204

100 Years of the Ising Model

1Hugo Duminil-CopinHugo Duminil-Copin

1Unige, 7-9 rue du Conseil-Général, 1205 Genève, Switzerland , IHES, 35 route
de Chartres, 91440 Bures-Sur-Yvette, France .

Dedicated to the memory of colleagues and friends Dmitry Ioffe and Vladas
Sidoravicius

60K35]82B20

# 100 Years of the (Critical) Ising Model on the Hypercubic Lattice

(2022)

###### Abstract

We take the occasion of this article to review one hundred years of the
physical and mathematical study of the Ising model. The model, introduced by
Lenz in 1920, has been at the cornerstone of many major revolutions in
statistical mechanics. We wish, through its history, to outline some of these
amazing developments. We restrict our attention to the ferromagnetic nearest-
neighbour model on the hypercubic lattice, and essentially focus on what
happens at or near the so-called critical point.

###### :

[

###### keywords:

Ising model, percolation theory, phase transition, statistical mechanics

††volume: 1

##  1 Short motivation

How to provide an introduction to (part of) statistical physics aimed at a
wide audience of mathematicians? The question is not easy, especially since
the domain is positioned halfway between theoretical physics and mathematics,
and that contrarily to (some) other fields of mathematics, it is hard to
identify a theory that would embrace most of statistical physics. A partial
answer may be to follow the standard approach of teaching by examples, and to
pick what is maybe the most classical model of statistical physics, namely the
Ising model. Through its history, one may trace many of the revolutions, both
on the theoretical physics and mathematical sides, that statistical physics
underwent in the last century.

We therefore chose to streamline this history, from the emergence of the model
to explain experimental results, to its modern applications in mathematics,
physics, and beyond. Obviously, the story will be tainted by the expertise of
the author since thousands of papers have been published on the subject, which
ranges over many subfields of mathematics and theoretical physics. A
subjective selection of papers has therefore been made, and the attention has
been restricted to the model on the hypercubic lattice, at equilibrium, (for
most of the review) at criticality, and always with nearest neighbour
ferromagnetic interactions (we are well aware that non-critical and dynamical
aspects, as well as long-range, random, or antiferromagnetic interactions also
are of prime importance).

We tried to respect the timing of the appearance of the different notions
pertaining to the model, and avoided as much as possible some tempting
anachronisms. As a result, certain readers may be surprised by some
statements, knowing that simpler and more natural versions exist nowadays.
Also, the large number of breakthroughs in the Ising model’s history –
Peierls’ argument, Onsager’s solution on the square lattice and the exact
integrability results that followed, Kadanoff’s scaling and universality
hypotheses, the Lee-Yang theorem, correlation inequalities, the Fortuin-
Kasteleyn representation, reflection positivity, Aizenman’s treatment of the
random current representation and use of differential inequalities, conformal
field theory, rigorous renormalisation group, Chelkak-Smirnov’s conformal
invariance, 3D conformal bootstrap, to cite but a few – forced us to be very
quick on some of these developments. References are added for the avid reader.
We also refer to [20, 87] for historical introductions, and [48] for a book on
statistical physics including a study of the Ising model.

##  2 The first twenty years: a laborious start

###  2.1 Ising model’s prehistory

In 1895, the French physicist Pierre Curie [30] noticed that magnets lose
their magnetic attraction when they are heated above a certain critical
temperature, now called the Curie temperature (the phenomenon seemed to have
been discovered before by the French physicist Pouillet in 1832, see [71] for
precise references and a discussion). While the Curie temperature varies from
slightly over 100 degrees Celsius for certain alloys, to 769.85 degrees
Celsius for magnets made of iron, the underlying phenomenon is always the
same: at a certain temperature, a magnet ceases to be able to keep a
spontaneous magnetisation and exhibits magnetisation only when an external
field is applied to it. This phenomenon is called a phase transition between a
paramagnetic phase above the Curie temperature, and a ferromagnetic phase
below it.

Curie also identified a law, now called Curie’s law, relating the magnetic
susceptibility of the system to the temperature and the external magnetic
field applied to the magnet. He noticed the similarity between the
ferromagnetic and paramagnetic phases of a magnet in terms of the temperature
and the external magnetic field applied to it, and respectively the liquid and
gas phases of a fluid in terms of the pressure and temperature. Pierre Weiss
[103] tried to produce an efficient physical explanation of this phenomenon by
introducing an assumption, referred to today as the mean-field approximation.
This mean-field model, called the Curie-Weiss model, gave rise to an
interesting yet not fully accurate description of the phase transition.

The German physicist Wilhelm Lenz got interested in Curie’s law. Lenz agreed
with one of Weiss’ suggestions that magnets are made of elementary pieces that
behave themselves as small magnets. Yet, he was at the same time in line with
his contemporary physicists, thinking that one of Weiss’ assumption, namely
that elementary magnets can rotate freely within a solid, was wrong. Taking
this into account, he challenged the rotational freeness. Observing that a
crystal selects certain directions corresponding to its symmetries, he made
the assumption that elementary magnets also behave in this way. By analogy, he
then suggested that a crystal-like mechanism for magnets should favour that
neighbouring elementary magnets are aligned, therefore corresponding to either
pointing in the same or opposite directions. At the end, the reasoning of Lenz
led to the assumption that elementary magnets were taking only two possible
directions that are opposite of each other. He formalized this reasoning in
[81].

At this stage, Lenz did not propose an explicit form for the interaction
between elementary magnets. Also, the paper approximately explained the
typical behaviour of a paramagnet having respectively zero magnetisation when
no magnetic field is applied, and a magnetisation when such a magnetic field
is applied, but Lenz made no mention of what will later be referred to as the
ferromagnetic behaviour.

Ernst Ising was a German physicist born in 1900, who was a PhD student of Lenz
in Hamburg. He graduated in 1924 and published a paper [70] on Lenz’s model in
1925. So, what did Ising actually achieve in his famous paper from 1925?

First of all, he went one step further than Lenz by specifying the interaction
between elementary magnets. He first made the assumption that interactions
"decay rapidly with distance, so that we, in general, to a first
approximation, only have to take the influence on neighbouring elements into
account". He also assumed that "of all the possible positions that the
neighbouring atoms can assume in relation to each other, the one that requires
the minimum energy is when they are both acting in the same direction". These
two assumptions led to the mathematical model that we will define formally in
the next section. In order to treat this model, Ising made a further
assumption: he assumed that the elementary magnets are positioned on a linear
chain.

From all of this, Ising could deduce Curie’s law in the paramagnetic phase.
While this was a source for optimism, the latter was severely challenged by
the observation that the magnetisation was tending to 0 as the magnetic field
vanishes, irrespective of the temperature. In other words, no explanation for
ferromagnetism was in sight. Even worse, despite a few attempts at
generalising the model (Ising considered non-nearest neighbour interactions,
more possible directions for the elementary magnets, and a hybrid three
dimensional model that would correspond to a limit in which only pairs of
neighbouring elementary magnets in one direction truly interact), the
ferromagnetism did not seem to be explainable. This led Ising to conjecture
that the model was not a good explanation for ferromagnetism (even when
considering higher dimensional base graph for the spins), a thought that he
gathered in a letter to American historian Stephen Brush years later: "I
discussed the result of my paper widely with Professor Lenz and with Dr.
Wolfgang Pauli, who at that time was teaching in Hamburg. There was some
disappointment that the linear model did not show the expected ferromagnetic
properties."

After his PhD, Ising left academia to become a teacher in Germany before being
forced to step down due to his Jewish origins. He fled Nazi Germany and
emigrated to the United States, where he became a Professor in Physics at
Bradley University. He never published after his first original paper, and
only later became aware of how famous the model had grown into.

###  2.2 Formal definition

Let us turn to the formal definition of the model for our magnet. Consider a
finite non-oriented subgraph $G=(V,E)$ of the hypercubic lattice
$\mathbb{Z}^{d}$ with vertex-set $V$ corresponding to the position of its
elementary magnet constituents, and edge-set $E$ modeling the links between
neighbouring ones. An edge $e\in E$ is often written $e=\\{x,y\\}$, where $x$
and $y$ are its endpoints. The elementary magnet at $x\in V$ will be a
quantity $\sigma_{x}\in\\{-1,+1\\}$, where $-1$ and $+1$ correspond to the two
opposite directions that it may take. The value $\sigma_{x}$ is called the
spin at $x$, and the collection $(\sigma_{x}:x\in V)\in\\{-1,1\\}^{V}$ of all
spins at vertices in $V$ is called the spin configuration, and should be
understood as the state of our magnet.

The energy – or Hamiltonian – of a configuration $\sigma$ on $G$ is given by

$H_{G,h}(\sigma):=-\sum_{\\{x,y\\}\in E}\sigma_{x}\sigma_{y}-h\sum_{x\in
V}\sigma_{x},$ (2.1)

where $h\in\mathbb{R}$ is called the magnetic field. Sometimes, one may want
to generalise the model to accommodate non-nearest neighbour and non-
ferromagnetic interactions by setting

$H_{G,h,(J_{x,y})}(\sigma):=-\sum_{x,y\in
V}J_{x,y}\sigma_{x}\sigma_{y}-h\sum_{x\in V}\sigma_{x}$ (2.2)

where the $(J_{x,y}:x,y\in V)$ are called the coupling constants of the model.
Except when otherwise stated, we focus here on the Hamiltonian $H_{G,h}$
corresponding to what is called the nearest neighbour ferromagnetic (n.n.f.)
Ising model on $G$.

Following Boltzmann, one considers the (grand) partition function of the Ising
model on $G$ at inverse-temperature $\beta$ and magnetic field $h$ defined by

$Z(G,\beta,h):=\sum_{\sigma\in\\{-1,1\\}^{V}}\exp[-\beta H_{G,h}(\sigma)].$
(2.3)

The quantity $\beta$ is interpreted as the inverse of the temperature, as the
latter corresponds to the thermal excitation of elementary magnets, for which
it is natural to predict that the larger their excitation, the less relevant
their interaction.

Physicists then consider the linear form defined for every function
$X:\\{-1,1\\}^{V}\rightarrow\mathbb{R}$ by the formula

$\langle
X\rangle_{G,\beta,h}:=\frac{1}{Z(G,\beta,h)}\sum_{\sigma\in\\{-1,1\\}^{V}}X(\sigma)\exp[-\beta
H_{G,\beta,h}(\sigma)].$ (2.4)

At this stage, we do not consider $\langle\cdot\rangle_{G,\beta,h}$ itself,
and instead focus on a thermodynamical quantity of the system called the free
energy. Consider a box $\Lambda_{n}:=[-n,n]^{d}\cap\mathbb{Z}^{d}$ and define
the free energy of the $d$-dimensional Ising model by the formula

$f(\beta,h):=-\frac{1}{\beta}\lim_{n\rightarrow\infty}\frac{1}{|\Lambda_{n}|}\ln
Z(\Lambda_{n},\beta,h)$ (2.5)

(the existence of the limit is justified by a sub-additivity argument left to
the reader).

Originally, Lenz and Ising were interested in a quantity

$m(\beta,h):=-\tfrac{\partial}{\partial h}f(\beta,h),$ (2.6)

which is interpreted as the magnetisation of the system in the presence of a
magnetic field of strength $h$. One may then define the spontaneous
magnetisation, which corresponds to the remaining magnetisation when removing
the magnet from the ambient magnetic field,

$m^{*}(\beta):=\lim_{h\searrow 0}m(\beta,h)$ (2.7)

(to justify the limit, one may prove that $m(\beta,h)$ decreases as $h$
decreases). The cases $m^{*}(\beta)=0$ and $m^{*}(\beta)>0$ are respectively
called the paramagnetic and ferromagnetic cases as they correspond to the
cases where the magnet respectively loses or keeps its magnetisation even
without external magnetic field.

###  2.3 What does the Ising model truly model?

The Ising model did not develop quickly after its introduction. The original
paper was cited very sporadically in the ten years that followed. In fact,
Ising himself was aware of one citation to his paper only, and this lack of
interest was one of the reasons that pushed him to abandon academia.

There are several explanations why the paper received little attention. The
first one is that the negative result of the paper, stating that the model did
not explain ferromagnetism, was a pretty disappointing one. The second is a
timing problem. A few years after Ising’s paper, Heisenberg introduced another
model of ferromagnetism [63] based on quantum mechanics, in which the
“classical” spins of the Ising model are replaced by the quantum spins of
electrons. In other words, Heisenberg’s model tries to explain ferromagnetism
via the interaction of the spin angular momentum of the electrons in the
atoms, while the Ising model was relying on their magnetic moments. In a
certain sense, the Ising model was a semi-classical version of Heisenberg
model, and as such was violating the latest developments of quantum mechanics.
The discrepancy between the great predictive successes of the Heisenberg
model, and the impossibility to reconcile the Ising model with the recent
advancements in modern physics almost entirely disqualified the model as a
good description of ferromagnetic materials.

At this point, one may wonder why this model, initially introduced in
theoretical physics to explain ferromagnetism but seemingly failed to do so,
did not simply fall into darkness after this rocky start. An element of answer
can be found in the developments of other fields of physics, which we now
review.

In 1919, the Russian-German chemical physicist Gustav Tamman presented an
interesting experiment in which atoms in alloys of copper and gold tend to be
surrounded by atoms of the other kind (to picture this, think of a chessboard
colouring of the square lattice). In Tamman’s experiment, the thermal
agitation has a direct impact on how much the atoms tend to be in the right
places. In 1935, Bragg and Williams [19] explained this phenomenon by a
statistical mechanics’s argument involving the energy cost of having an atom
in the wrong place. Hans Bethe simplified the model by assuming that only
nearest atoms interact.

In 1936, Ralph Fowler and his team in Cambridge introduced another theoretical
model to understand the adsorption of metal vapour on a glass. Fowler more
generally identified a class of experiments exhibiting similar behaviours,
that he named cooperative phenomena.

The German theoretical physicists Rudolf Peierls later noticed the similarity
between Bethe’s approximation of the Bragg-Williams model, Fowler’s theory of
adsorption, and the Ising model. While the original physical problems are
different, the mathematical treatment is in fact similar. In retrospect,
Peierls was perhaps the first person to identify that the Ising model could
treat a number of different phenomena, even though the model was a coarse
caricature for each one of the phenomena in question.

This observation was maybe what kept the Ising model alive for some years, but
it is mathematics that truly changed the nature of the model and made it what
it is today. We now turn to the first mathematical breakthrough in the model.

###  2.4 Peierls’ argument

While Peierls agreed with the majority of the physics community that the Ising
model was not a good model for ferromagnetism, he certainly recognised that
the model was of mathematical interest. Furthermore, he totally disagreed with
the naive generalisation, based on the few attempts of Ising, of the absence
of a ferromagnetic phase to higher dimensional lattices. This led him to
reconsider the problem of the Ising model in two and three dimensions. As a
result, he produced what is probably one of the most important papers in the
early Ising history [91], in which he developed a technique which is now
widely known in statistical physics as Peierls’ argument.

Roughly speaking, the argument runs as follows. When considering a
configuration $\sigma$ of the Ising model on $\mathbb{Z}^{2}$, or a finite
subgraph of it, one may associate a subset $E(\sigma)$ composed of the edges
$\\{x,y\\}$ of the graph with $\sigma_{x}\neq\sigma_{y}$. In a planar context,
one may draw these edges $e\in E(\sigma)$ by considering the dual edges
$(e^{*}:e\in E(\sigma))$ on the dual graph111The dual graph
$G^{*}=(V^{*},E^{*})$ of a planar graph $G=(V,E)$ is the planar graph with
vertex-set given by the faces of $G$ (including the exterior one) and edge-set
$E^{*}$ given by unordered pairs $\\{u,v\\}$, where $u$ and $v$ are two faces
that are bordered by the same edge. When this edge is $e$, we denote the dual
edge $\\{u,v\\}$ by $e^{*}$. The map $e\mapsto e^{*}$ is therefore a bijection
between $E$ and $E^{*}$. On the square lattice, the dual graph is nothing but
the translate by $(\tfrac{1}{2},\tfrac{1}{2})$ of the square lattice itself.;
see Figure 1. These dual edges and their endpoints form an even subgraph of
the dual graph (call $\mathrm{Even}(G^{*})$ the set of such even subgraphs)
which can be interpreted as a collection of loops on the dual graph. The
representation of configurations $\sigma$ in terms of even subgraphs is called
the low-temperature expansion. Using the mapping between $\sigma$ and
$E(\sigma)$, one may rewrite the partition function as

$Z(G,\beta,0)=\sum_{\sigma\in\\{-1,1\\}^{V}}e^{-\beta
H_{G,h}(\sigma)}=e^{\beta|E(G)|}\sum_{F\in\mathrm{Even}(G^{*})}e^{-2\beta|F|}.$
(2.8)

This formula immediately highlights the fact that $\beta$ large renders
configurations $F\in\mathrm{Even}(G^{*})$ with large loops unlikely. Building
on this observation, Peierls was able to obtain that $m^{*}(\beta)>0$ for
large values of $\beta$, see Frame 1 for more details.

The idea to introduce a model of “domain walls” separating the different
phases (here pure $+1$ and pure $-1$) from each other is not restricted to the
Ising model: it has been very fruitful to prove the existence of phase
transitions, and Peierls’ argument is now one of the most famous and robust
arguments in statistical physics.

Frame 1: a quick version of Peierls’ argument

We do not consider the magnetization $m^{*}(\beta)$ but rather the correlation
$\langle\sigma_{0}\sigma_{\mathbf{g}}\rangle_{\Lambda_{n}^{+},\beta,0}$, where
$\Lambda_{n}^{+}$ is the graph $\Lambda_{n}$ plus a vertex $\mathbf{g}$,
sometimes referred to as Griffiths’ “ghost” vertex, connected to all the
vertices on the boundary of $\Lambda_{n}$; see Figure 1 on the left. The limit
as $n$ tends to infinity can be shown to be $m^{*}(\beta)$, so it is
sufficient to prove that the quantity is bounded away from 0 uniformly in $n$.

If one denotes by $\mathbf{C}=\mathbf{C}(\sigma)$ the connected component of
$0$ in $\mathbb{R}^{2}\setminus\\{e^{*}:e\in E(\sigma)\\}$, one may decompose
the magnetisation depending on the value of $\mathbf{C}$ to get

$\langle\sigma_{0}\sigma_{\mathbf{g}}\rangle_{\Lambda_{n}^{+},\beta,0}=1-2\sum_{\mathbf{g}\notin
C\in\mathrm{Even}((\Lambda_{n}^{+})^{*})}\langle\mathbb{I}(\mathbf{C}=C)\rangle_{\Lambda_{n}^{+},\beta,0}.$
(2.9)

Now things become interesting. For every $C\notni\mathbf{g}$, consider the
configuration $\mathrm{Flip}_{C}(\sigma)$ obtained from $\sigma$ by flipping
the values of the spins inside $C$. This effectively corresponds to removing
the set $\partial_{e}C$ of edges in $E(\sigma)$ with exactly one endpoint in
$C$. Taking into account the cost of this operation leads to

$\displaystyle\langle\mathbb{I}(\mathbf{C}=C)\rangle_{\Lambda_{n}^{+},\beta,0}\leq
e^{-2\beta|\partial_{e}C|}$

for every $C\notni\mathbf{g}$. At this stage, the fact that $\partial_{e}C$ is
a loop and that there are at most $(k+1)4^{k}$ possible loops of length $k$
surrounding the origin gives

$\langle\sigma_{0}\sigma_{\mathbf{g}}\rangle_{\Lambda_{n}^{+},\beta,0}\geq
1-2\sum_{k\geq 1}k4^{k}e^{-2\beta
k}>1-\frac{8e^{-2\beta}}{(1-4e^{-2\beta})^{2}}.$ (2.10)

##  3 Onsager’s 1944 revolution and the integrability of the Ising model

###  3.1 Kramers-Wannier treatment of the Ising model and duality

While Peierls’ result is certainly one of the first key rigorous steps in the
understanding of the Ising model, the work [80] of Hans Kramers and Gregory
Wannier in 1941 propelled the Ising model in another dimension in terms of
mathematical interest. Indeed, the two physicists agreed that the Ising model
was not necessarily an accurate description of ferromagnetism, but they were
precursors in strongly believing that having mathematical models that can be
rigorously analysed was of crucial interest for the understanding of physical
phenomena, even if only approximate.

Kramers and Wannier’s goal was to understand what happens for the Ising model
at arbitrary inverse-temperature. Peierls’ argument shows that the model
behaves like a ferromagnet when $\beta$ is large. A fairly simple argument,
see Frame 4, shows that it behaves like a paramagnet when $\beta$ is small. It
is therefore tempting to think that there is an intermediate inverse-
temperature, playing the theoretical role of the inverse of Curie’s
temperature, that separates a paramagnetic phase from a ferromagnetic phase,
i.e. a critical inverse-temperature $\beta_{c}$ defined by the formula

$\beta_{c}=\beta_{c}(\mathbb{Z}^{d}):=\inf\\{\beta:m^{*}(\beta)>0\\}.$ (3.1)

Of course, the notion of critical inverse-temperature immediately leads to the
following question: can one compute the value of the critical point
$\beta_{c}$?

Figure 1: On the left. A picture of the low-temperature expansion on
$\Lambda_{1}^{+}$. The set $\Lambda_{1}^{+}$ is depicted in dashed gray, and
$(\Lambda_{1}^{+})^{*}$ in plain black. The edges in the dual configuration
are depicted in bold. On the right. The set $\mathbb{S}(5,3)$ with the bottom
and top sets depicted. In this case $\underline{\tau}$ and $\overline{\tau}$
are respectively constant equal to $-1$ and $+1$.

The work [80] represented an important historical step towards this
computation. It involved a number of ideas that deeply influenced the way
mathematicians and physicists approach the Ising model. The first key
observation is that Kramers and Wannier did not work with the Ising model in
the presence of a magnetic field (in other words, they set $h$ to be 0).
Instead, they proposed to look at the specific heat defined by

$C(\beta):=-\beta^{2}\tfrac{\partial^{2}}{(\partial\beta)^{2}}(\beta
f)(\beta,0).$ (3.2)

Kramers and Wannier argued that the critical point of the model on
$\mathbb{Z}^{2}$ should correspond to a value of $\beta$ at which $C(\beta)$
blows up. The next step is maybe the most interesting one. By assuming that
there exists a unique point at which $C(\beta)$ blows up, they were able to
predict the value of $\beta_{c}$. The reason behind this prediction is the
following duality relation for the free energy on $\mathbb{Z}^{2}$:

$\beta f(\beta,0)=\beta^{*}f(\beta^{*},0)-2\beta+\ln 2+2\ln\cosh(\beta^{*}),$
(3.3)

where $\beta$ and $\beta^{*}$ are related via the formula
$\tanh(\beta^{*})=e^{-2\beta}$. The uniqueness implies that $\beta_{c}$ must
be the self-dual point satisfying $\beta^{*}=\beta$, i.e. $\beta_{c}$ must be
equal to $\tfrac{1}{2}\ln(1+\sqrt{2})$. Of course, this reasoning is not a
formal proof as it is a priori non-obvious that the singular point is unique.

The proof of Kramers and Wannier of the duality relation is also of great
interest. Originally, they used so-called transfer matrices to do it; see
Frame 2 for details. While they did not invent those matrices (they already
appeared in the work of Montroll [86]), they probably made the first important
use of them. Today, the derivation of this relation is straightforward and
does not rely on transfer matrices. It involves relating the partition
functions $Z(G,\beta,0)$ and $Z(G^{*},\beta^{*},0)$ using, for the first one,
the expression given by the low-temperature expansion (2.8), and for the
second, an alternative representation called the high-temperature expansion,
obtained by van der Waerden [102] and briefly described in Frame 4. When
observing that the dual of a box in the square lattice is (except on the
boundary) a box of the square lattice, one obtains the identity above by
considering larger and larger boxes.

Frame 2: transfer matrices of the Ising model

To lighten the presentation we restrict our attention to the case $h=0$.
Consider the slices
$\mathbb{S}(N,M):=(\mathbb{Z}/N\mathbb{Z})^{d-1}\times\llbracket
0,M\rrbracket$ with no edges between the vertices of the bottom
$(\mathbb{Z}/N\mathbb{Z})^{d-1}\times\\{0\\}$ (we call
$(\mathbb{Z}/N\mathbb{Z})^{d-1}\times\\{M\\}$ the top of the slice); see
Figure 1 on the right. Let $\sigma_{|{\mathrm{bottom}}}$ and
$\sigma_{|{\mathrm{top}}}$ be the restrictions of $\sigma$ to the top and
bottom of $\mathbb{S}(N,M)$, considered as two elements of
$\\{-1,1\\}^{(\mathbb{Z}/N\mathbb{Z})^{d-1}}$. Introduce the quantity

$Z(N,M,\underline{\tau},\overline{\tau}):=\sum_{\sigma\in\\{-1,1\\}^{\mathbb{S}(N,M)}}\exp[-\beta
H_{\mathbb{S}(N,M),h}(\sigma)]\mathbb{I}(\sigma_{|\mathrm{bottom}}=\underline{\tau},\sigma_{|\mathrm{top}}=\overline{\tau}),$
(3.4)

where
$\underline{\tau},\overline{\tau}\in\\{-1,1\\}^{(\mathbb{Z}/N\mathbb{Z})^{d-1}}$,
as well as the so-called transfer matrix

$V_{N}(\underline{\tau},\overline{\tau}):=Z(N,1,\underline{\tau},\overline{\tau})=\exp\Big{[}-\beta\Big{(}\sum_{x\in(\mathbb{Z}/N\mathbb{Z})^{d-1}}\underline{\tau}_{x}\overline{\tau}_{x}+\sum_{\\{x,y\\}\in
E((\mathbb{Z}/N\mathbb{Z})^{d-1})}\overline{\tau}_{x}\overline{\tau}_{y}\Big{)}\Big{]}.$
(3.5)

One immediately finds that
$Z(N,M,\underline{\tau},\overline{\tau})=V_{N}^{M}(\underline{\tau},\overline{\tau}).$
Other quantities of the model may be written in terms of transfer matrices:
for instance the partition function of the model on the $d$-dimensional torus
$(\mathbb{Z}/N\mathbb{Z})^{d}$ becomes the trace of $V_{N}^{N}$.

One important aspect of those transfer matrices $V_{N}$ is that certain
questions on the behaviour of the model are rephrased as spectral questions on
the transfer matrix. For instance, by letting $M$ and then $N$ go to infinity,
one observes that the asymptotic behaviour of the partition function on
$(\mathbb{Z}/N\mathbb{Z})^{d-1}\times(\mathbb{Z}/M\mathbb{Z})$, and therefore
the value of the free energy, are connected to the asymptotic behaviour of the
leading eigenvalue of $V_{N}$ as $N$ tends to infinity. This can very well be
an intractable problem, but in some cases it is not.

###  3.2 Onsager’s result

Kramers and Wannier’s results unraveled the potential mathematical interest of
the Ising model, but the real revolution came only a few years after with one
of the most impressive achievements in mathematical physics. Lars Onsager,
Nobel prize winner in 1968, was a Norwegian specialist in theoretical
chemistry. He was particularly interested in mathematical problems and focused
his attention on the Ising model for the formidable challenge that its exact
solution represented more than for his physical relevance222His opinion on
this fact seemingly evolved: in his first 1944 paper [88] he presents it as a
poor model of ferromagnetism, but a fairly good model for binary alloys, while
in his paper with Kaufman in 1949 [77] he describes it as a model for
ferromagnetism..

To everyone’s surprise, Onsager announced at a conference of the New York
Academy of Sciences in 1942 that he obtained the following exact expression
for the free energy (at zero magnetic field) of the Ising model on the square
lattice $\mathbb{Z}^{2}$:

$-\beta f(\beta,0)=\ln
2+\frac{1}{8\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}\ln[\cosh(2\beta)^{2}-\sinh(2\beta)(\cos\theta_{1}+\cos\theta_{2})]d\theta_{1}d\theta_{2}.$
(3.6)

This implies, in physics jargon, that the model is exactly solvable. This
solvability is itself linked to a deep property of the model called
integrability. Onsager’s underlying idea was that the transfer matrices of the
2D Ising model are a product of two matrices that generate (by taking
successive brackets) a finite-dimensional Lie algebra. He used this
observation to derive the asymptotic behaviour of the leading eigenvalue of
these matrices in his famous 1944 paper [88]. In 1949, Bruria Kaufman [77]
provided an alternative and simpler derivation.

A few years later, Onsager surprised the world of theoretical physicists again
by claiming an exact expression for the spontaneous magnetisation on
$\mathbb{Z}^{2}$: for $\beta\geq\beta_{c}$,

$m^{*}(\beta)=(1-\sinh(2\beta)^{-4})^{1/8}.$ (3.7)

While the result was announced by Onsager first, it was a young physicist,
that would later become one of the most influential theoretical physicists of
the second half of the twentieth century, Chen-Ning Yang (from Yang-Baxter’s
equation, Yang-Mills’s theory, Lee-Yang’s theory, etc.), who provided a
mathematical proof [107] of this statement by achieving a mathematical tour de
force involving Toeplitz determinants. The proof relies on a computation,
again using transfer matrices but much more evolved than for the free energy,
of the two-point function
$\langle\sigma_{(0,0)}\sigma_{(n,0)}\rangle_{(\mathbb{Z}/N\mathbb{Z})^{2},\beta,0}$,
and the observation that

$m^{*}(\beta)^{2}=\lim_{n\rightarrow\infty}\lim_{N\rightarrow\infty}\langle\sigma_{(0,0)}\sigma_{(n,0)}\rangle_{(\mathbb{Z}/N\mathbb{Z})^{2},\beta,0}$
(3.8)

(at the time, such an identity was not obviously true, but nowadays this can
be proved easily using for instance the FK percolation, see Section 7.2).

In the forties and fifties, these successes were considered by physicists as a
mathematical curiosity rather than a truly crucial advance. Yet, they had a
revolutionary impact on theoretical physics for multiple reasons: First, the
level of sophistication of the mathematical tools used in the proofs is
without any common measure with what was previously used in such kinds of
problems, and these techniques created whole new types of mathematical
physics. Second, the behaviour of the model did not correspond to previous
mean-field approximations, thus invalidating rigorously the Curie-Weiss or
Landau theories and opening a new era in statistical mechanics. Third, the
results had many direct applications for the Ising model itself: for instance
the specific heat $C(\beta)$ can easily be shown to blow up logarithmically as
$\beta$ approaches $\tfrac{1}{2}\log(1+\sqrt{2})$, thus confirming rigorously
that this value is the critical point of the system (the logarithmic blow up
is one example of non mean-field behaviour).

Numerous alternatives have been proposed to the approach of Onsager-Kaufman-
Yang, often referred to as the algebraic method. As a joke, Baxter and Enting
named their 1978 paper [10], introducing a solution to the 2D Ising model
involving the notion of star-triangle transformation, the "399th solution of
the Ising model". This count is, of course, overestimated, but one can list a
large number of alternative strategies.

The first such strategy is called the combinatorial approach and is referring
to an original argument of Kac and Ward [72] rewriting the partition function
of the model in terms of the square root of the determinants of so-called Kac-
Ward’s matrices using a combinatorial expansion of the partition function
generalising the van der Waerden high-temperature expansion [102]. The
advantage of such an approach is that it does not rely on transfer matrices,
and therefore is applicable to every finite planar graph, even with arbitrary
nearest-neighbour coupling constants. Unfortunately, the original argument was
not entirely rigorous and one had to wait until 1999 [33] to finally obtain a
mathematical derivation of this approach. Nowadays, the method is very well
understood and especially useful in relation to discrete holomorphicity and
higher genus graphs, see [29] and references therein for a more complete
account.

The (nowadays) most classical method is probably the Pfaffian method. It came
as an attempt to go around the substantial difficulties to make the
combinatorial approach rigorous. Due to Hurst and Green [66], Kasteleyn [76],
and Fisher [46], the strategy consists in writing the Ising partition function
on a finite planar graph $G$ in terms of the dimer (a dimer configuration is a
subset of edges which covers every vertex exactly once) partition function on
a related graph $K(G)$ (the precise definition of the graph depends on the
implementation of the Pfaffian method). It is then possible to relate the
partition function to a skew-symmetric adjacency matrix and express the
partition as a Pfaffian, hence the name of the method. This strategy has been
the basis of a number of more refined results about the model, in particular
the computation of the spin-spin correlations of the model at and away from
criticality. For the deepest and most impressive results, we recommend that
the reader takes a look at the two books of McCoy-Wu [84] and Palmer [90].

Another approach of importance was proposed by Schultz-Mattis-Lieb in [96] to
tackle the cases for which a transfer matrix can be used. In this paper, they
connected the transfer matrix with the exponential of a quantum hamiltonian.
This connection to 1D quantum spin chains has been very fruitful and
understood in a number of alternative ways since then. As a byproduct, the
authors were able to express the partition function as a Grassmann "Gaussian"
integral. The advantage of this way of writing the partition function is that
the Pfaffians emerge naturally. This approach is at the basis of
renormalisation schemes in two dimensions; see Section 8.3.

Yet another approach dealing with the context in which transfer matrices can
be applied is worth mentioning, as it is by far the most generalisable to
other models. It is based on the commutation of the transfer matrices attached
to the model with different critical parameters. Pioneered by Rodney Baxter,
this approach consists in using the so-called Yang-Baxter equation. The
advantage is that the same strategy can be applied to a very large variety of
integrable systems, such as the six-vertex model, to cite only one example. We
refer to [9] and references therein for more details.

##  4 The fifties and sixties: the Ising model becomes a laboratory for
understanding critical phenomena

The fifties and sixties were probably the decades during which the Ising model
became an "unavoidable" model. The realisation that having a tractable model
of statistical physics could be a useful explanatory but also predicting tool
became more and more obvious. The Ising model, with Onsager’s solution, was a
prime example of a model with such qualities.

The model therefore developed tremendously in the postwar era in theoretical
physics as well as in a rapidly growing field called mathematical physics. The
latter gathered more and more physicists that were interested in rigorous
aspects of the objects they studied, and mathematicians willing to study
problems that were naturally emerging from physical modelling. The Ising model
offered a wonderful playground for such scientists, and the number of papers
mentioning the model started to be counted in the hundreds.

###  4.1 Progress in mathematical physics: from perturbative regions of the
phase diagram to the vicinity of the critical point

During this period, the newly developing community of mathematical physicists
recognised that the study of phase transitions, and in particular of the
critical phase (when $\beta$ is equal to $\beta_{c}$), was a vast field of its
own. While the previous developments mostly concerned values of $\beta$ and
$h$ that were far from the critical regime (Peierls’ argument [91] or Baker’s
use of Padé approximant [8] for instance), the situation changed drastically
around the fifties. The interest in the intermediate values of $\beta$ became
stronger and stronger. Onsager’s solution offers a precise understanding of
the critical behaviour of the 2D Ising model, yet it has clear downsides
related to the relative fragility of the integrability of the system. As a
consequence, mathematical physicists started using the Ising model not only as
a solvable system, but more generally as a good mathematical model that one
should not reduce to its integrability aspects. New rigorous techniques
emerged during this period to try to understand the vicinity of the critical
point for non-integrable cases, for instance in higher dimensions.

####  4.1.1 Correlation inequalities

It is natural to ask which monotonicity properties are satisfied by the
system, in particular by the spin-spin correlations
$\langle\sigma_{A}\rangle_{G,\beta,h}$ where $\sigma_{A}:=\prod_{x\in
A}\sigma_{x},$ when the parameters vary (for instance, $G$, $\beta$ or $h$).

To tackle such questions, mathematical physicists started proving what we now
call correlation inequalities using combinatorial arguments. Among the first
such examples are Griffiths’ inequalities [57]: for every $\beta,h\geq 0$ and
every $A,B\subset V$,

$\langle\sigma_{A}\rangle_{G,\beta,h}\geq 0\qquad\text{ and
}\qquad\langle\sigma_{A}\sigma_{B}\rangle_{G,\beta,h}\geq\langle\sigma_{A}\rangle_{G,\beta,h}\langle\sigma_{B}\rangle_{G,\beta,h}.$
(4.1)

A byproduct of the second inequality when applied to $B=\\{x,y\\}$ and summed
over all edges $\\{x,y\\}$, is that correlations
$\langle\sigma_{A}\rangle_{G,\beta,h}$ are increasing in $\beta$ (and also in
$G$ with a little bit of additional work). One may derive the same for the
spontaneous magnetisation $m^{*}(\beta)$, so that the definition of
$\beta_{c}$ can now be rephrased as

$\beta_{c}=\inf\\{\beta\geq 0:m^{*}(\beta)>0\\}=\sup\\{\beta\geq
0:m^{*}(\beta)=0\\}.$ (4.2)

This implies in particular that there is indeed a unique transition between
paramagnetic and ferromagnetic phases.

Other interesting correlation inequalities were obtained in subsequent years.
Let us contemplate a few examples (we do not write them in full generality,
and we drop the subscript after $\langle\cdot\rangle$):

  * •

GHS’s inequality [59]: for $h\geq 0$ and $x\in G$,

$\tfrac{\partial^{2}}{(\partial h)^{2}}\langle\sigma_{x}\rangle\leq 0.$ (4.3)

  * •

Simon-Lieb’s inequality [83]: for $S\ni 0$ and $x\notin S$, when
$\langle\cdot\rangle_{S}$ refers to the model in $S$,

$\langle\sigma_{0}\sigma_{x}\rangle\leq\sum_{y\in\partial
S}\langle\sigma_{0}\sigma_{y}\rangle_{S}\langle\sigma_{y}\sigma_{x}\rangle.$
(4.4)

  * •

Messager-Miracle-Solé’s inequality [85]: for $x,y\in\mathbb{Z}_{+}^{d}$ (below
$\langle\cdot\rangle$ is defined on $\mathbb{Z}^{d}$)

$\langle\sigma_{0}\sigma_{x+y}\rangle\leq\langle\sigma_{0}\sigma_{x}\rangle.$
(4.5)

  * •

FKG’s inequality [47]: for any increasing functions
$f,g:\\{-1,1\\}^{V}\rightarrow\mathbb{R}$,

$\langle fg\rangle\geq\langle f\rangle\langle g\rangle.$ (4.6)

This far from exhaustive list, which we did not discuss in detail, is intended
to show the variety of possible correlation inequalities. Clever use of these
inequalities provided the embryo of what would be considered later as the
theory of non-critical statistical physics systems at equilibrium, as the
correlation inequalities and their consequences often generalise in the same
(or slightly altered) form to a wider class of lattice spin models.

####  4.1.2 The Ising model with a magnetic field: the Lee-Yang theory

While studying the whole phase diagram is a Herculean task that was far beyond
the techniques developed at the time, a beautiful development enabled
mathematical physicists to understand the case $h\neq 0$.

The twin papers [82], referred to as the Lee-Yang theory, relate the
regularity properties of the free energy (and therefore the location of
singular points corresponding to places where a phase transition occurs) to
the locus of the complex zeroes of the partition function $Z(G,\beta,h)$ when
seen as a function of $h\in\mathbb{C}$. Beyond the result itself, the
philosophy consisting in studying the complex zeroes of the partition function
had a resounding effect on the field of mathematical physics. This can be put
in parallel with the analysis of zeroes of the Riemann zeta function: one
learns something about prime numbers by studying the zeroes of a generating-
type function associated with them.

The result of Lee and Yang is not restricted to the n.n.f. Ising model on
$G\subset\mathbb{Z}^{d}$, but the latter gives an important application of it.
In our context, let $Z(G,\beta,\mathbf{h})\in\mathbb{C}$ (for
$\mathbf{h}=(\mathbf{h}_{x}:x\in V)\in\mathbb{C}^{V}$) be the partition
function defined as in (2.3) with the difference that the magnetic field is
allowed to vary with the vertex, i.e. that the $\sum_{x\in V}h\sigma_{x}$ term
of the Hamiltonian in (2.2) is replaced by $\sum_{x\in
V}\mathbf{h}_{x}\sigma_{x}$. The result states that for this model, the zeroes
of the function $\mathbf{h}\mapsto Z(G,\beta,\mathbf{h})$ are satisfying
$\mathrm{Re}(\mathbf{h}_{x})=0$ for every $x\in V$.

As a consequence of this theorem, the free energy $f(\beta,h)$ (which we
recall from (2.5) is expressed in terms of the limit of the logarithm of
partition functions) is analytic as soon as $h\neq 0$. Other consequences
follow, such as exponential decay of so-called truncated correlations of the
system, as well as analyticity of the other thermodynamical quantities when
the magnetic field is non-zero. Roughly put, the Lee-Yang theory enables to
understand in full detail the part of the phase diagram corresponding to non-
zero magnetic field.

###  4.2 Revolutionary progress on the physics front

In parallel to these first successes in mathematical physics, revolutionary
progress was made during this period on the physical understanding of phase
transitions. Among other things, the scaling and universality hypotheses were
formulated, and the pillars of the renormalisation group were cast, in both
cases using the Ising model as an important source of inspiration.

####  4.2.1 Critical exponents and the success of scaling theory

A fundamental notion of physics is the assumption that thermodynamical
quantities of physical systems near criticality tend to take simple forms when
expressed in terms of the parameters of the system. A major advance was
achieved in the sixties by American chemist Benjamin Widom who proposed in
[104] that these quantities are powers in each parameter. For the Ising model,
the parameters are $\beta$ and $h$, and this scaling hypothesis translates
into the existence of so-called critical exponents. To give a few examples
related to already defined quantities, one may for instance predict that

$m^{*}(\beta)=(\beta-\beta_{c})_{+}^{\bm{\beta}+o(1)}\quad\quad
m(\beta_{c},h)=h^{1/\bm{\delta}+o(1)}\quad\quad\langle\sigma_{0}\sigma_{x}\rangle_{\beta_{c},0}=\frac{1}{|x|^{d-2+\bm{\delta}}}$
(4.7)

(notice that $\bm{\beta}$ and $\beta$ have nothing to do with each other),
where $o(1)$ is a quantity tending to 0 as $\beta$ tends to $\beta_{c}$, $h$
tends to 0, or $|x|$ tends to infinity, respectively. In fact, a whole family
of such exponents, denoted by
$\bm{\alpha},\bm{\beta},\bm{\gamma},\bm{\delta},\bm{\eta},\bm{\nu}$ (for the
most classical ones) can be defined for each model. Understanding the phase
transition boils down to, among other things, deriving those exponents.

Dealing with such exponents, one may naturally wonder how many degrees of
freedom truly exist in statistical physics models. For instance, could some of
these critical exponents be connected via direct relations that would
transcend the precise definition of each model? In the sixties, physicists
such as Essam, Fisher, and Widom himself, to cite only those three (see [42,
45, 104] for some early works on the subject), started unraveling systematic
connections between the exponents, thus hinting towards the fact that only two
degrees of freedom exist and that exponents are related by so-called scaling
relations

$\bm{\nu}d=2-\bm{\alpha}=2\bm{\beta}+\bm{\gamma}=\bm{\beta}(\bm{\delta}+1)=\bm{\gamma}\frac{\bm{\delta}+1}{\bm{\delta}-1}\qquad
2-\bm{\eta}=\frac{\bm{\gamma}}{\bm{\nu}}=d\frac{\bm{\delta}-1}{\bm{\delta}+1}.$
(4.8)

The scaling relations apply in a context which is far more general than just
the Ising model (see for instance [38] for a proof in the case of a large
family of two-dimensional percolation models). In the course of discovering
these different scaling relations, the Ising model in two and three dimensions
played the important role of a sanity check. While other experimental systems
were used as testing grounds, the Ising model was the only example of a
theoretical system which did not exhibit mean-field behaviour (and therefore
was not too “trivial”) and for which such exponents were available, either
rigorously thanks to the exact solution in 2D, or approximately thanks to
Baker’s use of Padé approximant [8] in 3D.

To conclude this section, let us mention an important quantity, called the
correlation length $\xi(\beta)$ of the system, that plays an important role in
the scaling hypothesis (it corresponds to the exponent $\bm{\nu}$). We
consider the case $\beta<\beta_{c}$ but a similar notion can be introduced for
$\beta>\beta_{c}$, with analogous interpretations.

When considering, say, spin-spin correlations at criticality, one expects an
algebraic decay as mentioned in (4.7). Yet, when $\beta<\beta_{c}$, the
scaling hypothesis cannot hold uniformly in $|x|$ and such a decay does not
occur. In fact, it was found in many systems that spin-spin correlations decay
exponentially fast (see Section 7.1 for more details) and the inverse-rate of
decay is the correlation length $\xi(\beta)$. This correlation length has an
interesting interpretation: it is the smallest scale at which the system with
$\beta<\beta_{c}$ is off-critical, meaning that when looking at a system with
a size which is much smaller than $\xi(\beta)$, the difference between the
system and a critical system will be invisible to the physicist’s eye, while
on the contrary when the size is much larger than $\xi(\beta)$, the model
looks similar to the case of $\beta\ll\beta_{c}$. In other words, when
approaching the critical point, a system becomes more and more “critical”. By
how much this is true depends on the size of the system, and the correlation
length separates between the sizes at which the system looks critical, and the
sizes at which it looks clearly non critical.

####  4.2.2 Kadanoff’s block-spin renormalisation and universality

While Widom’s scaling hypothesis provides compelling evidence that critical
exponents exist, the underlying justification of the hypothesis itself
remained slightly superficial until Russian physicist Leo Kadanoff provided an
illuminating argument for it. In his famous 1966 paper [73], Kadanoff
suggested that the block-spin renormalisation transformation – i.e. replacing
a block of neighbouring sites by one site having a spin equal to the dominant
spin in the block – corresponds to appropriately changing the scale and the
parameters $\beta$ and $h$ of the model. Assuming that iterating this
procedure somehow converges suggests that the asymptotic properties of the
system are described by a fixed point of a renormalisation map. As a result,
one ends up with the scale invariance of the model. This argument, inspired by
the study of the Ising model, turned out to be the basis of the monumental
theory of the renormalisation group (RG) that was put in a general framework a
few years later by Kenneth Wilson [106].

The block-spin argument of Kadanoff achieved much more than a physical
justification of the scaling hypothesis. Assuming uniqueness of the fixed
point also implies that the renormalisation of Ising models defined on
different $d$-dimensional lattices should converge to the same fixed point,
and therefore share the same critical exponents. This was already partially
realised in 2D by observing the Ising model on the square, hexagonal, and
triangular lattices (they are all exactly solvable) as well as in 3D by
approximations using series expansions [34], but the renormalisation argument
suggests that the few examples of equalities between exponents are, in fact,
the illustration of a much more general phenomenon.

What is now known as the universality hypothesis was explicitly formulated in
parallel by Robert B. Griffiths and Kadanoff in 1971 [58, 74]. Roughly
speaking, it states that the critical properties of a physical system only
depend on

  * •

the lattice dimension $d$;

  * •

the symmetry of the space of possible spins ($\mathbb{Z}/2\mathbb{Z}$ symmetry
for Ising);

  * •

the speed of decay of coupling constants (this is only relevant when the
$J_{x,y}$ are allowed to decay polynomially with $\|x-y\|$, which is not the
case in this text).

This realisation of universality is fundamental to the relevance of
statistical physics as a whole. To borrow from Kadanoff’s wording: “Why study
a simplified model like the Ising model? The strategy of studying physical
questions by using highly simplified models is made rewarding by a
characteristic of physical systems called “universality”, in that many systems
may show the very same qualitative features, and sometimes even the same
quantitative ones. To study a given qualitative feature, it often pays to look
for the simplest possible example.”

To summarise Section 4, by the end of the sixties it became clear to
mathematical physicists and theoretical physicists that the Ising model was
one of the most striking examples of a simple physical system which was rich
enough to grasp a large variety of phenomena falling in the range of
statistical physics. Results on the Ising model started to play a role similar
to experimental results in the sense that they could corroborate or, on the
contrary, invalidate the embryo of a theory. It is fair to say that the
importance of the model was never argued upon later on and that it was finally
recognised as one of the centerpieces of modern statistical physics.

##  5 The sixties and seventies: emergence of the probabilistic interpretation

Physicists and mathematical physicists think of the quantity
$\langle\cdot\rangle_{G,\beta,h}$ as a form attributing to each function
$X:\\{-1,1\\}^{V}\rightarrow\mathbb{R}$ (resp. $\mathbb{C}$) a value in
$\mathbb{R}$ (resp. $\mathbb{C}$). In the late sixties and seventies, the rise
of probabilistic methods led to an alternative interpretation of the Ising
model in which $\langle\cdot\rangle_{G,\beta,h}$ is now understood as (dual
to) a probability measure $\mu_{G,\beta,h}$. As a consequence of this
reinterpretation, it becomes natural to ask what the properties of a randomly
chosen spin configuration are, and what the possible measures on the infinite
lattice that can be obtained as limits of measures in finite volume are.

###  5.1 The random geometry of the spin configuration

As mentioned above, $\langle\cdot\rangle_{G,\beta,h}$ is the linear form
associated with the probability measure $\mu_{G,\beta,h}$ on $\\{-1,1\\}^{V}$
defined for every configuration $\sigma$ by the formula

$\mu_{G,\beta,h}[\\{\sigma\\}]:=\frac{1}{Z(G,\beta,h)}\exp[-\beta
H_{G,h}(\sigma)].$ (5.1)

Then, quantities like $\langle\sigma_{A}\rangle_{G,\beta,h}$ can be
interpreted as the correlations between the random variables $\sigma_{x}$ with
$x\in A$. Note that in this interpretation the partition function is a
normalising factor making the measure at hand a probability measure.

Let us assume for a moment that $h=0$ and interpret the phase transition in
terms of probability. The structure of the probability measure is such that
configurations have greater probability if they have more pairs of neighbours
with a similar spin. In this interpretation, the larger $\beta$ is the more
important it is that neighbours have the same spins. In particular, in the
limit as $\beta$ tends to infinity, one ends up with one of the two
configurations where all spins are the same. It becomes then natural to expect
that for $\beta$ large, typical configurations have an excess of one spin
compared to the other. On the other hand, when $\beta$ is very small, how much
the measure takes the agreements into account is fairly limited, and one may
expect that spins behave roughly independently, at least at large distance of
each other.

The interpretation in terms of random variables opens new uncharted
territories: one can interpret probabilistically natural thermodynamical
quantities such as magnetisation (which corresponds to the expectation of the
spin at a vertex) or surface tension. It also opens a way to new problems,
such as dynamics on the space of spin configurations or large deviations (for
instance for an Ising model at an inverse-temperature $\beta$, but with an
excess of $+1$ spins in a region and of $-1$ spins in another); see Frame 3.

Frame 3: sampling the Ising model – Glauber dynamics

The probabilistic interpretation naturally raises the question of sampling
random configurations according to $\mu_{G,\beta,0}$ (set $h=0$ for
simplicity). A classical method consists in expressing the measure as the
invariant measure of a Markovian dynamics $(\sigma(t):t\geq
0)\in(\\{-1,1\\}^{V})^{\mathbb{R}_{+}}$, called the Glauber dynamics and
defined as follows: attach an exponential clock to each vertex of $G$. Each
time a clock rings, say at time $t$ at $x\in V$,

  * •

If $\sigma_{x}(t)\sum_{y:\\{x,y\\}\in E}\sigma_{y}(t)<0$, switch the value of
the spin at $x$,

  * •

Otherwise, switch the value of the spin at $x$ with a probability equal to
$\exp[-2\beta\sum_{y:\\{x,y\\}\in E}\sigma_{y}(t)]$, and do not switch
otherwise.

Since $\mu_{G,\beta,0}$ is the only invariant measure for this dynamics, the
limit as $t$ tends to infinity, irrespectively of the initial value
$\sigma(0)$, is sampled according to $\mu_{G,\beta,h}$.

This dynamics was named after the American physicist Roy J. Glauber.
Alternative choices of dynamics are obtained by changing the jump
probabilities. In Figure 2, three simulations of the Ising model are shown
respectively below (on the left), at (in the middle) and above (on the right)
$\beta_{c}$.

###  5.2 Boundary conditions and the Gibbs formalism

An important output of the probabilistic interpretation of the model is that
it becomes natural to condition on spins in a subset of $V$. More precisely,
let $W\subset V$ and let $H$ be the graph with vertex-set $W$ and edge-set
induced by the edges of the graph $G$. Let $\tau\in\\{-1,1\\}^{V}$ be a spin
configuration on $G$. One may ask what is the law of the spins in $W$ when
conditioning $\sigma$ outside $W$ to be equal to $\tau$, i.e. what is
$\mu_{G,\beta,h}[\,\cdot\,|\sigma_{x}=\tau_{x},\forall x\notin W]$?

The answer to this question is best cast when introducing the notion of
boundary conditions. For a subgraph $G$ of $\mathbb{Z}^{d}$ and a
configuration $\tau\in\\{-1,1\\}^{\mathbb{Z}^{d}}$, introduce the measure
$\mu_{G,\beta,h}^{\tau}$ with $\tau$ boundary conditions defined like
$\mu_{G,\beta,h}$ except that $H_{G,h}$ is replaced by

$H_{G,h}^{\tau}(\sigma):=H_{G,h}(\sigma)-\sum_{\\{x,y\\}\in
E(\mathbb{Z}^{d}):x\in V,y\notin V}\sigma_{x}\tau_{y}.$ (5.2)

Note that the only values of $\tau$ that matter are on the exterior boundary
of $G$, i.e. on the vertices that are connected by an edge of $\mathbb{Z}^{d}$
to a vertex in $V$.

With this definition, we obtain the following important property of the Ising
model, called the spatial Markov property: for every finite subgraph $G$ of
$\mathbb{Z}^{d}$, every $W\subset V$, and every configuration
$\tau\in\\{-1,1\\}^{\mathbb{Z}^{d}}$, if $H$ denotes the graph induced by the
set $W$,

$\mu_{G,\beta,h}[\,\cdot\,|\sigma_{x}=\tau_{x},\forall x\notin
W]=\mu_{H,\beta,h}^{\tau}[\,\cdot\,].$ (5.3)

In words, when conditioning the Ising model on $G$ to coincide with a given
configuration outside $W$, one gets the measure in $H$ with the corresponding
boundary condition.

This property offers a natural consistency relation between measures
$\mu_{G,\beta,h}^{\tau}$ for varying $\tau$ and $G$. As a byproduct, one is
naturally led to postulate that any reasonable infinite-volume version of
Ising measures should satisfy the same consistency relation. One therefore
ends up with the following notion: a measure $\mu$ on
$(\\{-1,1\\}^{\mathbb{Z}^{d}},\mathfrak{F}_{\mathbb{Z}^{d}})$ is called a
Gibbs measure of the Ising model with parameters $\beta$ and $h$ if it
satisfies the Dobrushin-Lanford-Ruelle (DLR) property: for every finite
$V\subset\mathbb{Z}^{d}$ and $\tau\in\\{-1,1\\}^{\mathbb{Z}^{d}}$,

$\mu[\,\cdot\,|\mathfrak{F}_{\mathbb{Z}^{d}\setminus
V}]=\mu_{G,\beta,h}^{\tau}[\,\cdot\,]\text{ on }E_{\tau}\text{ $\mu$-almost
surely},$ (5.4)

where

  * •

$G$ is the graph induced by the vertex-set $V$;

  * •

$E_{\tau}$ is the event that $\sigma$ and $\tau$ agree on the exterior
boundary of $G$;

  * •

$\mathfrak{F}_{\mathbb{Z}^{d}\setminus V}$ is the $\sigma$-algebra generated
by the random variables $(\sigma_{x}:x\notin V)$.

The notion of Gibbs measure is not restricted to the Ising model (see [52] for
a book on the subject), but the classification of such Gibbs measures has been
the object of intense study in the specific case of the Ising model, with a
very successful outcome.

The first question that one may ask is the existence of Gibbs measures. At
least three such measures can be defined in a fairly straightforward way. By
taking limits as $G$ tends to $\mathbb{Z}^{d}$ of the measures
$\mu_{G,\beta,h}$, $\mu_{G,\beta,h}^{+}$, and $\mu_{G,\beta,h}^{-}$ (where $+$
and $-$ refer, with a slight abuse of notation, to $\tau$ equal to all $+1$ or
all $-1$), one ends up with three (possibly equal) Gibbs measures
$\mu_{\beta,h}$, $\mu_{\beta,h}^{+}$ and $\mu_{\beta,h}^{-}$. More generally,
one may construct measures by taking all possible sub-sequential limits of
measures of the form $\mu_{G,\beta,h}^{\tau}$, where one may even consider
$\tau$ as a random variable.

In general, the set of possible Gibbs measures on $\mathbb{Z}^{d}$ is a non-
empty simplex whose extremal measures are called extremal states. One can
therefore try to classify such extremal Gibbs measures.

Some cases are quite simple to treat: for $h\neq 0$ or $h=0$ and
$\beta<\beta_{c}$ the simplex is reduced to a singleton, i.e. there exists a
unique Gibbs measure. When $h=0$ and $\beta=\beta_{c}$, it was recently proved
that this is also the case [5]. On the contrary, when $h=0$ and
$\beta>\beta_{c}$, things are more interesting. It was realised very early on
that there may be more extremal states than the two obvious
$\mu_{\beta,0}^{+}$ and $\mu_{\beta,0}^{-}$, but examples that were found did
not exhibit translation invariance. The most important such specimen was
provided by Russian mathematical physicist Roland Dobrushin [31], who
explained that in three dimensions, the measure
$\mu_{\beta,0}^{\mathrm{dobr}}$ obtained by taking the limit of measures
$\mu_{[-n,n]^{3},\beta,0}^{\tau}$, where $\tau$ is all plus on the upper half-
space, and all minus on the lower half-space, was not translation invariant in
the vertical direction at high values of $\beta$. The existence of these
Dobrushin states is related to a very deep and still mysterious (at least on a
mathematical level) phenomenon in 3D statistical physics often referred to as
the roughening phase transition.

Leaving non-translation invariant measures aside, many efforts were made to
prove that every translation invariant Gibbs state is a convex combination of
$\mu_{\beta,0}^{+}$ and $\mu_{\beta,0}^{-}$. The first result in this
direction proved a stronger statement that draws a direct link with the
previous paragraph. In two dimensions, Aizenman [1] and Higuchi [64] proved in
the eighties that every Gibbs state, not only translation invariant ones, is a
mixture of $\mu_{\beta,0}^{+}$ and $\mu_{\beta,0}^{-}$. In particular,
$\mu_{\beta,0}=\tfrac{1}{2}\mu_{\beta,0}^{+}+\tfrac{1}{2}\mu_{\beta,0}^{-}$.
In higher dimensions, it took twenty more years to obtain the result for every
translation invariant Gibbs measure. We refer to the historical proof of
Bodineau [18] and to the recent generalisation of Raoufi [93].

Figure 2: On the left. Simulations at three different temperatures
($\beta<\beta_{c}$, $\beta=\beta_{c}$, and $\beta>\beta_{c}$) of the Ising
model with plus boundary conditions on the top and minus boundary conditions
on the bottom. Pluses are in gray and minuses in black. Credit: S. Smirnov. On
the right. An example of a bubble of minuses in an environment of pluses at
$\beta>\beta_{c}$. Credit: Y. Velenik.

###  5.3 Phase coexistence and Wulff shape

The classification of Gibbs states naturally raises the question of the
coexistence of different so-called phases. When $h=0$ and $\beta>\beta_{c}$,
$\mu_{\beta,0}^{+}$ and $\mu_{\beta,0}^{-}$ are not equal: they correspond to
two extremal states, sometimes referred to as the plus and minus phases. Now,
what happens when one tries to “mix” the two states? For instance, how does it
look if one asks that part of the space is in one state, and the other part is
in the other one?

In 2D, an interface is created between the two phases (see Figure 2 for
simulations at different temperatures). While it is not obvious to define such
an object in general, let us consider the simple example of the Ising model on
a finite box $[-n,n]^{2}$ of the triangular lattice with plus spins on the
part of the boundary above the $x$-axis, and minus spins on the rest of the
boundary. In this case, one can draw a unique interface going from $(-n,0)$ to
$(n,0)$ winding between pluses and minuses. It was understood heuristically
early on that above criticality this interface should have the same
fluctuations as Brownian motions, but it took decades to turn this intuition
into a rigorous proof, first in the large $\beta$ regime and then in the whole
$\beta>\beta_{c}$ regime; see [56] and references therein. The techniques
involved also enabled mathematicians to understand precise asymptotics of
spin-spin correlations in the non-critical regimes. The theory, known under
the coined name of Ornstein-Zernike theory, is now an area of intense research
and spans over a large variety of statistical physics models. We refer to [23]
for details on the Ising case.

When conditioning on the neighbourhood of the origin to be in a plus phase
inside a minus phase, one ends up with a “bubble” (see Figure 2 on the right)
converging when taking larger and larger volume to the so-called Wulff shape.
In 2D, this bubble was analysed in detail, see the book [32] and the article
[67]. In 3D, the story is even more complex. The boundary between the plus and
minus phases is a kind of two-dimensional surface. The study of this object is
quite intricate, and the fluctuations of the surface are still widely open. We
refer to [17, 24, 15] and references therein.

##  6 The seventies and eighties: the Ising model and field theory

###  6.1 Constructive quantum field theory

Quantum field theories with local interaction are central in most subfields of
theoretical physics, from high energy to condensed matter physics. The
mathematical challenge of the proper formulation of this concept led to the
program of constructive quantum field theory (CQFT). A path towards that goal
was charted through the proposal to define quantum fields satisfying Wightman
axioms [105] using the Osterwalder-Schrader theorem [89], in which case the
construction boils down to producing relevant random distributions defined
over the corresponding Euclidean space that meet a number of conditions such
as suitable analyticity, permutation symmetry, Euclidean covariance, and
reflection-positivity.

Finding these _Euclidean fields_ boils down to constructing probability
averages over random distributions $\Phi(x)$ of the form

$\langle F(\Phi)\rangle\approx\frac{1}{\mathrm{norm}}\int
F(\Phi)\exp[-H(\Phi)]\prod_{x\in\mathbb{R}^{d}}d\Phi(x),$ (6.1)

where

  * •

$F(\Phi)$ is a smeared average of the form
$T_{f}(\Phi):=\int_{\mathbb{R}^{d}}f(x)\Phi(x)dx$ associated with continuous
functions of compact support $f$.

  * •

$H(\Phi)$ is a Hamiltonian
$H(\Phi):\approx(\Phi,A\Phi)+\int_{\mathbb{R}^{d}}P(\Phi(x))\,dx$ with
$(\Phi,A\Phi)$ a positive definite and reflection-positive (see Section 6.2)
quadratic form, and $P(\Phi(x))$ an even polynomial whose terms of order
$\Phi(x)^{2k}$ are interpreted heuristically as representing $k$-particle
interactions.

By linearity, the expectation values of products of such variables can be
rewritten as

$\Big{\langle}\prod_{j=1}^{n}T_{f_{j}}(\Phi)\Big{\rangle}:=\int_{(\mathbb{R}^{d})^{n}}S_{n}(x_{1},\dots,x_{n})\,\prod_{j=1}^{n}f(x_{j})\,dx_{1}\dots
dx_{n},$ (6.2)

where the $S_{n}(x_{1},\dots,x_{n})$ are the Schwinger functions of the
corresponding Euclidean field theory which can be interpreted heuristically as
pointwise correlations $\langle\prod_{j=1}^{n}\Phi(x_{j})\rangle.$
Interpreting (6.1) raises a number of problems of varying difficulty.

The simplest example of Euclidean fields are the reflection-positive (see
Section 6.2 again) Gaussian fields, for which $H(\Phi)$ contains only
quadratic terms. Gaussian fields are alternatively characterised by $2n$-point
Schwinger functions satisfying Wick’s law:

$S_{2n}(x_{1},\dots,x_{2n})=\sum_{\pi\text{
pairings}}\prod_{j=1}^{n}S_{2}(x_{\pi(2j-1)},x_{\pi(2j)}).$ (6.3)

The field theoretical interpretation of (6.3) is the absence of interaction.
Due to that and to their algebraically simple structure, such fields are
referred to as trivial.

The next level of difficulty is to add the next lowest order even term, i.e.
$\lambda\Phi^{4}$ for $\lambda>0$. Note that, if it exists at all, the
corresponding field is a random distribution so making sense of this fourth
power is not straightforward. The heuristic RG approach to the problem by
Wilson [106] indicates that in low enough dimensions, the problem could be
tackled through a renormalisation procedure. The CQFT program has successfully
yielded non-trivial scalar field theories over $\mathbb{R}^{2}$ [55] and
$\mathbb{R}^{3}$ [54, 44], and is still a lively field of mathematical
physics.

A natural example aimed at constructing a $\Phi^{4}_{d}$ functional integral
is to regularise it with a pair of cutoffs: at a short distance
(_ultraviolet_) scale and a large distance (_infrared_) scale. A lattice
version of that is the restriction of $\Phi(\cdot)$ to the vertices of a
finite graph $\Lambda_{R}^{(a)}:=(a\mathbb{Z})^{d}\cap[-R,R]^{d}$, where $a$
and $R$ play respectively the roles of the ultraviolet and infrared cutoffs.
For the corresponding finite collection of variables
$(\phi_{x}:x\in\Lambda_{R}^{(a)})$, the Hamiltonian is then interpreted in
terms of a Riemann-sum style discrete analog of the integral expressions,
leading to the following statistical-mechanics Gibbs equilibrium state average

$\langle
F(\phi)\rangle=\frac{1}{\mathrm{norm}}\int_{\mathbb{R}^{\Lambda_{R}^{(a)}}}F(\phi)\exp{[-H(\phi)]}\prod_{x\in\Lambda_{R}^{(a)}}d\rho(\phi_{x}),$
(6.4)

with a Hamiltonian $H(\phi)$ and an a-priori measure $\rho$ of the form

$H(\phi)=-\sum_{\\{x,y\\}\subset
E(\Lambda_{R}^{(a)})}\,\phi_{x}\phi_{y}\,,\qquad
d\rho(\phi_{x})=e^{-\lambda\phi_{x}^{4}-b\phi^{2}_{x}}d\phi_{x}\,,$ (6.5)

where $d\phi_{x}$ is the Lebesgue measure on $\mathbb{R}$. This is called the
$\phi^{4}$ lattice model.

The cutoffs are removed through the limit $R\nearrow\infty$ followed by
$a\searrow 0$. Parameters may be added to adjust in the process the spin-spin
correlations $\langle\phi_{x_{1}}\dots\phi_{x_{n}}\rangle$ in such a way that
they stabilise to the Schwinger functions $S_{n}(x_{1},\dots,x_{n})$ in the
continuum limit scale.

The Ising model can be thought of as a limiting case of a $\phi^{4}$ lattice
model as it is obtained by letting $\lambda=b/2$ tend to infinity (the limit
of the measures $\rho$ then forces the spins $\phi_{x}$ to take the values
$\pm 1$). Actually, the discrete approximations of the $\phi^{4}$ functional
integral and the Gibbs states of an Ising model are always connected. This
relation is based on a construction which was initiated by Griffiths to obtain
the Lee-Yang theorem for the $\phi^{4}$ lattice models, and was advanced
further by Griffiths and Simon [60]. A probability measure on $\rho(d\phi)$ on
$\mathbb{R}$ is said to belong to the Griffiths-Simon class if the expectation
values with respect to $\rho$ can be represented as an Ising model on the
complete graph with well-chosen coupling constants, or as a limit of such
models (satisfying some mild tail conditions). The $\phi^{4}$ lattice model
belongs to the Griffiths-Simon class. For this reason, most techniques that
are at our disposal for the Ising model apply to the Griffiths-Simon class.
This makes the Ising model an object of major interest when working on CQFT.
The developments of the model have therefore been deeply connected to CQFT in
the eighties, and we now discuss some examples of such interactions.

###  6.2 Reflection positivity

The notion of reflection positivity was introduced in Quantum Field Theory in
the work of Osterwalder-Schrader [89], and we refer to [16] for a review.
While reflection positivity did not emerge initially as a property of the
Ising model, the model remains one of the most natural instances of a
reflection positive model, and some of the most striking applications of
reflection positivity are indeed dealing with the Ising model.

Consider the Ising model on a $d$-dimensional torus
$\mathbb{T}_{L}:=(\mathbb{Z}/L\mathbb{Z})^{d}$ with $L$ even and split equally
the torus into two pieces $\mathbb{T}_{L}^{+}$ and $\mathbb{T}_{L}^{-}$ using
hyperplanes (the two pieces are isomorphic to
$[0,L/2]\times(\mathbb{Z}/L\mathbb{Z})^{d-1}$) and consider a reflection
$\vartheta$ with respect to one of these hyperplanes mapping
$\mathbb{T}_{L}^{+}$ to $\mathbb{T}_{L}^{-}$. We say that
$\langle\cdot\rangle$ is reflection positive if for all
$f,g:\mathbb{T}_{L}^{+}\rightarrow\mathbb{R}$,

$\langle f\vartheta g\rangle=\langle g\vartheta
f\rangle\qquad\text{and}\qquad\langle f\vartheta f\rangle\geq 0,$ (6.6)

or, in other words, that $f,g\mapsto\langle f\vartheta g\rangle$ is a positive
semi-definite symmetric bilinear form. The archetypical examples of reflection
positive measures are the Ising n.n.f. measures
$\langle\cdot\rangle_{\mathbb{T}_{L},\beta,0}$, but many other examples exist,
including some Ising models with long-range interactions.

Reflection positivity has two important implications, namely gaussian
domination leading to the infrared bound, and the chessboard estimate. By lack
of space, and since most of the applications of reflection positivity to the
specific example of the Ising model rely on the infrared bound, let us focus
on it and gaussian domination.

Gaussian domination is a statement linking the partition function of the Ising
model with magnetic field to the partition function of the model without it.
Formally, it states that for every function
$\mathbf{h}:V\rightarrow\mathbb{R}$, $Z_{L}(\mathbf{h})\leq Z_{L}(0)$, where

$Z_{L}(\mathbf{h}):=\sum_{\sigma\in\\{-1,1\\}^{V}}\exp\Big{[}-\beta\sum_{\\{x,y\\}\in
E(\mathbb{T}_{L})}(\sigma_{x}-\sigma_{y}+\mathbf{h}_{x}-\mathbf{h}_{y})^{2}\Big{]}.$
(6.7)

Gaussian domination can be proved via reflection positivity through the two
hyperplanes mentioned above to show that for each $\mathbf{h}$, a symmetric
version of $\mathbf{h}$ with respect to a hyperplane has a larger value of
$Z_{L}(\cdot)$. Gaussian domination immediately implies a Fourier version of
the infrared bound by using a second-order expansion of $Z_{L}(\mathbf{h})$
near 0: for $d>2$ and every $(a_{x})\in\mathbb{C}^{\mathbb{T}_{L}}$ summing to
zero,

$\sum_{x,y\in\mathbb{T}_{L}}a_{x}\overline{a}_{y}\langle\sigma_{x}\sigma_{y}\rangle_{\mathbb{T}_{L},\beta,0}\leq\tfrac{2}{\beta}\sum_{x,y\in\mathbb{T}_{L}}a_{x}\overline{a}_{y}G(x,y),$
(6.8)

where $G(x,y)$ is the Green function of the simple random walk on
$\mathbb{Z}^{d}$.

In the specific case of the Ising model, the Messager-Miracle-Solé inequality
enables to turn this Fourier estimate into a pointwise estimate on the two-
point function: there exist $C,C^{\prime}>0$ such that for every $\beta>0$ and
every $x,y\in\mathbb{Z}^{d}$,

$\langle\sigma_{x}\sigma_{y}\rangle_{\beta,0}-m^{*}(\beta)^{2}\leq\tfrac{C}{\beta}G(x,y)\leq\tfrac{C^{\prime}}{\|x-y\|_{2}^{d-2}}.$
(6.9)

This is particularly interesting when $\beta$ approaches $\beta_{c}$ from
below, as it implies that the spin-spin correlations decay algebraically fast
at $\beta_{c}$, with an exponent at least $d-2$.

###  6.3 The random current revolution

The context of CQFT was also at the origin of one of the most important
revolutions in our understanding of the Ising model that we will describe in
Section 6.4. The technique, called the random current representation, was
introduced by Aizenman (inspired by combinatorial identities from [59]). It
became one of the most powerful and robust tools available to mathematicians
to study the Ising model. We describe it now (see [35] for a review).

The whole story starts with the observation that the component
$\exp[\beta\sigma_{x}\sigma_{y}]$ of the Hamiltonian term attached to each
edge can be rewritten using Taylor’s expansion to get

$Z(G,\beta,0)=\sum_{\sigma\in\\{-1,1\\}^{V}}\prod_{\\{x,y\\}\in
E}\sum_{{\mathbf{n}}_{\\{x,y\\}}=0}^{\infty}\frac{(\beta\sigma_{x}\sigma_{y})^{{\mathbf{n}}_{\\{x,y\\}}}}{{\mathbf{n}}_{\\{x,y\\}}!}=\sum_{{\mathbf{n}}\in\mathbb{Z}_{+}^{E}}w_{\beta}({\mathbf{n}})\sum_{\sigma\in\\{-1,1\\}^{V}}\prod_{x\in
V}\sigma_{x}^{\Delta_{x}(\mathbf{n})},$ (6.10)

where

$w_{\beta}(\mathbf{n}):=\prod_{\\{x,y\\}\in
E}\frac{\beta^{\mathbf{n}_{\\{x,y\\}}}}{\mathbf{n}_{\\{x,y\\}}!}\qquad\text{
and }\qquad\Delta_{x}(\mathbf{n}):=\sum_{y\in V:\\{x,y\\}\in
E}\mathbf{n}_{\\{x,y\\}}.$ (6.11)

Now, the involutions on spin configurations switching the spins at a vertex
immediately imply that the sum on $\sigma$ on the right-hand side is either
equal to $2^{|V|}$ if $\Delta_{x}(\mathbf{n})$ is even for all $x\in V$, or 0
otherwise (this seems like a very elementary observation, but it bears at the
heart of it the $+/-$ symmetry of the space of possible spins).

Call a function from $E$ to $\mathbb{Z}_{+}$ a current. A source of the
current will be a vertex $x$ with $\Delta_{x}(\mathbf{n})$ odd. The set of
sources will be denoted by $\partial\mathbf{n}$. The previous discussion and
the notation lead to the identity

$Z(G,\beta,0)=2^{|V|}\sum_{\partial\mathbf{n}=\emptyset}w_{\beta}(\mathbf{n}),$
(6.12)

where from now on we omit to specify that we consider currents when using the
notation $\mathbf{n}$.

A current $\mathbf{n}$ with $\partial\mathbf{n}=A$ can be interpreted as the
occupation time of a collection of paths pairing vertices of $A$ and loops –
or equivalently the number of times the collection of paths and loops goes
through an edge. The decomposition into loops and paths is not unique,
nonetheless it remains interesting to interpret currents in terms of them.

Proceeding in a similar fashion with the numerator of the spin-spin
correlations, we get that

$\langle\sigma_{A}\rangle_{G,\beta,0}=\frac{\sum_{\partial\mathbf{n}=A}w_{\beta}(\mathbf{n})}{\sum_{\partial\mathbf{n}=\emptyset}w_{\beta}(\mathbf{n})}.$
(6.13)

In words, one may write spin-spin correlations in terms of weighted sums of
currents with specific source constraints $\partial\mathbf{n}=A$ and
$\partial\mathbf{n}=\emptyset$. Note that the source constraint is not the
same for the numerator and denominator.

Frame 4: the high-temperature expansion and $\beta_{c}>0$

The high-temperature expansion of the Ising model, due to van der Waerden
[102], can be neatly defined here as the set of edges with an odd current (it
can also be obtained by a direct expansion using that
$\exp[\beta\sigma_{x}\sigma_{y}]=\cosh(\beta)+\sinh(\beta)\sigma_{x}\sigma_{y}$).
One ends up with another expression of the partition function in terms of even
subgraphs

$Z(G,\beta,0)=\cosh(\beta)^{|E|}\sum_{F\in\mathrm{Even}(G)}\tanh(\beta)^{|F|},$
(6.14)

which resembles the low-temperature expansion, except that it is on $G$
instead of $G^{*}$ and that it is valid for arbitrary graphs and not only
planar ones. In particular, one may easily deduce the Kramers-Wannier duality
between the low and high temperature expansions at temperatures $\beta$ and
$\beta^{*}$ satisfying $\tanh(\beta)=e^{-2\beta^{*}}$ in the case of the
square lattice.

One application of currents (or alternatively high-temperature expansion) is
obtained by considering a mapping from currents with
$\partial\mathbf{n}=\\{x,y\\}$ to currents with $\partial\mathbf{n}=\emptyset$
setting the current on a path from $x$ to $y$ of odd current (such a path
necessarily exists) to $0$. This many-to-one mapping (one has to keep track of
the path and the value of the current on it to reconstruct the preimage)
increases drastically the weight of the current as soon as $\beta\ll 1$, which
shows that the spin-spin correlations
$\langle\sigma_{x}\sigma_{y}\rangle_{G,\beta,0}$ are decaying exponentially
fast in this regime. This implies in particular that $\beta_{c}>0$.

A key observation of Aizenman is that the so-called switching lemma, see Frame
5, pertaining to combinatorial properties of the random current model, could
be used to reinterpret spin-spin correlations as well as many other properties
in terms of probabilities involving multiple independent currents. This lemma
completely changed the point of view on currents, as it transforms them from a
combinatorial type object into a probabilistic one. In particular, intuitions
coming from probabilistic models such as random walks and percolation was
later used to prove new theorems on the Ising model; see Sections 6.4, 6.6,
and 7.1.

Frame 5: the switching lemma for random currents

Write ${\mathbf{n}}\in\mathcal{F}_{A}$ if there exists
$\mathbf{k}\leq\mathbf{n}$ with $\partial\mathbf{k}=A$. Note that if
$A=\\{x,y\\}$, this is equivalent to the existence of a path from $x$ to $y$
which is made of edges with a positive current. Recall that $A\Delta B$
denotes the symmetric difference of the sets $A$ and $B$. With this notation,
the switching lemma [59] states that for every
$F:\mathbb{Z}_{+}^{E}\rightarrow\mathbb{R}$ and every two sets of vertices
$A,B\subset V$,

$\sum_{\begin{subarray}{c}\partial\mathbf{n}_{1}=A\\\
\partial\mathbf{n}_{2}=B\end{subarray}}w(\mathbf{n}_{1})w(\mathbf{n}_{2})F(\mathbf{n}_{1}+\mathbf{n}_{2})=\sum_{\begin{subarray}{c}\partial\mathbf{n}_{1}=A\Delta
B\\\
\partial\mathbf{n}_{2}=\emptyset\end{subarray}}w(\mathbf{n}_{1})w(\mathbf{n}_{2})F(\mathbf{n}_{1}+\mathbf{n}_{2})\mathbb{I}(\mathbf{n}_{1}+\mathbf{n}_{2}\in\mathcal{F}_{B}).$
(6.15)

The name of the lemma is fairly self-explanatory, as it consists, when
considering sums of two currents, of a recipe to switch the sources from the
second one to the first one. The proof is a very entertaining combinatorial
problem that is left to the reader.

A direct application (to illustrate the strength of the lemma) is the case
$A=B$, which gives immediately that

$\langle\sigma_{A}\rangle_{G,\beta,0}^{2}=\mathbb{P}^{\emptyset}_{G}\otimes\mathbb{P}^{\emptyset}_{G}[\mathbf{n}_{1}+\mathbf{n}_{2}\in\mathcal{F}_{A}],$
(6.16)

where $\mathbb{P}^{B}_{G}$ is the measure on currents $\mathbf{n}$ on $G$ with
$\partial\mathbf{n}=B$ attributing to each such $\mathbf{n}$ a probability
that is proportional to $w(\mathbf{n})$, and $\otimes$ denotes the product for
probability measures. In words, one may interpret the square of spin-spin
correlations $\langle\sigma_{A}\rangle_{G,\beta,0}$ as the probability, for
the sum of two independent random currents, of pairing the elements of $A$ by
paths of positive current. One may also try as an exercise to recover
Griffiths’ inequalities from the switching lemma.

###  6.4 Triviality in dimension $d>4$

In 1982, Michael Aizenman and Juerg Fröhlich [2, 49] independently proved that
the scaling limit of the Ising model is trivial in dimension five and more in
the following sense. Consider discrete smeared averages defined by

$T_{f,L}(\sigma):=\frac{1}{\sqrt{\Sigma_{L}}}\sum_{x\in\mathbb{Z}^{d}}f(x/L)\,\sigma_{x}\,,$
(6.17)

where $f$ ranges over compactly supported continuous functions, and
$\Sigma_{L}:=\big{\langle}\big{(}\sum_{x\in\Lambda_{L}}\sigma_{x}\big{)}^{2}\big{\rangle}$
denotes the variance of the sum of spins over the box of size $L$. The theorem
states that when $d>4$, these smeared averages $T_{f,L}(\sigma)$ are
approximately Gaussian of variance $\langle
T_{f,L}(\sigma)^{2}\rangle_{\beta}$ in the sense that there exists an explicit
constant $C_{f}>0$ such that for every $\beta\leq\beta_{c}$, every
$L\leq\xi(\beta)$, and every $z>0$,

$\Big{|}\,\big{\langle}\exp[z\,T_{f,L}(\sigma)-\tfrac{z^{2}}{2}\langle
T_{f,L}(\sigma)^{2}\rangle_{\beta}]\big{\rangle}_{\beta}\,-\,1\,\Big{|}\leavevmode\nobreak\
\leq\leavevmode\nobreak\ \frac{C_{f}z^{4}}{L^{d-4}}.$ (6.18)

In words, the previous statement claims that the characteristic function of
$T_{f,L}(\sigma)$ is close to the one of a Gaussian random variables.

As a direct consequence of this result, one obtains that any well-defined
scaling limit of the Ising model, and in fact more generally of the $\phi^{4}$
lattice model, is inevitably Gaussian. The result marked a brutal stop in the
CQFT program outlined in Section 6.1 as the proofs suggested, while not
proving, that the model should also be trivial in four dimensions.

As mentioned above, one of the most striking applications of the random
current representation is related to CQFT. Indeed, Aizenman’s proof of this
theorem relies on a beautiful parallel between random walks and the paths
joining sources in currents. We do not resist discussing this link below. But
before doing so, let us mention that the approach of Fröhlich in [49], based
on the Brydges-Fröhlich-Spencer (BFS) walk representation of spin-spin
correlations [21], is deeply connected to the random current as well. The
walks in the BFS representation play the roles of the paths between sources in
the random current. The advantage of this alternative approach is that it
works for more general models, at the cost of losing the switching lemma and
its benefits.

Let us focus on the four-point function and define the corresponding Ursell
function given, for $x_{1},\dots,x_{4}\in\mathbb{Z}^{d}$, by

$U_{4}^{\beta}(x_{1},\dots,x_{4}):=\langle\sigma_{x_{1}}\cdots\sigma_{x_{4}}\rangle_{\beta}-\sum_{\pi\text{
pairing}}\prod_{i=1}^{2}\langle\sigma_{x_{\pi(2i-1)}}\sigma_{x_{\pi(2i)}}\rangle_{\beta}.$
(6.19)

A simple exercise involving the switching lemma shows that

$U_{4}^{\beta}(x_{1},\dots,x_{4})=-2\langle\sigma_{x_{1}}\sigma_{x_{2}}\rangle\langle\sigma_{x_{3}}\sigma_{x_{4}}\rangle\mathbb{P}^{\\{x_{1},x_{2}\\}}\otimes\mathbb{P}^{\\{x_{3},x_{4}\\}}[x_{1},\dots,x_{4}\text{
all connected in }\mathbf{n}_{1}+\mathbf{n}_{2}],$ (6.20)

where connected in $\mathbf{n}_{1}+\mathbf{n}_{2}$ means being connected by a
path of edges with $\mathbf{n}_{1}+\mathbf{n}_{2}$ not equal to zero. If one
remembers that one can think of a current with sources $x_{1}$ and $x_{2}$ as
a path connecting the two vertices together with a collection of loops, one
can reinterpret the right-hand side of the previous identity at the light of
so-called random walks (a random walker traces his way through the vertices of
a graph by picking its next steps at random among neighbours of where it
currently stands – this Markov process is one of the most fundamental objects
of probability theory). It is a classical result that two random walks
connecting two pairs of points that are at a mutual distance of order $L$
intersect with a probability bounded away from 0 as $L$ tends to infinity in
dimensions $d<4$, and tending to zero in dimension $d\geq 4$.

At this stage, it is totally unclear why the paths linking the points $x_{1}$
and $x_{2}$ in $\mathbf{n}_{1}$, and $x_{3}$ and $x_{4}$ in $\mathbf{n}_{2}$,
would behave as random walks. It is also unclear what would be the impact of
the additional loops. Still, it is tempting to think that if an analogy with
random walks was valid, then it would single out dimensions $d\geq 4$ as being
dimensions for which $U_{4}^{\beta}$ becomes much smaller than products of
two-point correlations or, in other words, for which Wick’s law would become
asymptotically valid, thus hinting at triviality.

When the dimension is strictly larger than 4, the story for random walks
becomes even simpler, as the expected number of intersections is also tending
to zero with $L$. Using the infrared bound to estimate the spin-spin
correlations of the Ising model, one may go around the difficulty of proving a
random walk type behaviour for currents to show that the intersection
probability is tending to 0.

Making the argument work for currents in dimension 4 is more subtle because,
contrarily to larger dimensions, the expected number of intersections does not
tend to 0 when $L$ tends to infinity. Hence, in order to prove that the
intersection probability goes to 0, one inevitably has to go deeper in the
understanding of the analogy between currents and random walks.

###  6.5 Rigorous renormalisation group in 4D Ising

The triviality of the Ising model in dimension $d>4$ naturally raises the
question of its triviality in dimension $d=4$, which is not only the pertinent
physical dimension for CQFT, but also for the so-called $4-\varepsilon$
expansions providing information on dimension 3. In the eighties, Wilson’s
renormalisation group method was already in every physicists’ toolbox, yet the
challenges to overcome to cast the general theory in a mathematical framework
seemed out of reach. Interestingly, a very relevant case became an important
exception.

Consider the lattice version of the $\phi^{4}$ model discussed in Section 6.1.
The case $b=\lambda=0$ corresponds to a Gaussian field known under the name of
discrete Gaussian Free Field (GFF), which enjoys a number of striking
features. One of them is that the model converges, when rescaling the lattice,
to the continuum GFF. In a series of impressive papers [51, 43, 62],
mathematical physicists proved in the eighties that, when starting from a
weakly coupled $\phi^{4}$ lattice model (meaning that $\lambda$ is small), one
may apply a multi-scale analysis to prove convergence of the model to the
continuum GFF.

Several methods were used at the time, but let us mention that the method of
Gawedski and Kupiainen [51] can be thought of as a rigorous version of
Kadanoff block-spin renormalisation procedure. It consists of writing the
model in terms of averages of spins over large blocks of size $L^{k}$, and to
average them out scale by scale. At leading order, each step of the procedure
boils down to modifying the parameters of the model. Of course, the reality is
much more complicated than the first order analysis suggests, and the
renormalisation scheme is quite complex.

An alternative to this block-spin renormalisation was later developed by
Bauerschmidt, Brydges, and Slade [11] in order to obtain refined results, as
well as to treat more general models. In these alternative approaches, the
block-spin analysis is replaced by the following strategy: one thinks of
quantities in the $\phi^{4}$ lattice model as being expressed in terms of the
discrete GFF itself. In order to control the asymptotic behaviour of such
quantities, one decomposes the covariance of the discrete GFF into a sum of
finite-range covariances that one integrates out one by one. At each step a
change of the parameters of the system is required to keep things converging
towards a limit. Doing so enables the authors to focus their attention on how
the parameters evolve under this procedure. This evolution can be thought of
as the renormalisation map in the renormalisation group.

The level of sophistication of these techniques is quite astonishing, and the
precision of the results outstanding. As one may guess, this comes at a price.
At the bottom of both strategies lies the fact that the original $\phi^{4}$
lattice model is in the “vicinity of a model”, the Gaussian Free Field, that
enjoys a number of nice properties. As a result, the technique is (as for
today) perturbative in nature, which is somehow its main limitation. We will
see another instance of such a renormalisation scheme, this time near another
fixed point, when discussing the 2D Ising model.

###  6.6 Forty years later: the random current strikes back

While renormalisation techniques provided impressive rigorous results in
dimension 4, they remained as we mentioned perturbative, meaning that they
required that the lattice $\phi^{4}$ model one starts from has a small
$\phi^{4}$ term. Yet, if one would like to construct a non-trivial 4D quantum
field theory, one would definitely try to start with a strongly coupled
$\phi^{4}$ lattice model (meaning with a $\phi^{4}$ terms which is not a
priori small), for instance working with the Ising model which in some sense
can be thought of as the model with the strongest possible coupling, thus
excluding existing renormalisation group techniques.

This asks for another approach, and this is probably why one had to wait for
forty years to finally obtain a proof of the triviality of the 4D Ising and
$\phi^{4}$ lattice models, which states [4] that there exists $c>0$ such that
for the n.n.f. $\phi^{4}$ lattice model on $\mathbb{Z}^{4}$ with parameters
$b,\lambda$, and a compactly supported continuous function $f$, there exists
$C_{f}>0$ such that for every $\beta\leq\beta_{c}=\beta_{c}(b,\lambda)$, every
$L\leq\xi(\beta)$, and every $z>0$,

$\Big{|}\,\big{\langle}\exp[z\,T_{f,L}(\varphi)-\tfrac{z^{2}}{2}\langle
T_{f,L}(\varphi)^{2}\rangle_{\beta}]\big{\rangle}_{\beta}\,-\,1\,\Big{|}\leavevmode\nobreak\
\leq\leavevmode\nobreak\ \frac{C_{f}z^{4}}{(\log L)^{c}}.$ (6.21)

The strategy of the proof uses a more delicate probabilistic perspective on
the random current than in [2], still keeping in mind the interpretation in
terms of random walks of the paths joining the sources of the current. Indeed,
it can be proved that two random walkers in four dimensions going from points
to points that are all at a mutual distance of order $L$ intersect with
probability of order $(\log L)^{-c}$ for some universal constant $c>0$. The
reason is that while the expected number of intersections is of order 1, the
number of intersections, when such intersections exist, is with high
probability quite large in $L$ (and is growing with $L$). The core of the
paper is to apply a similar argument to the paths in the random current. Of
course, challenges emerge when trying to handle the highly non-Markovian paths
obtained by considering the paths joining the sources in currents.
Nevertheless, guided by the random walk intuition, one can build a multi-scale
analysis to prove that conditioned on intersecting, random currents intersect
a large number of times, and ultimately deduce from this the triviality
result.

##  7 The last fifty years: Ising model and percolation

Percolation theory gathers under its umbrella a variety of random graphs
systems. A configuration on $G=(V,E)$ is an element $\omega=(\omega_{e}:e\in
E)\in\\{0,1\\}^{E}$ which is interpreted as a subgraph with vertex-set $V$ and
edge-set $\\{e\in E:\omega_{e}=1\\}$. Then, different percolation models can
be defined by considering different measures on $\\{0,1\\}^{E}$. Historically,
the original model, called Bernoulli percolation, is defined in such a way
that the $\omega_{e}$ are independent Bernoulli random variables. It was
introduced to understand the behaviour of liquid in a porous medium.
Nevertheless, the theory of non-Bernoulli models has been found to be related
to a variety of other models of statistical physics explaining various
physical phenomena.

As often, the Ising model has played an essential role in the development of
percolation theory, and conversely certain advances in percolation theory have
been fundamental to our understanding of the Ising model. Sometimes, the link
between the two models is simply an analogy between their behaviours, but
sometimes the connection is much more direct. For instance, spin-spin
correlations can be rewritten in terms of a percolation model, in which case
we speak of the percolation model as being a graphical representation of the
Ising model. We now propose to discuss some examples of these links between
the Ising model and percolation.

###  7.1 Percolation interpretation of random currents

We have seen one example of a graphical representation in Frame 5 where the
squares of spin-spin correlations get rephrased as connectivity properties of
the sum of two currents. One may easily define a percolation model out of the
pair of currents above by saying that for an edge $\\{x,y\\}$,
$\omega_{\\{x,y\\}}=1$ if $(\mathbf{n}_{1}+\mathbf{n}_{2})_{\\{x,y\\}}>0$.
Then, the square of the spin-spin correlations between two points becomes the
probability, for this percolation model, that $x$ and $y$ are connected in
$\omega$.

The best illustration of how intuition from percolation or the Ising model can
drive developments on the other model is provided by an important result on
the Ising model in the regime $\beta<\beta_{c}$. This result from 1987, due to
Aizenman, Barsky and Fernandez [3] (see [39] for an alternative argument),
states that correlations of the n.n.f. Ising model decay exponentially fast as
soon as $\beta<\beta_{c}$ in the sense that for each such $\beta$, there
exists $\tau>0$ such that for every $x,y\in\mathbb{Z}^{d}$,

$\langle\sigma_{x}\sigma_{y}\rangle_{\beta,0}\leq\exp(-\tau\|x-y\|).$ (7.1)

We say that the phase transition is sharp: there is no intermediate phase
$(\beta_{\mathrm{exp}},\beta_{c})$ in the Ising model in which spin-spin
correlations would decay polynomially. Let us mention that a similar
exponential decay was obtained recently for truncated correlations
$\langle\sigma_{x}\sigma_{y}\rangle_{\beta,0}-m^{*}(\beta)^{2}$ when
$\beta>\beta_{c}$, see [36].

This theorem is of fundamental importance for the following reason.
Perturbative results, which are combinatorial in nature, are valid under the
assumption that certain quantities decay exponentially fast, and in fact with
a rate of decay which is sufficiently large. While this hypothesis is
important to apply the techniques, it happens to be of little relevance from a
physical point of view. In fact, one expects that most of the phenomenology
remains unchanged as long as spin-spin correlations decay exponentially fast.
As a consequence, (7.1) can be thought of as a bottleneck in the understanding
of the phase $\beta<\beta_{c}$: as soon as it is obtained, a number of
important results can be derived from it. As an example, the results on
fluctuations of interfaces and Ornstein-Zernike estimates were proved to hold
in the whole regime $\beta<\beta_{c}$. The result also provides meaning to the
correlation length $\xi(\beta)$ mentioned in Section 4.1, as it proves that it
is finite as soon as $\beta<\beta_{c}$.

Let us now comment on the proof. The argument relies on a fruitful idea
consisting in deriving differential inequalities between thermodynamical
quantities of the Ising model. The archetypical example of such differential
inequalities are given, for the problem at hand, by (recall that the
magnetisation $m=m(\beta,h)$ is a function of $\beta$ and $h$)

$m\leq\tanh(\beta h)\tfrac{\partial}{\partial(\beta
h)}m+m^{2}(\beta\tfrac{\partial}{\partial\beta}m+m)\qquad\text{ and }\qquad
m\tfrac{\partial}{\partial\beta}m\geq c.$ (7.2)

The interesting feature here is that similar differential inequalities appear
when studying Bernoulli percolation. In fact, a number of results were
obtained in parallel during the eighties, where each result for Ising had its
pendant for Bernoulli percolation, and vice versa. As an example, critical
exponents for $d>4$ were obtained by Aizenman and Fernandez [7] using
differential inequalities that can be adapted to Bernoulli percolation. These
techniques are useful to transform qualitative results (e.g. a quantity tends
to 0) to quantitative ones (e.g. exponentially fast). We do not resist
mentioning one of them: for $h=0$ and $\beta<\beta_{c}$,

$\Big{(}1-\frac{B}{\chi}\Big{)}\frac{2d\chi^{2}}{1+B}\leq\tfrac{\partial}{\partial\beta}\chi\leq
2d\chi^{2},$ (7.3)

where $\chi(\beta):=\sum_{x}\langle\sigma_{0}\sigma_{x}\rangle_{\beta,0}$ is
the susceptibility, and $B(\beta)$ is the Bubble diagram and is given by

$B(\beta):=\sum_{x\in\mathbb{Z}^{d}}\langle\sigma_{0}\sigma_{x}\rangle_{\beta,0}^{2}.$
(7.4)

Since the Infrared Bound implies that $B(\beta)$ remains bounded uniformly in
$\beta<\beta_{c}$ as soon as $d>4$, $\chi(\beta)$ must blow up like
$1/|\beta-\beta_{c}|$ as $\beta$ approaches $\beta_{c}$ from below.

Another striking instance of how fruitful the connection between percolation
models and the Ising model was for the development of both models is the
following continuity result of the phase transition of the 3D Ising model, due
to [5], stating that the n.n.f. Ising model satisfies $m^{*}(\beta_{c})=0$ for
every $d\geq 3$.

The argument relies on percolation methods applied to the double random
current representation of an argument of Burton and Keane proving the
uniqueness of the infinite connected component of percolation. The whole
argument can be improved and extended to study all translation-invariant Gibbs
measures, obtaining the classification result already mentioned in Section
5.2.

###  7.2 Fortuin-Kasteleyn percolation

Another (and in fact older) example of a graphical representation is provided
by a special case of the Fortuin-Kasteleyn (FK) percolation. In this model,
introduced in [47], the measure $\phi_{G,p,q}$ is given, for $G=(V,E)$ finite
and $\omega\in\\{0,1\\}^{E}$, by

$\phi_{G,p,q}[\\{\omega\\}]:=\frac{1}{Z(G,p,q)}p^{|\omega|}(1-p)^{|E|-|\omega|}q^{k(\omega)},$
(7.5)

where $p\in[0,1]$ and $q>0$ are the parameters of the model, called
respectively the edge-weight and the cluster-weight, $|\omega|:=\sum_{e\in
E}\omega_{e}$ is interpreted as the number of edges in $\omega$, and
$k(\omega)$ is the number of connected components of $\omega$.

When $q=1$, one ends up with the classical Bernoulli percolation model in
which the $\omega_{e}$ are independent. When $q\neq 1$, the state of edges is
no longer independent and one ends up with a dependent percolation model whose
study is central in modern probability theory. From now on, we focus on the
case $q=2$, which we call the FK Ising model. We confine our discussion to two
features of this percolation model, namely its link to the Ising model, and
the FKG inequality.

Let us start with the former, which provides a recipe to obtain the Ising
model configuration out of FK Ising; see Figure 3. Consider a random variable
$\omega\in\\{0,1\\}^{E}$ with the law of FK Ising with parameter $p\in[0,1]$
and construct $\sigma\in\\{-1,1\\}^{V}$ by

  * •

choosing for every connected component $\mathcal{C}$ of $\omega$ a spin
$\sigma_{\mathcal{C}}$ uniformly between $-1$ and $+1$, and independently of
the other connected components.

  * •

defining $\sigma_{x}=\sigma_{\mathcal{C}}$ for every $\mathcal{C}$ and every
$x\in\mathcal{C}$.

Then, $\sigma$ has the law of the Ising model on $G$ with parameter
$\beta=\tfrac{1}{2}\log[1/(1-p)]$ and $h=0$. This coupling, due to Fortuin and
Kasteleyn and often referred to as the Edwards-Sokal coupling due to the paper
[40], enables to express correlation functions of the Ising model in terms of
FK Ising. For instance, by decomposing on the events that $x$ is connected to
$y$ or not in $\omega$, one easily gets that

$\langle\sigma_{x}\sigma_{y}\rangle_{G,\beta,0}=\phi_{G,1-e^{-2\beta},2}[x\text{
connected to }y\text{ in }\omega].$ (7.6)

Similarly,
$\langle\sigma_{A}\rangle_{G,\beta,0}=\phi_{G,1-e^{-2\beta},2}[\mathcal{F}_{A}]$,
where $\mathcal{F}_{A}$ is the event that each connected component of $\omega$
contains an even number (possibly equal to 0) of vertices in $A$. Another
interesting feature of this coupling is that it is at the basis of so-called
cluster algorithms due to Swendsen and Wang, who used it to speed up the
Glauber dynamics and the simulation of the Ising model, in particular near the
critical point.

Figure 3: The Edwards-Sokal coupling, with on the left a picture of the FK
Ising configuration (bold edges are those with $\omega_{e}=1$), in the middle,
spins are attached to each cluster (one example in black and others in grey),
and on the right, the spins without the FK Ising configuration.

The interest of FK Ising and more generally FK percolation models with $q\geq
1$ is that they enjoy some nice monotonicity properties (dependent percolation
models satisfying these properties have been an object of intense study in the
past ten years). Let us mention two such properties. The Fortuin-Kasteleyn-
Ginibre (FKG) inequality states that for every increasing functions
$f,g:\\{0,1\\}^{E}\rightarrow\mathbb{R}$,

$\phi_{G,p,q}[fg]\geq\phi_{G,p,q}[f]\phi_{G,p,q}[g].$ (7.7)

This inequality is often used for indicator functions of increasing events
(i.e. events for which the indicator function is an increasing function), in
which case the inequality states that increasing events are positively
correlated. Another manifestation of the monotonicity properties is the
monotonicity in $p$: for every increasing function
$f:\\{0,1\\}^{E}\rightarrow\mathbb{R}$ and $p^{\prime}\geq p$,

$\phi_{G,p^{\prime},q}[f]\geq\phi_{G,p,q}[f].$ (7.8)

These monotonicity properties are particularly useful. The second one applied
to FK Ising and the indicator function of $\mathcal{F}_{A}$ implies that
$\langle\sigma_{A}\rangle_{G,\beta,0}$ is increasing in $\beta$, and the first
one applied to indicator functions of $\mathcal{F}_{A}$ and $\mathcal{F}_{B}$
implies the second Griffiths inequality
$\langle\sigma_{A}\sigma_{B}\rangle_{G,\beta,0}\geq\langle\sigma_{A}\rangle_{G,\beta,0}\langle\sigma_{B}\rangle_{G,\beta,0}$.

###  7.3 The broader impact of the Ising model on dependent percolation models

In the first fifty years that followed its introduction, the theory of
percolation was much more advanced for Bernoulli percolation than for other
dependent percolation models. The past ten years have seen tremendous progress
in bridging the gap between our understanding of the Bernoulli case and the
others. The interplay between dependent percolation models and the Ising model
has been fundamental for these developments.

We already saw that the Ising model is related to FK Ising and a percolation
model created out of random currents. It does not come as a surprise that one
of the first dependent percolation models to see significant progress in its
understanding was the FK Ising. Of course, the Edwards-Sokal coupling enables
to transfer immediately certain known facts about the Ising model to its
percolation representation (for instance, the critical point of the FK Ising
on $\mathbb{Z}^{2}$ is $1-e^{-2\beta_{c}}=\sqrt{2}/(1+\sqrt{2})$ thanks to
Onsager’s result). Also, the model enjoys some specific features that make its
direct analysis simpler than for other dependent percolation models.

For all these reasons, the FK Ising became the entrance gate to a new realm of
results on dependent percolation models. A perfect illustration of this is
provided by the study of crossing probabilities for planar dependent
percolation models. Let us provide slightly more detail.

One important feature of critical dependent percolation models in two
dimensions is that they satisfy the box-crossing property (BCP), and its
connected notion the Russo-Seymour-Welsh theory (RSW). More precisely, if for
a rectangle $R$, the event $\mathrm{Cross}(R)$ corresponds to the existence of
a path in $\omega$ between the left and right sides of $R$, the properties
(BCP) and (RSW) for a percolation model on $\mathbb{Z}^{2}$ with measure
$\mathbb{P}$ are the following:

  * •

(BCP) for all $\rho>0$, there exists $c>0$ such that for every $n\geq 1$,

$c\leq\mathbb{P}[\mathrm{Cross}([0,\rho
n]\times[0,n])|\omega_{|[-n,(\rho+1)n]\times[-n,2n]^{c}}]\leq 1-c\text{ almost
surely}.$ (7.9)

  * •

(RSW) for all $\rho>0$, there exists $C>0$ such that for every $n\geq 1$,

$\mathbb{P}[\mathrm{Cross}([0,\rho
n]\times[0,n])]\geq\mathbb{P}[\mathrm{Cross}([0,n]\times[0,\rho n])]^{C}.$
(7.10)

These two properties have been the driving force of the progress in our
understanding of the 2D dependent percolation models. The FK-Ising model
played an essential role in these developments, as it was the first dependent
percolation model for which (BCP) could be proved [37]. This development
triggered a whole new direction of research that led to substantial progress
in our understanding of (BCP) and (RSW) for various percolation models.

##  8 Over the last ten years: conformal invariance of the Ising model

###  8.1 What is conformal invariance?

As mentioned before, Kadanoff used his block-spin renormalisation to predict
that the large scale properties of the critical Ising model were invariant
under scaling. The same argument also leads to postulate translation and
rotation invariance. In 1970, Polyakov [92] suggested a much stronger
invariance of the model. Since we saw that it is natural to associate a QFT
with the large scale properties of the critical Ising model, and since this
QFT is a local field, these properties should be invariant under any map which
is locally a composition of translation, rotation and homothety. As a
corollary one predicts full conformal invariance, i.e. invariance under all
one-to-one holomorphic maps. This prediction was turned into a classification
of possible conformal field theories (CFT) in 2D in seminal papers by Belavin,
Polyakov and Zamolodchikov [12] that generated an explosion of activity,
allowing non-rigorous explanations of many critical phenomena.

From a mathematical perspective, the notion of conformal invariance of a model
is not straightforward to define. A number of interpretations of the limit of
large scale properties – called the scaling limit – can be taken, and we
mention a few now.

For clarity of the exposition, we focus on the critical Ising model on
$\mathbb{Z}^{d}$ and its rescaled versions $a\mathbb{Z}^{d}$ for $a>0$. We
drop the subscript referring to $\beta$ and $h$ as they are fixed to be equal
to $\beta_{c}$ and $0$ respectively. Consider a simply connected domain
$\Omega\subsetneq\mathbb{R}^{d}$.

(Spins) The most natural approach is to consider the spin-spin correlations
defined for every $a>0$ and $x_{1},\dots,x_{n}\in\Omega$ by

$S_{\Omega}^{(a)}(x_{1},\dots,x_{n}):=\langle\sigma_{[x_{1}]_{a}}\dots\sigma_{[x_{n}]_{a}}\rangle_{a\mathbb{Z}^{d}\cap\Omega},$
(8.1)

where $[x]_{a}$ is the vertex of $a\mathbb{Z}^{d}\cap\Omega$ closest to $x$.
These Schwinger functions already appeared as the key players in CQFT. One is
then interested in the limit as $a$ tends to 0 of these properly renormalised
quantities. If the limit exists, we call it $S_{\Omega}(x_{1},\dots,x_{n})$.

(Energies) Another object of interest is the energy-energy correlations. For
$a>0$ and $x_{1},\dots,x_{n}\in\Omega$, one considers at the quantities

$T_{\Omega}^{(a)}(x_{1},\dots,x_{n}):=\langle\varepsilon_{(x_{1})_{a}}\dots\varepsilon_{(x_{n})_{a}}\rangle_{a\mathbb{Z}^{d}\cap\Omega},$
(8.2)

where
$\varepsilon_{\\{u,v\\}}:=\sigma_{u}\sigma_{v}-\langle\sigma_{u}\sigma_{v}\rangle_{a\mathbb{Z}^{d}}$
and $(x)_{a}$ is the edge closest to $x$. The quantity $\varepsilon_{x}$ is
called the energy. One is again interested in the limit
$T_{\Omega}(x_{1},\dots,x_{n})$ as $a$ tends to 0 of these properly rescaled
quantities.

(Geometry of interfaces) In two dimensions, another direction was proposed in
the nineties. It consists in considering the low-temperature representation,
i.e. the interfaces between plus and minus spins. In a domain $\Omega$, it
creates a family of non-intersecting loops together with arcs from boundary to
boundary. Let $\mathfrak{C}_{\Omega}$ be the set of such collections of loops
and arcs. The set $\mathfrak{C}_{\Omega}$ can be turned into a metric space by
attaching a distance $d_{\Omega}$ which, heuristically, states that two
configurations are close to each other when the large loops and arcs are close
to each other. Let us call $\mathcal{C}^{(a)}_{\Omega}$ the random variable
obtained by considering the low-temperature expansion of a critical Ising
model configuration in $a\mathbb{Z}^{d}\cap\Omega$. Here, we are interested in
the limit of $\mathcal{C}^{(a)}_{\Omega}$ as a random object.

Now, what do we mean by conformal invariance? Roughly speaking, we mean that
certain quantities of the model are conformally covariant/invariant. With the
definitions above, it would for instance mean that there exists a way of
renormalising the $S_{\Omega}^{(a)}(x_{1},\dots,x_{n})$ and
$T_{\Omega}^{(a)}(x_{1},\dots,x_{n})$ in such a way that they converge to
quantities $S_{\Omega}(x_{1},\dots,x_{n})$ and $T_{\Omega}(x_{1},\dots,x_{n})$
that satisfy that there exist $\bm{\Delta}_{\sigma},\bm{\Delta}_{\varepsilon}$
such that for every conformal (i.e. holomorphic and one-to-one) map
$f:\Omega\rightarrow f(\Omega)$, we have

$\displaystyle S_{f(\Omega)}(f(x_{1}),\dots,f(x_{n}))$
$\displaystyle=|f^{\prime}(x_{1})|^{-\bm{\Delta}_{\sigma}}\cdots|f^{\prime}(x_{n})|^{-\bm{\Delta}_{\sigma}}\,S_{\Omega}(x_{1},\dots,x_{n}),$
(8.3) $\displaystyle T_{f(\Omega)}(f(x_{1}),\dots,f(x_{n}))$
$\displaystyle=|f^{\prime}(x_{1})|^{-\bm{\Delta}_{\varepsilon}}\cdots|f^{\prime}(x_{n})|^{-\bm{\Delta}_{\varepsilon}}\,T_{\Omega}(x_{1},\dots,x_{n}).$
(8.4)

For the geometry of interfaces, the situation is even simpler as one means
that the family of loops and arcs $\mathcal{C}_{\Omega}^{(a)}$ converges to a
limit $\mathcal{C}_{\Omega}$ as $a$ tends to $0$ and that this limit satisfies
that $\mathcal{C}_{f(\Omega)}$ and $f(\mathcal{C}_{\Omega})$ have the same law
for every conformal map $f:\Omega\rightarrow f(\Omega)$.

###  8.2 Conformal invariance of the 2D Ising model

Around fifteen years ago, Smirnov [100] and Chelkak and Smirnov [28] obtained
a major breakthrough towards proving conformal invariance of 2D Ising model.
This fundamental proof, that we discuss below, opened the way to a very deep
understanding of the scaling limit of the model.

A few years later, Chelkak-Izyurov-Hongler [27] proved conformal covariance of
the spin-spin correlations (with $\bm{\Delta}_{\sigma}=1/8$). It was later
proved in [22] that the quantities $S_{\Omega}(x_{1},\dots,x_{n})$ are the
Schwinger functions of a random distribution, that can be understood as the
spin-field that physicists sometimes refer to. In the same spirit, conformal
covariance of the energy-energy correlations was proved in [65] (with
$\bm{\Delta}_{\varepsilon}=1$). In this case, one may prove that the
correlations are not the Schwinger functions of a random distribution. Turning
to interfaces, the following result was the culmination of the theory: the
arcs in $\mathcal{C}_{\Omega}$ are given by the so-called free arc ensemble of
parameter 3 and the loops by conformal loop ensembles of parameter $3$ in the
simply connected domains obtained as the complements of the arcs (see [13,
14]). In particular, the scaling limit is conformally invariant. This body of
work uses the ideas from [100, 28] together with the theory of the Schramm-
Loewner evolution and its consequences.

As mentioned above, an important breakthrough came from the works [100, 28]
where conformal covariance of so-called fermionic observables
$f_{\Omega}^{(a)}$ is proved. Those observables are linear combinations of
order-disorder operators (see Frame 6) considered by Kadanoff and Ceva in
[75], see also [26] for several connections to other classical objects.

Frame 6: Fermionic observable

Consider a simply connected domain $\Omega\subset\mathbb{C}$ and for $a>0$,
let $\mathbf{\Omega}$ be the largest connected component of
$a\mathbb{Z}^{2}\cap\Omega$. Consider $n$ vertices $x_{1},\dots,x_{n}$ of
$\mathbf{\Omega}$, and $n$ faces $f_{1},\dots,f_{n}$ of $\mathbf{\Omega}$ such
that $f_{i}$ is bordered by $x_{i}$ for every $1\leq i\leq n$. Choose $n$
disjoint cuts $\ell_{1},\dots,\ell_{n}$, i.e. families of dual edges
$(e_{i}^{*}(j))$ forming self-avoiding paths in the dual from the unbounded
face to the center of $f_{i}$. Define the disorder operator $\mu_{\ell}$ for a
cut $\ell$ as the observable that effectively switches the coupling constants
of the edges $e_{i}(j)$ associated with the $e_{i}^{*}(j)$ in the cut (it can
be written as a product of terms of the form
$\exp[-2\beta\sigma_{x}\sigma_{y}]$ over edges appearing in the family of
edges $\\{e_{i}(j):i,j\\}$). Then, the order-disorder correlations are given
by the formula

$F_{\Omega}^{(a)}(x_{1},f_{1},\dots,x_{n},f_{n}):=\langle\sigma_{x_{1}}\mu_{\ell_{1}}\dots\sigma_{x_{n}}\mu_{\ell_{n}}\rangle_{\mathbf{\Omega}}.$
(8.5)

Let us mention that these quantities can be expressed in terms of correlations
of Grassmann variables in the Schultz-Mattis-Lieb representation [96].

Smirnov introduced a fermionic observable $f_{\Omega}^{(a)}$ defined at
centers of edges $\\{x,y\\}$ of $\mathbf{\Omega}$ that can be written as a
linear combination (with complex coefficients) of the $F_{\Omega}^{(a)}$ with
$x_{1}$ equal to $x$ or $y$, and $f_{1}$ to one of the two faces bordered by
$\\{x,y\\}$. The details of the definition are unimportant here and the take-
home message is that Chelkak and Smirnov proved that the limit (as $a$ tends
to 0) of these fermionic observables is conformally covariant.

The conformal covariance of the fermionic observable should be understood as
the first brick among the conformal covariance results of spin-spin, energy-
energy correlations, and even of the conformal invariance of interfaces. Let
us mention that these results require substantial additional ideas compared to
[100, 28]. In fact, conformal covariance/invariance of virtually all
quantities one may be interested in the 2D Ising model can be recovered today.

The proof of the theorem relies on the observation that $f_{\Omega}^{(a)}$ is
the solution of a discrete version of a Riemann-Hilbert boundary value
problem. More precisely, the function can be proved, via combinatorial
arguments involving the van der Waerden high-temperature expansion, to be
preholomorphic (see Frame 7), and to satisfy certain boundary conditions.
These special features are connected to the integrability of the model. From
general principles on preholomorphic functions, the limit as $a$ tends to 0 of
these objects must be the holomorphic solution of a continuum Riemann-Hilbert
boundary value problem, which can be computed and proved to be conformally
covariant. Such reasoning has been used in several existing proofs of
conformal invariance, for instance for dimers or Bernoulli site percolation on
the triangular lattice. It has created an explosion of results in the field as
many quantities can be proved to converge using a similar strategy.

Frame 7: Preholomorphic observables

The notion of preholomorphic function on a planar graph $G$ appeared
implicitly in the work of Kirchhoff on electrical networks [78]. It was
explicitly linked to holomorphicity in the work of Isaacs [68, 69], in which
the author proposed to discretise the Cauchy-Riemann equation to get to the
definition (on the square lattice)

$F(NW)-F(SE)=i[F(NE)-F(SW)],$ (8.6)

where $NW$, $SW$, $SE$, and $NE$ are the four corners found in
counterclockwise order around each face, when starting from the top left
vertex.

The properties of preholomorphic functions have been the object of a renewed
interest with the emergence of the question of conformal invariance in
connections to boundary value problems. Indeed, general theorems stating that
preholomorphic functions satisfying certain boundary value conditions converge
when taking finer and finer meshsize to holomorphic solutions of the continuum
version of the boundary value problem took a central place in the theory.

In the case of the Ising model, the complexity of the boundary value problem
(involving a condition on the argument of the fermionic observable) pushed
Smirnov to introduce a stronger notion of preholomorphicity, called
$s$-holomorphicity, which is also satisfied by fermionic observables. The
advantage of this notion is that it enables one to define the imaginary part
of the primitive of the square of the observable, which roughly speaking
becomes the discrete solution of a Dirichlet boundary value problem, a much
more tractable problem for which convergence (when $a$ tends to 0) can be
proved very elegantly.

###  8.3 Towards universality of the 2D Ising model

As mentioned in Section 4.2.2, the large-scale properties of the critical
Ising model should not depend on the precise properties of the underlying
graph. With the tremendous successes that have been achieved over the years in
the case of the Ising model on $\mathbb{Z}^{2}$ and more generally on planar
graphs, it is natural to test the validity of the universality hypothesis in
this context. Several advances have been made in this direction in the last
fifteen years.

The first impressive progress can be found in the work of Chelkak and Smirnov
themselves [28]. They observed that the preholomorphicity argument leading to
conformal invariance can be articulated naturally in the setting of so-called
isoradial graphs. An isoradial graph is an embedding of a graph $G$ in the
plane such that every face of the graph is inscribed in a circle of radius 1.
In this context, one may define special coupling constants $J_{x,y}$ depending
on the graph in such a way that $\beta_{c}=1$ and that the fermionic
observable is naturally preholomorphic on this graph. Then, the strategy of
Chelkak and Smirnov on the square lattice applies to isoradial graphs with the
same conclusions. Note that this result can be understood as a universality
result on the graph (isoradial graphs are a fairly large family of planar
graphs, even though not fully general), but that the choice of $J_{x,y}$ is
determined by the embedded graph itself. Moreover, a striking feature of this
theorem is that no transitivity or quasi-transitivity is required for this to
work.

In recent developments, Chelkak generalised the conformal invariance result to
a wider class of Ising models, namely those defined on planar locally-finite
doubly periodic weighted graphs $(G,J)$, i.e. weighted graphs which are
invariant under the action of some lattice
$\Lambda\approx\mathbb{Z}\oplus\mathbb{Z}$ (in such case $G/\Lambda$ is a
finite graph embedded in the torus). For such models, Chelkak proved in [25]
that there exists an embedding in the plane, called an $s$-embedding, with the
property that the scaling limit of the critical model defined on this
embedding is conformally invariant.

This result is a strong indication of universality for planar graphs. Now what
happens beyond planar graphs? The universality conjecture asserts that the
scaling limit depends on the large scale geometry of the graph (for instance a
planar Euclidean geometry). In particular, one may consider the graph obtained
with the vertex-set $\mathbb{Z}^{2}$ and edge-set given by pairs of vertices
at a distance at most $R$ of each other. This model, called the finite-range
model on $\mathbb{Z}^{2}$, should have a behaviour that is similar to the
nearest-neighbour case as it is “almost planar”. The additional difficulty is
that non-planarity immediately breaks the integrability of the system. The
universality of such Ising models has been investigated in two different
directions.

First, one may consider finite-range models that are perturbations of the
nearest-neighbour integrable case, meaning that non-nearest neighbour
interactions are very weak, i.e. that $J_{x,y}$ is small when
$1<\|x-y\|_{2}\leq R$. Using the Schultz-Mattis-Lieb Grassmann representation
[96] of the nearest neighbour case, one may express the partition function and
more generally the energy-energy and spin-spin correlations in terms of
Grassmann variables, and therefore at the end in terms of the nearest-
neighbour model. Using an elaborate multi-scale analysis and studying the
renormalisation of parameters induced by this multi-scale analysis, Giuliani-
Greenblatt-Mastropietro derived in [53] the large-scale behaviour of energy-
energy correlations in the full plane. While the previously mentioned
renormalisation schemes in dimension 4 were enabled by the fact that the model
is a small perturbation of the discrete GFF (which is a gaussian process), the
two-dimensional case relies on a similar connection, this time to the n.n.f.
Ising model on $\mathbb{Z}^{2}$ (which has a Grassmannian structure). As a
consequence, the strategy suffers from the same limitations as the 4D case in
the sense that it is restricted to small perturbations of the n.n.f. Ising
model on $\mathbb{Z}^{2}$.

A totally different approach explaining the emergence of planarity in finite
range Ising models was proposed in [6] based on the random current
representation. The underlying idea relies on the fact that thanks to the
switching lemma, intersection properties of random currents with sources are
related to the structure of $n$-point correlations in the model. Yet, the
intersection properties of long paths on the graph induced by $\mathbb{Z}^{2}$
and the edges between vertices at a distance $R$ of each other resemble the
ones that can be obtained for planar graphs. As an example of a possible
application, one can obtain that spin-spin correlations on the boundary of a
domain $\Omega$ have a Pfaffian structure, a result which is specific to the
universality class of the 2D Ising model. More precisely, for any collection
of points $x_{1}=(k_{1},0),\dots,x_{2n}=(k_{2n},0)$ satisfying
$k_{1}<k_{2}<\dots<k_{2n}$ on the boundary of the upper half-plane
$\mathbb{H}:=\mathbb{Z}\times\mathbb{Z}_{+}$,

$\langle\sigma_{x_{1}}\cdots\sigma_{x_{2n}}\rangle_{\mathbb{H},\beta_{c}}\ =\
\mathrm{Pfaff}_{n}\big{(}\big{[}\langle\sigma_{x_{i}}\sigma_{x_{j}}\rangle_{\mathbb{H},\beta_{c}}\big{]}_{1\leq
i<j\leq 2n}\big{)}\big{[}1+o(1)\big{]}\,,$ (8.7)

where $o(1)$ is a function of the points $x_{1},\dots,x_{2n}$ which tends to
zero for configuration sequences with $\min\\{|x_{i}-x_{j}|:1\leq i<j\leq
2n\\}$ tending to infinity.

This is, to the author’s knowledge, the first property witnessing the 2D Ising
universality class that can be obtained in a level of generality that is not
restricted to planar graphs and their perturbations. Also, the proof relies on
the key properties of the Ising model that one would like to use: the $\pm$
spin symmetry (entering the story through the use of the random current
representation) and the large scale planarity of the underlying graph (which
for finite range models on $\mathbb{Z}^{2}$ is the reason behind the “almost”
intersection properties of long paths). The trade-off is that full conformal
invariance of this family of models is still out of reach.

###  8.4 Conformal bootstrap in 3D Ising model

At this point, we already mentioned that the 1D Ising model was trivially
solved in the original paper of Ising [70], and that it took 20 more years to
achieve a solution of the 2D Ising model [88]. We also saw that the model in
dimensions 4 and higher is much simpler as its large-scale properties should
be Gaussian. This singles out 3D as the remaining challenging dimension. To
the best of our knowledge [101], it is not known whether the model is
integrable or not. This is particularly problematic as the third dimension is
probably the most relevant one physically (for instance the model should be in
the universality class of liquid–vapour systems, and totally anisotropic
magnets).

In recent years, a striking progress has been made on the physics side using
the so-called conformal bootstrap. A conformal field theory (CFT) is
characterised by the correlation functions $\langle--\rangle$ of an infinite
number of local operators $\mathcal{A}(x)$, which in the case of Ising should
be understood as the objects obtained by taking the limit of random variables
defined in terms of spins next to a given position of space. For example, the
scaling limit of spin and energy observables $\sigma_{x}$ and
$\varepsilon_{\\{x,y\\}}=\sigma_{x}\sigma_{y}-\langle\sigma_{x}\sigma_{y}\rangle$
give such local operators in the case of Ising, but one may think of more
complicated ones, such as the scaling limit of (products of) the gradient
$\sigma_{x+y}-\sigma_{x}$ of the spins.

Conformal invariance already forces huge constraints on the correlations of
operators in the theory. Oversimplifying slightly, for scalar local operators
there must exist exponents $\bm{\Delta}_{\mathcal{A}}$ and coefficients
$f_{\mathcal{A}\mathcal{B}\mathcal{C}}$ such that

$\displaystyle\langle\mathcal{A}(x)\mathcal{A}(y)\rangle$
$\displaystyle=\frac{1}{\|x-y\|_{2}^{\bm{\Delta}_{\mathcal{A}}}},$ (8.8)
$\displaystyle\langle\mathcal{A}(x)\mathcal{B}(y)\mathcal{C}(z)\rangle$
$\displaystyle=\frac{f_{\mathcal{A}\mathcal{B}\mathcal{C}}}{\|x-y\|_{2}^{\bm{\Delta}_{\mathcal{A}}+\bm{\Delta}_{\mathcal{B}}-\bm{\Delta}_{\mathcal{C}}}\|y-z\|_{2}^{\bm{\Delta}_{\mathcal{B}}+\bm{\Delta}_{\mathcal{C}}-\bm{\Delta}_{\mathcal{A}}}\|z-x\|_{2}^{\bm{\Delta}_{\mathcal{C}}+\bm{\Delta}_{\mathcal{A}}-\bm{\Delta}_{\mathcal{B}}}}$
(8.9)

(in (8.8), we adopted without loss of generality the normalization of
$\mathcal{A}$ that makes the constant in the numerator equal to 1). The
exponents and coefficients depend a priori on the CFT, but a striking feature
is that there exists a way, called the conformal block decomposition, to
express multi-point correlations of local operators in terms of three-point
functions by gluing points together using the so-called operator product
expansion. This theoretically shows that all the information in a CFT can be
encoded in terms of the $\Delta_{\mathcal{A}}$ and the
$f_{\mathcal{A}\mathcal{B}\mathcal{C}}$. Of course, determining these
coefficients is very difficult.

While in 2D this was done in the eighties, the analogous question remains
widely open in 3D. Nevertheless, one can proceed in a slightly different way
by asking which choices of these quantities can lead to a consistent CFT. This
approach, called the conformal bootstrap, was shown to be amazingly powerful
in 3D. The underlying idea is that one is facing an infinite family of
consistency relations coming from different ways of applying the conformal
block decomposition (which is not unique). For instance, one may start with
$\langle\mathcal{A}(x_{1})\mathcal{A}(x_{2})\mathcal{A}(x_{3})\mathcal{A}(x_{4})\rangle$
and proceed by gluing first $x_{1}$ and $x_{2}$ or, on the contrary, $x_{3}$
and $x_{4}$. This leads to two decompositions of the same object as a linear