any guide, it could be of help to consider the two models in
parallel.)

Progress on this question may be closely related to the goal of non-
perturbative bounds on the _beta function_ which in the renormalization group
analysis guides the flow of the scale-dependent renormalized coupling
constant. It would be of interest to link in this manner the current rigorous
analysis of the scaling limits with the “renormalization group” perspective on
the subject.

3) Can $P(\varphi)_{d}$ field theory make sense beyond the realm of convergent
functional integrals?

It is an intriguing observation that in the diagrammatic layout of relations
such as (8.2) and (11.2) one typically finds cubic diagrams for percolation in
place of quartic diagrams for Ising models. This suggests a relation with a
$\varphi^{3}$ versus $\varphi^{4}$ field theory, and through that lead one to
guess that that the upper critical dimension for percolation would be $6$.
While the euclidean functional integral with the cubic polynomial in place of
(2.10) does not converge in any dimension, various aspects of this prediction
were proven correct (cf. [54] and references therein).

One wonders: could there be another perspective on field theory, in which it
makes sense beyond the somewhat timid formulation which is outlined in the
introduction? Indeed different proposals, motivated by other observations,
continue being made cf. [30].

Acknowledgments I would like to thank Hugo Duminil-Copin for our fruitful
collaboration. It was advanced through mutual visits to Princeton and Geneva
University, sponsored by a Princeton-UniGe partnership grant. I am also
grateful for the earlier successful collaborations on the subject with R.
Graham, D. Barsky, R. Fernandez and S. Warzel, and thank J. Shapiro and S.
Warzel for useful comments on the initial draft of this article. The author’s
work reported here was supported in parts by NSF grants, and the Weston
visiting professorship at the Weizmann Institute.

## References

  * [1] M. Aizenman, _Geometric analysis of $\varphi^{4}$ fields and Ising models. I, II_, Comm. Math. Phys., 86(1):1–48 (1982).
  * [2] M. Aizenman, _The Intersection of Brownian Paths as a Case Study of a Renormalization Group Method for Quantum Field Theory._ , Comm. Math. Phys., 97:91–110 (1985).
  * [3] M. Aizenman, D.J. Barsky, _Sharpness of the phase transition in percolation models._ Comm.Math. Phys. 108, 489–526 (1987).
  * [4] M. Aizenman, D.J. Barsky, and R. Fernández, _The phase transition in a general class of Ising-type models is sharp_, J. Stat. Phys., 47 (3-4):343–374 (1987).
  * [5] M. Aizenman, J. T. Chayes, L. Chayes and C. M. Newman, _Discontinuity of the magnetization in one-dimensional $1/|x-y|^{2}$ Ising and Potts models._ J. Stat. Phys. 50, 1–40 (1988).
  * [6] M. Aizenman and H. Duminil-Copin _Marginal triviality of the scaling limits of critical $4D$ Ising and $\varphi^{4}_{4}$ models_, Annals of Mathematics, 194 (1): 163–235:719 (2021).
  * [7] M. Aizenman, H. Duminil-Copin, and V. Sidoravicius, _Random Currents and Continuity of Ising Model’s Spontaneous Magnetization_ , Comm. Math. Phys., 334:719–742 (2015).
  * [8] M. Aizenman, H. Duminil-Copin, V. Tassion and S. Warzel, Emergent Planarity in two-dimensional Ising Models with finite-range Interactions, Inventiones Mathematicae, 216 (3):661–743 (2019).
  * [9] M. Aizenman, H. Duminil-Copin, S. Warzel, _Dimerization and Néel Order in Different Quantum Spin Chains Through a Shared Loop Representation._ Ann. H. Poincaré 21, 2737–2774 (2020).
  * [10] M. Aizenman and R. Fernández, _On the critical behaviour of the magnetization in high-dimensional Ising models_ , J. Stat. Phys., 44 (3-4):393–454 (1986).
  * [11] M. Aizenman and R. Graham, _On the renormalized coupling constant and the susceptibility in $\varphi^{4}_{4}$ field theory and the Ising model in four dimensions_, Nucl. Phys. B, 225:261–288 (1983).
  * [12] M. Aizenman and B. Nachtergaele, _Geometric aspects of quantum spin states_ , Comm. Math. Phys. 164, 17–63 (1994).
  * [13] M. Aizenman and C.M. Newman _Tree graph inequalities and critical behavior in percolation models._ J. Stat. Phys. 36, 107–143 (1984).
  * [14] C. Aragao de Carvalho, S. Caracciolo, J. Fröhlich, _Polymers and $g\varphi^{4}$ theory in four dimensions_, Nucl. Phys. B215 [FS7], 209-248 (1983).
  * [15] M. Aizenman and S. Warzel, _Kac-Ward Formula and Its Extension to Order-Disorder Correlators Through a Graph Zeta Function_ , J. Stat. Phys. 173, 1755–1778 (2018).
  * [16] P. W. Anderson, G. Yuval, and D. R. Hamann, _Exact Results in the Kondo Problem. II. Scaling Theory, Qualitatively Correct Solution, and Some New Results on One-Dimensional Classical Statistical Models_ , Phys. Rev. B 1:4464 (1970).
  * [17] D. Barsky and M. Aizenman, Percolation Critical Exponents Under the Triangle Condition. Annals of Probab. 19 (4), 1520–1536 (1991).
  * [18] R. Bauerschmidt, D.C. Brydges, and G. Slade, Scaling limits and critical behaviour of the 4-dimensional $n$-component $|\varphi^{4}|$ spin model, J. Stat. Phys., 157:692–742 (2014).
  * [19] M. Biskup, _Reflection positivity and phase transitions in lattice spin models_ , in Methods of contemporary mathematical statistical physics, Lecture Notes in Math., vol. 1970, pp. 1–86 (Springer, Berlin, 2009).
  * [20] D.C. Brydges, J. Fröhlich and A.D. Sokal, _A new proof of the existence and nontriviality of the continuum $\varphi^{4}_{2}$ and $\varphi^{4}_{3}$ quantum field theories_, Comn. Math. Phys. 91, 141–186 (1983).
  * [21] D. Brydges, J. Fröhlich, and T. Spencer, _The random walk representation of classical spin systems and correlation inequalities_ , Comm. Math. Phys., 83 (1):123–150 (1982).
  * [22] D. Chelkak and S. Smirnov, _Universality in the 2D Ising model and conformal invariance of fermionic observables_ , Inventiones mathematicae 189, 515–580 (2012).
  * [23] F. Camia and C.M. Newman, _Ising (conformal) fields and cluster area measures_ , PNAS 106 (14), 5457–5463 (2009).
  * [24] R. L. Dobrushin and S. B. Shlosman, _Completely analytical interactions: Constructive description._ J. Stat. Phys. 46, 983–1014 (1987).
  * [25] H. Duminil-Copin, Random currents expansion of the Ising model, in European Congress of Mathematics, Eur. Math. Soc. (Zürich, 2018), pp. 869-889.
  * [26] H. Duminil-Copin, Lectures on the Ising and Potts models on the hypercubic lattice.. In Random Graphs, Phase Transitions, and the Gaussian Free Field, Springer Proc. Math. Stat. 304 pp. 35–161 (Springer, Cham, 2020),
  * [27] H. Duminil-Copin, S. Goswami, and A. Raoufi, Exponential decay of truncated correlations for the Ising model in any dimension for all but the critical temperature, Comm. Math. Phys., 374(2):891–921 (2020).
  * [28] H. Duminil-Copin, A. Raoufi, and V. Tassion, _Sharp phase transition for the random-cluster and Potts models via decision trees_ , Ann. Math. 189 75–99 (2019).
  * [29] S. El-Showk, M. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, _Solving the 3D Ising model with the conformal bootstrap._ Phys. Rev. D, 86 (2), 25022 (2012).
  * [30] R. Fantoni and J. Klauder, _Affine Quantization of $\varphi^{4}_{4}$ Succeeds While Canonical Quantization Fails_, Phys. Rev. D 103, 076013 (2021).
  * [31] J. Feldman, J. Magnen, V. Rivasseau, and R. Sénéor, Construction and Borel Summability of Infrared $\varphi_{4}^{4}$ by a Phase Space Expansion, Comm. Math. Phys., 109:437–480 (1987).
  * [32] M.E. Fisher, _Statistical mechanics of dimers on a plane lattice._ Phys. Rev. 124 1664–1672 (1961).
  * [33] C.M. Fortuin and P.W. Kasteleyn, _On the random-cluster model: I. Introduction and relation to other models._ Physica 57 (4): 536 (1972).
  * [34] P.W. Kasteleyn, _Dimer statistics and phase transitions._ J. Math. Phys. 4, 287–293 (1963).
  * [35] C.M. Fortuin, P.W. Kasteleyn and J. Ginibre, ”Correlation inequalities on some partially ordered sets”, Comm. Math. Phys. 22 (2), 89–103 (1971).
  * [36] J. Fröhlich, _On the triviality of $\lambda\varphi^{4}_{d}$ theories and the approach to the critical point in $d_{\begin{subarray}{c}>\\\ (-)\end{subarray}}4$ dimensions_, Nuclear Physics B, 200(2):281–296 (1982).
  * [37] J. Fröhlich, R. Israel, E.H. Lieb, B. Simon, Phase transitions and reflection positivity. I. General theory and long range lattice models, Comm. Math. Phys., 62:1–34 (1978).
  * [38] J. Fröhlich, B. Simon, and T. Spencer, _Infrared bounds, phase transitions and continuous symmetry breaking_ , Comm. Math. Phys., 50(1):79–95 (1976).
  * [39] K. Gawedzki and A. Kupiainen, Massless Lattice $\varphi_{4}^{4}$ Theory: Rigorous Control of a Renormalizable Asymptotically Free Model, Comm. Math. Phys., 99:197–252 (1985).
  * [40] A. Giuliani, Scaling limits and universality of Ising and dimer models, (2021 preprint) arxiv.org/pdf/2110.09372
  * [41] A. Giuliani, V. Mastropietro, Universal Finite Size Corrections and the Central Charge in Non-solvable Ising Models, Comm. Math. Phys. 324, 179–214 (2013).
  * [42] J. Glimm and A. Jaffe, _Positivity of the $\varphi_{3}^{4}$ hamiltonian._, Fortschritte der Physik, 21(7):327–376 (1973).
  * [43] R. Griffiths, Correlation in Ising ferromagnets I, J. Math. Phys., 8:478–489 (1967).
  * [44] R. Griffiths, Rigorous Results for Ising Ferromagnets of Arbitrary Spin, J. Math. Phys., 10:1559 (1969).
  * [45] R.B. Griffiths, C.A. Hurst, and S. Sherman, _Concavity of magnetization of an Ising ferromagnet in a positive external field_ , J. Math. Phys., 11:790–795 (1970).
  * [46] G. Grimmett, _The random-cluster model_ , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 333, Springer-Verlag, Berlin (2006).
  * [47] J. Groeneveld, R.J. Boel and P.W. Kasteleyn, _Correlation-function identities for general planar Ising systems_ , Physica A: Stat. Mech. and its App., 93 (1), 138–154 (1978).
  * [48] F. Guerra, L. Rosen and B. Simon, _The $P(\varphi)^{2}$ Euclidean Quantum Field Theory as Classical Statistical Mechanics_, Annals of Math., 101:111–189 (1975).
  * [49] M. Hairer, A theory of regularity structures, Inventiones Mathematicae, 198 (2), 269–504 (2014).
  * [50] J.M. Hammersley.  Percolation processes: Lower bounds for the critical probability. Ann. Math. Statist., 28:790–795 (1957).
  * [51] T. Hara and H. Tasaki, A Rigorous Control of Logarithmic Corrections in Four-Dimensional $(\varphi_{4})^{4}$ Spin Systems. II. Critical Behaviour of Susceptibility and Correlation Length, J. Stat. Phys., 47(1/2):99-121 (1987).
  * [52] T. Hara, R. van der Hofstad and G. Slade. _Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models._ Ann. Probab. 31, 349–408. (2003)
  * [53] G.C. Hegerfeldt,  Correlation inequalities for Ising ferromagnets with symmetries,  Comm. Math. Phys., 57(3):259–266 (1977).
  * [54] M. Heydenreich and R. van der Hofstad, Progress in High-Dimensional Percolation and Random Graphs. CRM Short Courses. (Springer, Cham. 2017).
  * [55] C.A. Hurst and H.S. Green, _New Solution of the Ising Problem for a Rectangular Lattice_ , J. Chem. Phys. 33, 1059 (1960).
  * [56] A. Jaffe and E. Witten, _Quantum Yang-Mills Theory_ , The millennium prize problems, (1):129 (2006), www.claymath.org/sites/default/files/yangmills.pdf.
  * [57] L. Kadanoff, _Correlations along a Line in the Two-Dimensional Ising Model._ Phys. Rev. 188, 859 (1969).
  * [58] L.P. Kadanoff and H. Ceva, _Determination of an Operator Algebra for the Two-Dimensional Ising Model_ , Phys. Rev. B 6, 3918–3939 (1971).
  * [59] H. Kesten, _The incipient infinite cluster in two-dimensional percolation_ , Probab. Theory Related Fields, 73(3):369–394 (1986).
  * [60] M.G. Krein, M.A. Rutman, _Linear operators leaving invariant a cone in a Banach space_ , Transl. Amer. Math. Soc.26, 199 (1950) (cf. p.279).
  * [61] G.F. Lawler, _Intersections of random walks_ , Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2013, Reprint of the 1996 edition.
  * [62] J.L. Lebowitz, _GHS and other inequalities_ , Comm. Math. Phys. 35:87–92 (1974).
  * [63] T.D. Lee and C.N. Yang, Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model. Phys. Rev., 87 (3): 410–419 (1952).
  * [64] E. H. Lieb, _A refinement of Simon’s correlation inequality_ , Comm. Math. Phys. 77(2):127–135 (1980).
  * [65] S. Miracle-Solé A. Messager, _Correlation functions and boundary conditions in the Ising ferromagnet_ , J. Stat. Phys. 17(4):245–262 (1977).
  * [66] C.M. Newman, _Inequalities for Ising models and field theories which obey the Lee-Yang theorem_ , Comm. Math. Phys., 41(1) (1975).
  * [67] L. Onsager, _Crystal statistics. I. A two-dimensional model with an order-disorder transition_ , Physical Review, Series II, 65(3–4):117–149 (1944).
  * [68] K. Osterwalder, R. Schrader, _Axioms for Euclidean Green’s functions I._ , Comm. Math. Phys. 31:83–112 (1973).
  * [69] K. Osterwalder, R. Schrader, _Axioms for Euclidean Green’s functions II._ , Comm. Math. Phys. 42:281–305 (1975).
  * [70] R. Peierls. _On Ising’s model of ferromagnetism_ , Math. Proceedings Cambridge Phil. Soc., 32 (3), 477–481 (1936).
  * [71] V. Riva and J. Cardy _Holomorphic parafermions in the Potts model and stochastic Loewner evolution._ J. Stat. Mech. P12001 (2006).
  * [72] A. Sakai, _Lace expansion for the Ising model._ Comm. Math. Phys. 272 283–344 (2007).
  * [73] A. Sakai, _Correct bounds on the Ising lace-expansion coefficients_ , (2021 preprint) https://arxiv.org/pdf/2003.09856.
  * [74] R. Schrader. _New correlation inequalities for the Ising model and $P(\varphi)$ theories_, Phys. Rev., B 15:2798 (1977).
  * [75] T.D. Schultz, D.C. Mattis, E.H. Lieb, _Two-dimensional Ising model as a soluble problem of many fermions_ , Reviews of Modern Physics 36(3), 856 (1964).
  * [76] B. Simon, The $P(\varphi)_{2}$ Euclidean (Quantum) Field Theory, Princeton Univ. Press, 1974).
  * [77] B. Simon, _Correlation inequalities and the decay of correlations in ferromagnets_ , Comm. Math. Phys. 77, 111-126 (1980).
  * [78] B. Simon and R.B. Griffiths, _The $\varphi^{4}_{2}$ field theory as a classical Ising model_, Comm. Math. Phys. 33(2):145–164 (1973).
  * [79] A.D. Sokal, _A rigorous inequality for the specific heat of an Ising or $\varphi_{4}^{4}$ ferromagnet._ Phys. Lett. bf 71 A, 451–453 (1979)
  * [80] A.D. Sokal, _An alternate constructive approach to the $\varphi^{4}_{3}$ quantum field theory, and a possible destructive approach to $\varphi_{4}^{4}$._, Ann. Inst. Henri Poincaré Phys. Théorique, 37:317–398 (1982).
  * [81] K. Symanzik, _Euclidean quantum field theory_ , Local quantum theory. Jost, R. (ed.). New York: Academic Press (1969).
  * [82] D. J. Thouless, _Long-Range Order in One-Dimensional Ising Systems_ , Phys. Rev. 187:732 (1969).
  * [83] J. van den Berg and H. Kesten, _Inequalities with applications to percolation and reliability._ , J. Appl. Probab, 22 556–569 (1985).
  * [84] R. van der Hofstad and A. Járai, _The incipient infinite cluster for high-dimensional unoriented percolation_ , J. Stat. Phys., 114(3-4):553–625 (2004).
  * [85] A.S. Wightman, _Quantum Field Theory in Terms of Vacuum Expectation Values_ , Phys. Rev. 101, 860 (1956).
  * [86] K.G. Wilson, _Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture_ , Phys. Rev. B 4 (1971.
  * [87] K.G. Wilson and M.E. Fisher, _Critical Exponents in $3.99$ Dimensions_ Phys. Rev. Lett. 28, 240 (1972).