# Continuous eigenfunctions of the transfer operator for the Dyson model

Anders Johansson, Anders Öberg, and Mark Pollicott

###### Abstract.

In this article we prove that there exists a continuous eigenfunction for the
transfer operator corresponding to potentials for the classical Dyson model in
the subcritical regime for which the parameter $\alpha$ is greater than $3/2$,
and we conjecture that this value is sharp.

This is a significant improvement on previous results where the existence of a
continuous eigenfunction of the transfer operator was only established for
general potentials satisfying summable variations, which would correspond to
the parameter range $\alpha>2$. Moreover, this complements as result by
Bissacot, Endo, van Enter and Le Ny [8], who showed that there is no
continuous eigenfunction at low temperatures.

Our approach to obtaining these new results involves a novel approach based on
random cluster models.

###### Key words and phrases:

Dyson model, transfer operator, eigenfunction, long-range Ising model

###### 2020 Mathematics Subject Classification:

Primary 37D35, 37A60, 82B20, 82B26, 82C27

##  1\. Introduction

It is well-known [30] that there exists a continuous and strictly positive
eigenfunction of a transfer operator defined on a symbolic shift space with a
finite number of symbols such that the potential has summable variations. Here
we prove the existence of a continuous eigenfunction for the important special
class of Dyson potentials in the subcritical regime up to at least the
critical line for Bernoulli percolation, i.e., in particular when the
potential does not satisfy the condition of summable variations.

For the Dyson model a continuous eigenfunction means that there is a
continuous Radon-Nikodym derivative between the two-sided equilibrium measure
(a translation invariant Gibbs measure) and the one-sided Gibbs measure.

A complementary paper to ours is the one by Bissacot, Endo, van Enter, and Le
Ny [8], where they show that there is no continuous eigenfunction in the
context of the Dyson model for low enough temperatures, although they phrase
their result in terms of the lack of the $g$-measure property. We will
describe the connection further below.

###  1.1. Preliminaries and results

Let $T$ be the left shift on the space $X=\mathscr{A}^{\mathbb{N}}$, where
$\mathscr{A}=\\{-1,+1\\}$. We denote by $C(X)$ the space of continuous
functions and by $\mathscr{M}(X)$ the space of probability measures on $X$.
Let $\phi:X\to\mathbb{R}$ be a continuous function which we refer to as the
_one-point potential_. The transfer operator $\mathscr{L}=\mathscr{L}_{\phi}$
is a positive operator $\mathscr{L}:C(X)\to C(X)$ defined by

(1) $\mathscr{L}f(x)=\sum_{y\in T^{-1}x}e^{\phi(y)}\,f(y).$

From the Ruelle-Perron-Frobenius theorem, we can deduce the existence of an
_eigenmeasure_ $\nu\in\mathscr{M}(X)$ to the dual operator
$\mathscr{L}^{*}:\mathscr{M}(X)\to\mathscr{M}(X)$ that satisfies
$\mathscr{L}^{*}\nu=\lambda\nu$, or equivalently

$\int\mathscr{L}f\,d\nu=\lambda\int f\,d\nu,f\in C(X),$

for the maximum eigenvalue $\lambda>0$.

In this paper, we want to establish the existence of a corresponding
continuous _eigenfunction_ $h(x)$, $0<h<\infty$, for the _long-range Dyson
model_ where

(2)
$\phi(x_{0},x_{1},\ldots)=x_{0}\cdot\beta\sum_{j=1}^{\infty}\frac{x_{j}}{j^{\alpha}},$

for parameters $\alpha>1$ and $\beta>0$.

An _equilibrium measure_ $\mu\in\mathscr{M}(X)$ corresponding to $\phi$ is a
minimiser of the free energy $P(\mu;\phi)=\mu(\phi)-H(\mu)$ among the set
$\mu\in{\mathscr{M}}_{T}(X)$ of all translation invariant probability
measures, that is measures such that $\mu=\mu\circ T^{-1}$. If there is an
eigenfunction $h$ with $\int h\,d\nu=1$ then the equilibrium measure $\mu$ can
be written as $\mu=h\nu$.

It is well-known ([18, 1]) that the long-range Dyson model with potential (2)
has a critical value of the parameter $\beta$, $\beta_{c}=\beta_{c}(\alpha)$,
such that we have a unique equilibrium state $\mu$ and a unique eigenmeasure
$\nu$ for $0<\beta<\beta_{c}$ and two ergodic states for $\beta_{c}<\beta$.
This $\beta_{c}(\alpha)$ is also the critical $\beta$ for _percolation_ in the
corresponding _random cluster_ model with $q=2$. There is also the critical
parameter $\beta_{c}^{1}(\alpha)$ for percolation in the Bernoulli random
graph model.

We can now present our main result.

######  Theorem 1.

For $3/2<\alpha\leq 2$ there exists a continuous eigenfunction of
$\mathscr{L}$ whenever $0<\beta\leq\beta_{*}$. Here,
$\beta_{*}=\beta_{*}(\alpha)$ is a critical value satisfying
$0<\beta^{1}_{c}\leq\beta_{*}\leq\beta_{c}$.

We can define $\beta_{*}$ as the supremum of $\beta\leq\beta_{c}$ for which
the corresponding random cluster model has a cluster size distribution with an
exponentially decreasing tail, see (28). Note that $\beta_{*}$ could well be
equal to $\beta_{c}$.

Let $\operatorname{var}_{n}f=\sup\\{|f(x)-f(y)|:x_{i}=y_{i},\,0\leq i\leq
n-1\\}$. If one assumes _summable variations_
$\sum_{n=1}^{\infty}\operatorname{var}_{n}(\phi)<\infty$ then the existence of
a continuous eigenfunction $h(x)$ for $\mathscr{L}$ follows from a classical
“cone-argument” used in, for example, Walters [30]. For the Dyson model,
summability of variations means that $\alpha>2$ and that the eigenfunction
$h(x)$ is Hölder continuous.

In Theorem 1, we have a continuous eigenfunction in a context when
$\alpha>3/2$ and $0<\beta<\beta_{*}$, and thus with the variations
$\operatorname{var}_{n}\phi=O(1/n^{\alpha-1})$, as $n\to\infty$. Note that the
condition $\alpha>3/2$ means that
$\sum_{n=1}^{\infty}(\operatorname{var}_{n}\phi)^{2}<\infty$.

If we have a strictly positive continuous eigenfunction $h$ of the transfer
operator, then we can recover an equilibrium measure $\mu$ ([29, 30]) as the
_Doeblin measure_ ([7]; $g$-measure in Keane’s terminology [27]) for the
Doeblin function ($g$-function) $g(x)$ defined by

(3) $g(x)=\frac{e^{\phi(x)}}{\lambda}\cdot\frac{h(x)}{h(Tx)},$

since $\sum_{y\in T^{-1}x}g(y)=1$ for all $x$. The corresponding transfer
operator $\mathscr{L}_{g}$ is a Markov transition operator and
$\mu\in\mathscr{M}(X)$ is a Doeblin measure for $g(x)$ if
$\mathscr{L}_{g}^{*}\mu=\mu$, i.e., it is the invariant distribution for a
stationary Markov process on the state space $X$. We refer to [23, 25, 16, 6,
21, 19]) for results on Doeblin measures.

In Bissacot, Endo, van Enter and Le Ny [8], they show that at low temperatures
for the Dyson model, there is no continuous Doeblin function $g(x)$ that
represents the Gibbs measure, and this gives a counterexample to the existence
of a continuous eigenfunction of the transfer operator for the Dyson model.
Fernandez and Maillard [17] proved that a Gibbs measure can be represented by
a Doeblin measure in the Dobrushin regime.

There were some extensions made to establish the existence of a measurable
eigenfunction bounded away from zero and infinity [10]. Walters proved some
regularity (but not continuity) for an eigenfunction ([30], Theorem 5.1) under
the so-called Bowen condition.

We conjecture that the condition $\alpha>3/2$ is sharp in the sense that we do
not have a continuous eigenfunction $h$, $0<h<\infty$, for the transfer
operator for the Dyson model when $\alpha<3/2$. We are grateful to Aernout van
Enter for pointing out that there is support in the mathematical physics
literature for such a conjecture, see Endo, van Enter, and Le Ny [14]. One
possible approach to prove sharpness might be to use the theory of Gallesco,
Gallo, and Takahashi [19] in combination with the observation that
$\alpha>3/2$ means that
$\sum_{n=1}^{\infty}(\operatorname{var}_{n}\phi)^{2}<\infty$.

###  1.2. The method of proof

Assume that $\nu$ is an eigenmeasure for $\mathscr{L}^{*}$ and that there is
some translation invariant $\mu$ which is absolutely continuous with respect
to $\nu$, i.e. such that $\mu=h\nu$. We use that the Radon-Nikodym derivative
$h(x)=d\mu/d\nu$ is an eigenfunction of the transfer operator $\mathscr{L}$.
This follows from the identities

$\displaystyle\int g\cdot h\,d\nu=\int(g\circ T)\cdot
h\,d\nu=\int\frac{1}{\lambda}\mathscr{L}(g\circ T\cdot h)\,d\nu=\int
g\cdot\left(\frac{1}{\lambda}\mathscr{L}h\right)\,d\nu,$

where the last equality follows from the definition of $\mathscr{L}$. This
hold for all $g\in C(X)$ if only if $\mathscr{L}h=\lambda h$ as elements of
$L^{1}(\nu)$.

Starting with the translation invariant $\mu$ and the eigenmeasure $\nu$, we
can try to construct the Radon-Nikodym derivative $h(x)$ as the limit of the
likelihood ratios $d\mu|_{\mathscr{F}_{n}}/d\nu|_{\mathscr{F}_{n}}$, i.e.

(4) $h(x)=\lim_{n\to\infty}h_{n}(x)\quad\text{where}\quad
h_{n}(x)=\frac{\mu[x_{0},\ldots x_{n}]}{\nu[x_{0},\ldots,x_{n}]}.$

The limit (4) is well-defined $\nu$-almost everywhere by the martingale
convergence theorem. If it exists in $L^{1}(\nu)$ then it equals $h=d\mu/d\nu$
and we can deduce the existence of an eigenfunction $h$ in $L^{1}(\nu)$. By
studying the associated _random cluster model_ , we can prove that the limit
in (4) is actually _continuous_ for the relevant Dyson models.

The proof goes roughly as follows. Let $\nu(x)$ be the eigenmeasure of
$\mathscr{L}_{\phi}^{*}$ and let $\mu(\bar{x})$,
$\bar{x}\in\mathscr{A}^{\mathbb{Z}}$, denote the natural extension of the
equilibrium measure $\mu$. We can represent $\nu(x)$ and $\mu(\bar{x})$ as
Gibbs measures for the Ising model corresponding to potentials $\Phi(x)$ and
$\Phi(\bar{x})$, respectively. Let $\Gamma(V)$ denote the space of graphs on
vertex set $V$. We lift $\nu(x)$ to a spin-cluster model $\nu(\gamma_{+},x)$
and $\mu(\bar{x})$ to a spin-cluster model $\mu(\gamma,\bar{x})$, where
$\gamma_{+}\in\Gamma(\mathbb{N})$ and $\gamma\in\Gamma(\mathbb{Z})$ are random
graphs. The bipartition $\mathbb{Z}=\mathbb{Z}_{<0}\uplus\mathbb{N}$
decomposes the graph $\gamma$ as
$\gamma=(\gamma_{-},{\gamma_{0}},\gamma_{+})$. It follows from the properties
of the random cluster model that we can factorise the distribution of $\gamma$
as

$\mu(\gamma_{-},{\gamma_{0}},\gamma_{+})=e^{R(\gamma)}\cdot\nu(\gamma_{-})\otimes\eta({\gamma_{0}})\otimes\nu(\gamma_{+}),$

where we prove in Lemma 2 that $e^{R}$ is an element of $L^{1}$. This
factorisation gives us an opening to compute the likelihood ratios in (4) and
to prove the continuity of the limit using the dominated convergence theorem.

Acknowledgements. We would like to thank Noam Berger and Evgeny Verbitskiy for
valuable discussions, and in particular Aernout van Enter for a very important
correspondence for this work. The second author wishes to thank the Knut and
Alice Wallenberg Foundation for financial support. The third author
acknowledges the ERC Grant 833802–Resonances.

##  2\. The proof of Theorem 1

###  2.1. Configurations, graphs and potentials

####  2.1.1. Configurations and potentials

By a configuration space we mean a set of form
$\mathscr{A}^{S}=\\{x:S\to\mathscr{A}\\}$, where $\mathscr{A}$ is a finite set
(the “alphabet”) and $S$ (the “sites”) is a countable set. We refer to
elements $x\in X$ as configurations and we give the space $X$ the usual
product topology and sigma-algebra $\mathscr{F}$. For $G\subset S$, we write
$x_{G}$ for the restriction $x|_{G}:G\to A\in\mathscr{A}^{G}$ of $x$ to $G$
and $\mathscr{F}_{G}$ for the $\sigma$-algebra generated by $x_{G}$.

We use ${\bar{F}}$ to denote the complement $S\setminus F$ of $F$. For all $F$
we can decompose $x\in X$ as $x=(x_{F},x_{\bar{F}})$. We express that $F$ is a
finite subset of $S$ by writing $F\Subset S$. Writing $F\uparrow S$ signifies
that $F\Subset S$ runs through an arbitrary increasing sequence of finite sets
with limit $S$. We denote $[x]_{F}$ the cylinder set $[x]_{F}=\\{y\mid
y_{F}=x_{F}\\}$ at $F$ containing $x$.

Implicitly, we assume all functions introduced are measurable. For a function
$f:X\to\mathbb{R}$ the _variation_ at $F\subset S$ is
$\operatorname{var}_{F}f=\sup\\{|f(x)-f(y)|:x_{F}=y_{F}\\}$. A function $f$ is
_local_ at $F$ if $f$ is $\mathscr{F}_{U}$-measurable for some finite subset
$U\Subset S$ and it is _continuous_ if $\lim_{F\uparrow
S}\operatorname{var}_{F}f=0$.

For two sequences $x,y\in X$, let $\Delta(x,y)\subset S$ denote the set where
they are different, i.e. $\Delta(x,y):=\\{i:x(i)\not=y(i)\\}$. With a
_potential_ $\phi(x)$ on $X$, we mean a formal limit $\phi(x)=\lim_{F\uparrow
S}\phi_{F}(x)$ of local functions, $\operatorname{var}_{F}\phi=0$, such that
the difference

$\Delta\phi(x,y):=\lim_{F\uparrow S}\phi_{F}(x)-\phi_{F}(y)$

is finite and well defined for any pair of configurations $x$ and $y$ such
that $\Delta(x,y)$ is a finite set. We can formally add potentials as long as
it is clear that the underlying limits give a well defined equality
$\Delta(\phi+\psi)(x,y)=\Delta\phi(x,y)+\Delta\psi(x,y)$ for the differences.
Note that, we can consider two potentials $\phi=\lim\phi_{F}$ and
$\psi=\lim\psi_{F}$ to be equal when it holds for all $F\Subset S$ that
$\phi_{F}(x)-\psi_{F}(x)$ does not depend on $x$.

####  2.1.2. Graphs

Let $V^{(2)}$ denote the set of unordered pairs $ij$ of elements $i,j\in V$,
i.e. $V^{(2)}$ is $V\times V$ modulo the equivalence relation with equivalence
classes $ij=\\{(i,j),(j,i)\\}$. We consider a _graph_ on the vertex set
$V=V(G)$ to be a map $G:E\to V$ from a set of edges $E=E(G)$ to $V^{(2)}$.
Edges $e$ of the form $G(e)=ii$, $i\in V$, are _loops_. We thus allow for
multiple edges and loops. The _complete graph_ on $V$, $K(V)$, is the
inclusion map of the non-loops in $V^{(2)}$. Given a bipartition
$V=V_{+}\uplus V_{-}$ of $V$ the _complete bipartite_ graph $K(V_{+},V_{-})$
is the inclusion of $V_{+}\times V_{-}$ in $V^{(2)}$.

A spanning subgraph $\gamma$ of $G$ is a restriction of $G$ to a subset
$E(\gamma)\subset E(G)$. We denote by $\Gamma(G)$ the space of spanning
subgraphs $\gamma$ of $G$ and we can represent $\gamma\in\Gamma(G)$ with a
configuration $\gamma\in\\{0,1\\}^{E(G)}$ or, equivalently, a subset
$\gamma\subset E(G)$. If $G$ is the complete graph $K(V)$ or the complete
bipartite graph $K(V_{+},V_{-})$, we write $\Gamma(V)$ or
$\Gamma(V_{+},V_{-})$ instead. We denote by $G[F]$ the subgraph _induced_ on
$F\subset V$, which means the restriction $G[F]:\,G^{-1}(F^{(2)})\to F^{(2)}$
of $G$ in both the domain and codomain. All spanning subgraphs
$\gamma\in\Gamma(G)$ we consider will have finite degrees, i.e.

$D(F,\gamma):=\sum_{i\in F}\sum_{j\in V}\gamma(ij)<\infty,$

for all $F\Subset V$.

Consider an equivalence relation ${\sim}$ on $V$ or, equivalently, a partition
$\tilde{V}=V/{\sim}$ into equivalence classes and identifying projection
$\pi:V\to\tilde{V}$. A _contraction_ $G/{\sim}$ of $G$ along ${\sim}$ is the
graph $G/{\sim}:E(G)\to{\tilde{V}}^{(2)}$ obtained from $G$ by identifying
pairs in the codomain along ${\sim}$. Note that $E(G)=E(G/{\sim})$ so
$\Gamma(G)$ and $\Gamma(G/{\sim})$ are equal as sets. If $F\Subset V$ then we
write $G^{F}$ for the local contraction at $F$ obtained from the equivalence
relation “$x,y\in F$ or $x=y$”, i.e. by contracting all vertices in $F$.

####  2.1.3. Clusters and decomposition along a cut $(V_{+},V_{-})$

For a graph $\gamma\in\Gamma(V)$, let $\mathscr{C}(\gamma)=\\{C\\}$ denote the
partition of $V$ into connected components (“clusters”). If $V$ is countably
infinite, we consider the number of clusters
$\omega(\gamma)=|\mathscr{C}(\gamma)|$ as a potential: The difference

$\Delta\omega(\gamma,\gamma^{\prime})=\lim_{F\uparrow
V}\omega(\gamma[F])-\omega(\gamma^{\prime}[F])$

for induced subgraphs along $F\uparrow V$ is well defined and finite. We have
$|\Delta\omega(\gamma,\gamma^{\prime})|\leq|\Delta(\gamma,\gamma^{\prime})|$
for any two graphs $\gamma$ and $\gamma^{\prime}$ with a finite symmetric
difference and the limit is eventually constant. Moreover, the potential
$\omega(\gamma)$ is continuous at $\gamma$ unless $\gamma$ contains more than
one infinite cluster.

The analysis of the likelihood ratios in (4) leads us to consider contractions
$\gamma^{F}$ of a random graphs $\gamma$ at the finite sets $F=[0,n-1]$. We
see that, given $F\Subset V$, the number of clusters
$\omega(\gamma)=|\mathscr{C}(\gamma)|$ satisfies the potential equality

(5) $\omega(\gamma)=\omega_{F}(\gamma)+\omega(\gamma^{F})-1$

where $\omega_{F}(\gamma)$ is the number of clusters intersecting $F$ and
$\omega(\gamma^{F})$ is the number of clusters in the contraction
$\gamma^{F}$. The constant difference of one is irrelevant for equality
between potentials.

We will on occasion consider a bipartition (or “cut”) $V=V_{+}\uplus\,V_{-}$
of the vertex set in two sets. Such a cut decomposes a given graph
$\gamma\in\Gamma(V)$ into three graphs

(6)
$\gamma=(\gamma_{+},{\gamma_{0}},\gamma_{-})\in\Gamma(V_{+})\times\Gamma(V_{+},V_{-})\times\Gamma(V_{-})$

and the analysis of the two-sided model with the one-sided rely on analysing
the decomposition for random graphs $\gamma$. The graphs
$\gamma_{\pm}=\gamma[V_{\pm}]$ are $\gamma$ induced on the two vertex parts
and the subgraph ${\gamma_{0}}$ is the bipartite subgraph $\gamma\cap
K(V_{+},V_{-})$ connecting vertices $i\in V_{-}$ with vertices $j\in V_{+}$.

A cut gives the potential identity

(7)
$\omega(\gamma)=\omega(\gamma_{+})+\omega(\gamma_{-})-|{\gamma_{0}}|+R(\gamma),$

where the cardinality $|{\gamma_{0}}|$ of ${\gamma_{0}}$ is taken as a linear
potential and where $R(\gamma)$ is a correction term. Since
$\omega(\gamma\setminus{\gamma_{0}})=\omega(\gamma_{+})+\omega(\gamma_{-})$,
we obtain the following expression for $R$

(8)
$R(\gamma)=|{\gamma_{0}}|-(\omega(\gamma\setminus{\gamma_{0}})-\omega(\gamma))=|{\gamma_{0}}|-(\omega(\gamma_{+})+\omega(\gamma_{-})-\omega(\gamma)).$

We could treat $R(\gamma)$ as a potential, but, in the cases of our interest,
we shall see that $R(\gamma)$ is a regular function which is small in a
certain sense.

The corank of a graph $\gamma$ is the maximum number of edges that one can
remove from $\gamma$ without increasing the number of components
$\omega(\gamma)$. It is also the minimum number one must remove in order to
make the graph acyclic. If we consider the bipartite (multi-)graph
$\tilde{\gamma}_{0}\in\Gamma(V_{+}/\mathscr{C}(\gamma_{+}),V_{-}/\mathscr{C}(\gamma_{-}))$
obtained from contracting ${\gamma_{0}}$ along the clusters in
$\mathscr{C}({\gamma_{+}})$ and $\mathscr{C}(\gamma_{-})$, then $R(\gamma)$ in
(8) equals the corank of $\tilde{\gamma}_{0}$.

If we contract the terms in the equality (7) at a finite set $F\Subset V_{+}$,
we obtain

(9)
$\omega(\gamma^{F})=\omega(\gamma_{+}^{F})+\omega(\gamma_{-})-|{\gamma_{0}}|+R_{F}(\gamma).$

Here $R_{F}(\gamma)$ equals
$\omega(\gamma^{F})-\omega(\gamma^{F}\setminus{\gamma_{0}}^{F})=\operatorname{corank}\tilde{\gamma}_{0}^{F}$,
where $\tilde{\gamma}_{0}^{F}$ denotes the bipartite graph ${\gamma_{0}}$
contracted first at $F$ and then along
$\mathscr{C}(\gamma_{+})\uplus\mathscr{C}(\gamma_{-})$.

For some fixed chosen order on the edges in ${\gamma_{0}}$, we say an edge
$ij$ is _irrelevant_ if there is an edge in ${\gamma_{0}}$ from the same
cluster $C$ in $\mathscr{C}(\gamma_{-})$ that precedes it in the chosen order.
We define $Q(\gamma_{-},{\gamma_{0}})$ as the total number of irrelevant
edges. That is,

(10)
$Q(\gamma_{-},{\gamma_{0}})=\sum_{C\in\mathscr{C}(\gamma_{-})}{\left(D(C,{\gamma_{0}})-1\right)}_{+},$

where $D(C,{\gamma_{0}})=\sum_{j\in C}\sum_{j\in V_{+}}{\gamma_{0}}(ij)$ is
the degree of $C\subset V_{-}$ in ${\gamma_{0}}$. Note that $Q$ does not
depend on $\gamma_{+}$.

From the interpretation of $R_{F}$ as the co-rank in $\tilde{\gamma}_{0}^{F}$,
it follows that for all $F$

(11) $R_{F}(\gamma)\leq Q(\gamma_{-},{\gamma_{0}})$

since, the graph where all vertices in one part has degree $1$, is acyclic.
Assuming $Q<\infty$, we also have

(12) $R_{F}(\gamma)\to Q(\gamma_{-},{\gamma_{0}})\quad\text{as $F\uparrow
V_{+}$},$

since all cycles in $\tilde{\gamma}_{0}^{F}$ eventually become 2-cycles when
$F\uparrow V_{+}$.

###  2.2. Random configurations and random graphs

Let $\mathscr{M}(X)$ denote the space of probability distributions of
configurations. Elements $\alpha\in\mathscr{M}(X)$ are usually written as
$\alpha(x)$ in order to make it clear that $\alpha$ is a distribution of the
random configuration $x\in X$. We also use $x\sim\alpha$ to denote that $x$
has distribution $\alpha$. When we need to specify a parameter $p$ of a
distribution $\alpha$ we use the form $\alpha(x;p)$. We denote the marginal
distribution of $x_{A}$ by $\alpha(x_{A})$. For a partition $A\uplus B=S$, we
denote by $\alpha(x_{A})\otimes\beta(x_{B})$ the product measure
$(\alpha\otimes\beta)(x)$ of $\alpha(x_{A})$ and $\beta(x_{B})$. A measure
$\eta$ is a _Bernoulli measure_ if and only if
$\eta(x)=\eta(x_{A})\otimes\eta(x_{B})$ for every bipartition $S=A\uplus B$.
To parametrise a general Bernoulli distribution, we use a function
$p:S\to\mathscr{M}(\mathscr{A})$ such that $p(s)$ is a probability
distribution on $\mathscr{A}$. The product distribution $\bigotimes_{s\in
S}p(s)(x_{s})$ is the corresponding Bernoulli measure $\eta(x;p)$. We write
$\mu(x)\prec\nu(x^{\prime})$ to state _stochastic domination_ between elements
in $\mathscr{M}(X)$, which means that we can couple $\mu(x)$ and
$\nu(x^{\prime})$ so that with probability one $x\leq x^{\prime}$ with respect
to the partial order $\cdot\leq\cdot$.

By a _gibbs measure_ (small $g$), we mean Gibbs measures, sufficiently
generalised to cover the random cluster setting below. It is a probability
distribution $\mu(x)\in\mathscr{M}(X)$ consistent with an associated
_specification_. That is, for every finite set $F\Subset S$ and for every
(instead of just for $\mu$-almost every) exterior configuration $x_{\bar{F}}$,
we have a well defined conditional probability $\mu(x_{F}\mid x_{\bar{F}})$ of
$x_{F}\in\mathscr{A}^{F}$ given $x_{\bar{F}}\in\mathscr{A}^{\bar{F}}$. Hence,
for each finite set $F\subset S$, the map $x_{\bar{F}}\mapsto\mu(\cdot\mid
x_{\bar{F}})$ is a well defined function of the exterior configuration
$x_{\bar{F}}$, satisfying the obvious consistency conditions for conditional
probabilities. An immediate class of unique gibbs measures are the Bernoulli
measures. The corresponding specification $\eta(x_{F}\mid
x_{\bar{F}};p)=\eta(x_{F};p|_{F})$ is independent of the boundary
$x_{\bar{F}}$.

Note that we do not require the specification of a gibbs measure to be
continuous and we are thus talking about gibbs measures in an extended sense.
The same specification may have multiple consistent gibbs measures, but, in
our context, we can consider all gibbs measure of this form to be _unique_ by
the subcriticality assumption on $\beta$. The reference [15] gives a more
thorough and rigourous introduction of Gibbs measures and some of their
extensions.

We can _modulate_ a gibbs measure $\alpha(x)$ with an “exponential of a
potential” $e^{\phi(x)}$ to obtain a set of new gibbs measures denoted by
$e^{\phi}\ltimes\alpha$. Let $\mu={\gamma_{0}}^{\phi}\ltimes\alpha$ denote an
element of this set. For all finite subsets $F\Subset S$, the specification of
$e^{\phi}\ltimes\alpha$ at $F$, i.e. $\mu(x_{F}\mid x_{\bar{F}})$, is well-
defined by the relation

(13) $\frac{\mu(x_{F}\mid x_{\bar{F}})}{\mu(y_{F}\mid
x_{\bar{F}})}=\exp\left(\Delta\phi(x,y)\right)\cdot\frac{\alpha(x_{F}\mid
x_{\bar{F}})}{\alpha(y_{F}\mid x_{\bar{F}})},$

where $y=(y_{F},x_{\bar{F}})$ and $\alpha(\cdot|x_{\bar{F}})$ is the
specification of $\alpha$. We write $e^{\phi}\ltimes\alpha$ to denote the set
of weak limit points, as $F\uparrow S$, of $\mu(x_{F}\mid
x_{\bar{F}})\cdot\alpha(x_{\bar{F}})$ where $\alpha(x_{\bar{F}})$ denotes the
marginal distribution of $x_{\bar{F}}$. In our context, this limit will be a
unique measure and we usually write $\mu=e^{\phi}\ltimes\alpha$. If $g(x)>0$
is a regular positive function such that $g(x)\in L^{1}(\alpha)$ then the
modulation $g\ltimes\alpha$ simply means taking the product with $g(x)$ and
normalising with a constant, i.e.

(14) $g\ltimes\alpha=\frac{g\cdot\alpha}{\int g(x)\,d\alpha(x)}.$

We can for example construct the Bernoulli measure $\eta(x;p)$ as the
modulation $e^{\phi(x)}\ltimes\upsilon(x)$ where $\upsilon(x)$ denotes the
_uniform_ Bernoulli measure on $X$ and $\phi(x)$ is the linear potential

$\phi(x)=\sum_{i\in S}\log p(i)(x_{i}).$

Modulation of a Bernoulli measure with a linear potential results in a new
Bernoulli measure.

The following rule shows that composition of modulation behaves naturally,
i.e.

(15) $e^{\psi}\ltimes(e^{\phi}\ltimes\alpha)=e^{\psi+\phi}\ltimes\alpha,$

provided $e^{\phi}\ltimes\alpha$ is unique. Another rule of computation is
that of distributivity over products of measures, i.e.

(16)
$e^{\psi(x_{A})+\phi(x_{B})}\ltimes(\alpha(x_{A})\otimes\beta(x_{B}))=(e^{\psi}\ltimes\alpha)(x_{A})\otimes(e^{\phi}\ltimes\beta)(x_{B}).$

Both rule (15) and (16) are immediate when applied to the specification
determined by (13). Since we assume that gibbs measures are uniquely specified
by their specifications, we can state the rules above as equalities between
gibbs measures.

####  2.2.1. The random cluster model

A _random graph model_ $\alpha(\gamma)$ is a probability distribution
$\alpha(\gamma)\in\mathscr{M}(\Gamma(G))$ on the space $\Gamma(G)$ of
(spanning) subgraphs $\gamma$ of a fixed ambient graph $G$. We can identify
$\gamma$ with the corresponding configuration $\gamma:E(G)\to\\{0,1\\}$ so
that $\Gamma(G)\cong\\{0,1\\}^{E(G)}$. In our context of “long range models”,
we will use the complete graph $G=K(V)$ on a countable vertex set $V$ as the
ambient graph and we write $\gamma\in\Gamma(V)$. However, we will almost
surely have finite degrees $D(F,\gamma)$.

The Bernoulli graph model $\eta(\gamma;p)$ is uniquely specified by its _edge
probabilities_ $p:G\to[0,1]$, so that $\gamma_{ij}=1$ with probability $p(ij)$
independently at each edge $ij\in E(G)$. The finite degree condition holds
whenever $\sum_{j:ij\in E(G)}p(ij)<\infty$ for all $i$. Note that the
Bernoulli graph model is independent of the graph structure and is the same
for all ambient graphs with the same set of edges.

The _random cluster model_ $\mathsf{RC}_{q}(\gamma;p)$ (or FK-model, see [20])
is the random graph distribution $\mu=q^{\omega(\gamma)}\ltimes\eta(\gamma;p)$
on $\Gamma(G)$ that one obtains if one modulates the Bernoulli graph
$\eta(\gamma;p)$ with $q^{\omega(\gamma)}$. Note that,
$\mathsf{RC}_{1}(\gamma;p)=\eta(\gamma;p)$. Since we focus on the Ising model,
we will assume that $q=2$ unless otherwise stated. We will only work in the
sub-critical regimes where the uniqueness of the random cluster model
$\mu=q^{\omega(\gamma)}\ltimes\eta(p)$ is well established [20].

It is well-known (see [20]) that the random cluster models satisfy a
stochastic domination relation, so that

$\mathsf{RC}_{q}(\gamma;p)\prec\mathsf{RC}_{q^{\prime}}(\gamma^{\prime};p^{\prime})\quad\text{when
$p\leq p^{\prime}$ and $q\geq q^{\prime}$.}$

In particular it holds that $\mathsf{RC}(\gamma;p)\prec\eta(\gamma;p)$. For a
fixed vertex $\mathrm{o}\in V$, we use $C_{\mathrm{o}}(\gamma)$ to denote the
cluster containing the vertex $\mathrm{o}$. It is a fact, see [5], that if we
condition on the cluster $C_{\mathrm{o}}$ then the distribution of the
remaining graph $\gamma\setminus C_{\mathrm{o}}$ is the random cluster model
with edge probabilities $p^{\prime}(ij)=p(ij)\mathsf{1}_{i,j\not\in
C_{\mathrm{o}}}$. It follows that the conditional distribution of
$\gamma\setminus C_{\mathrm{o}}$ is dominated by the unconditional
distribution of $\gamma\setminus C_{\mathrm{o}}$, so that

(17) $\mu(\gamma\setminus C_{\mathrm{o}}\mid C_{\mathrm{o}})\prec\mu(\gamma).$

The _random spin-cluster model_ $\mathsf{RC}((x,\gamma);p)$ is the joint
distribution of a random graph $\gamma$ together with an Ising _spin_
configuration, $x\in X={\\{+1,-1\\}}^{V}$, on the vertex set. One can obtain
the distribution

$\mu(x,\gamma)=\mathsf{RC}((x,\gamma);p)$

of $(x,\gamma)$ by first considering the product distribution
$\upsilon(x)\otimes\eta(\gamma)$ of the uniform distribution of $x\in X$ and
the Bernoulli distribution $\eta(\gamma;p)$ and then conditioning on the event
that $x$ and $\gamma$ are compatible in the sense that no edge in $\gamma$
connects vertices of opposite spin. An alternate, perhaps more direct, way to
derive the spin-cluster distribution $\mu(x,\gamma)$ is to first choose the
random graph $\gamma$ according to the random cluster model $\mu(\gamma)$ and
then to choose a spin $x(C)\in\\{+1,-1\\}$ to each cluster
$C\in\mathscr{C}(\gamma)$ uniformly at random.

For our purposes, one should note that the marginal distribution $\mu(x)$ of
the spins $x\in X$ is the Gibbs measure corresponding to the potential

(18) $\Phi(x)=\sum_{ij\in E(G)}-\log(1-p(ij))x_{i}x_{j}.$

The marginal distribution $\mu(\gamma)$ of $\gamma$ is the random cluster
model $\mathsf{RC}(\gamma;p)$. Percolation is the event that the random graph
$\gamma$ contains a cluster of infinite size. The almost sure existence of an
infinite cluster coincide with the existence of multiple Gibbs measures for
the spins $x\in X$.

####  2.2.2. Cylinder probabilities

Let $F$ be a finite subset of $V$ and consider the cylinder ${[x]}_{F}$ of
spins. Let $B_{F}(x,\gamma)\in\\{0,1\\}$ indicate the event that the graph
$\gamma$ is compatible with the cylinder: That is, that no _path_ in $\gamma$
connects $i,j\in F$ such that the spins $x_{i}$ and $x_{j}$ have opposite
signs. Recall that $\omega_{F}(\gamma)$ denote the number of clusters in
$\gamma$ that intersects $F$. From the alternate way to derive the spin-
cluster distribution, we deduce that the probability of the cylinder is

(19) $\mu({[x]}_{F})=\int
2^{-\omega_{F}(\gamma)}\,B_{F}(x,\gamma)\,d\mu(\gamma),$

since the probability that the cluster-wise assignment of spins $\\{x(C)\\}$
give rise to the cylinder ${[x]}_{F}$ equals $2^{-\omega_{F}(\gamma)}$
provided the graph $\gamma$ is compatible with $[x]_{F}$.

Note that $\omega_{F}(\gamma)\leq|F|$ is a bounded function. From (14), (19)
and the potential equality (5), we arrive at the following expression for the
cylinder probability

(20) $\mu({[x]}_{F})=\frac{1}{\int 2^{-\omega_{F}}\,d\mu}\cdot\int
B_{F}(x,\gamma)\,d\mu^{F}(\gamma),$

where

$\mu^{F}(\gamma)=2^{\omega(\gamma^{F})}\ltimes\eta(\gamma^{F};p).$

denote the random cluster model $\mu$ _contracted at_ $F$. Since
$\Gamma(G)\cong\Gamma(G^{F})$ as sets, we may choose to consider
$\mu^{F}(\gamma)$ as a perturbed random cluster distribution for
$\gamma\in\Gamma(G)$ or the random cluster distribution for the contraction
$\gamma^{F}\in\Gamma(G^{F})$.

####  2.2.3. Decomposition of the random cluster model across a cut

Consider the decomposition in (6) of a graph $\gamma$ across a cut
$(V_{+},V_{-})$. Let $\mu=2^{\omega(\gamma)}\ltimes\eta(\gamma)$ be the full
“two-sided” random cluster model. Similary, let
$\nu(\gamma_{\pm})=2^{\omega(\gamma_{\pm})}\ltimes\eta(\gamma_{\pm})$ be the
“one-sided” random cluster models for the graphs $\gamma_{\pm}$ on vertex sets
$V_{\pm}$. Assume that $F\Subset V_{+}$ is a fixed finite subset of $V_{+}$.
Let also $\mu^{F}(\gamma)$ and $\nu^{F}(\gamma_{+})$ be the contractions at
$F$ of $\mu(\gamma)$ and $\nu(\gamma_{+})$, respectively. From (6), it is
clear that the Bernoulli distribution $\eta(\gamma)=\eta(\gamma;p)$ factorises
into three Bernoulli measures

(21)
$\eta(\gamma)=\eta(\gamma_{+})\otimes\eta({\gamma_{0}})\otimes\eta(\gamma_{-}).$

We construct these Bernoulli measures by restricting the given edge
probability.

A similar factorisation for the contracted random cluster measure $\mu^{F}$
uses (16). From (21) and (9), we obtain that

$\displaystyle\mu^{F}(\gamma)$
$\displaystyle=2^{\omega(\gamma_{+}^{F})+\omega(\gamma_{-})-|{\gamma_{0}}|+R_{F}(\gamma)}\ltimes\left(\eta(\gamma_{+})\otimes\eta({\gamma_{0}})\otimes\eta(\gamma_{-})\right)=$
$\displaystyle=2^{R_{F}}\ltimes\left((2^{\omega(\gamma_{+}^{F})}\ltimes\eta(\gamma_{+}))\otimes(2^{-|{\gamma_{0}}|}\ltimes\eta({\gamma_{0}}))\otimes(2^{-\omega(\gamma_{-})}\ltimes\eta(\gamma_{-}))\right)$

and, hence, we have the following factorisation

(22)
$\mu^{F}(\gamma)=2^{R_{F}(\gamma)}\ltimes\left(\nu^{F}(\gamma_{+})\otimes\tilde{\eta}({\gamma_{0}})\otimes\nu(\gamma_{-})\right).$

where the measure $\tilde{\eta}$ is the Bernoulli measure

(23)
$\tilde{\eta}({\gamma_{0}})=2^{-|{\gamma_{0}}|}\ltimes\eta({\gamma_{0}})=\eta({\gamma_{0}};\tilde{p})\quad\text{where
$\tilde{p}=p/(2-p)$.}$

Clearly, $\tilde{\eta}({\gamma_{0}})\prec\eta({\gamma_{0}})$.

Specialising (22) with $F=\emptyset$, allow us to write the random cluster
model $\mu(\gamma)$ as

(24)
$\mu(\gamma)=2^{R(\gamma)}\ltimes\left(\nu(\gamma_{+})\otimes\tilde{\eta}({\gamma_{0}})\otimes\nu(\gamma_{-})\right).$

We shall see that, for the Dyson model, $2^{R(\gamma)}\in L^{1}(\mu)$, which
shows that the marginal distribution $\mu(\gamma_{+})$ of $\gamma_{+}$ under
the two-sided model is absolutely continuous with respect to the one-sided
measure $\nu(\gamma_{+})$.

###  2.3. The Dyson model

####  2.3.1. The one-sided and two-sided models

Let $\bar{X}=\mathscr{A}^{\mathbb{Z}}$ with projection $\bar{X}\to X$ given by
$\bar{x}\mapsto x=\bar{x}|_{\mathbb{N}}$. For the analysis of the long range
one-dimensional Dyson model, we consider the “two-sided” random spin-cluster
model

$\mu(\bar{x},\gamma)=\mathsf{RC}((\bar{x},\gamma);p)$

with vertex set $V=\mathbb{Z}$ where the edge probabilities are

(25) $p(ij)=1-e^{-J(ij)}\quad\text{where}\quad
J({ij})=\frac{\beta}{|i-j|^{\alpha}}.$

By (18), the marginal spin distribution $\mu(\bar{x})$ is the Gibbs measure

(26)
$\bar{\Phi}(\bar{x})=\sum_{i,j}\frac{\beta}{|i-j|^{\alpha}}\bar{x}_{i}\bar{x}_{j}=\sum_{k=-\infty}^{\infty}\bar{\phi}(T^{k}\bar{x})$

where $\bar{\phi}$ is the lift to $\bar{X}$ of one point potential $\phi$ from
(2). By symmetry and uniqueness, $\mu(\bar{x},\gamma)$ is translation
invariant with respect the left shift $T$ on $\mathbb{Z}$. In particular, so
is the marginal distribution $\mu(x)$ of $x\in X$.

Taking the cut of $\mathbb{Z}=V_{+}\uplus V_{-}$ where
$V_{+}=\mathbb{N}=\\{0,1,2,\dots\\}$ and $V_{-}=\\{\dots,-2,-1\\}$, we also
consider the two “one-sided” spin-cluster models

$\nu(x_{\pm},\gamma_{\pm})=\mathsf{RC}(x_{\pm},\gamma_{\pm};p_{\pm}).$

By the vertex map $j\mapsto-1-j$, $j\in\mathbb{N}$, we have an isomorphism
$\nu((x_{+},\gamma_{+}))\cong\nu((x_{-},\gamma_{-}))$. For this one-sided
model, the spin distribution $\nu(x)$ is the Gibbs measure corresponding to
the one-sided potential $\Phi(x)$

(27) $\Phi(x)=\sum_{k=0}^{\infty}\phi(T^{k}x)$

for $x\in X$, where we drop the subscripting with $\pm$ on the spin sequences.
The Gibbs measure $\nu(x)$ for the potential $\Phi(x)$ in (27) is also the
eigenmeasure for $\mathscr{L}_{\phi}^{*}$ since the definition of
$\mathscr{L}^{*}_{\phi}$ gives that

$\mathscr{L}^{*}\nu(x)=e^{\phi(x)}\cdot\nu(Tx)$

and the right hand side is, up to the normalising constant
$\frac{1}{\lambda}$, the Gibbs measure for $\Phi(x)$ due to the identity
$\Phi(x)=\phi(x)+\Phi(Tx)$.

####  2.3.2. Cluster size distribution

It is well-known (see e.g. [1], [20] or [12]) that for all $\alpha$,
$1<\alpha\leq 2$, and $q\geq 1$ there exists a critical parameter
$\beta_{c}=\beta_{c}(\alpha,q)$, such that percolation does not occur with
probability one for $0\leq\beta<\beta_{c}$ (the “sub-critical” regime), while
it occurs with probability one for $\beta_{c}<\beta<\infty$. The random
cluster model is moreover unique except for possibly at $\beta=\beta_{c}$.

We claim that there is a $\beta_{*}>0$ such that, for $0<\beta<\beta_{*}$, the
moment generating function of the cluster size has a positive radius of
convergence. In other words: For some $t_{0}=t_{0}(\alpha,\beta)>0$, such that
for $0<t<t_{0}$

(28)
$\mathsf{E}\left(e^{t\cdot|C_{\mathrm{o}}(\gamma)|}\right)=\sum_{k=0}^{\infty}\frac{\mathsf{E}(|C_{\mathrm{o}}|^{k})}{k!}t^{k}<\infty,$

where $\mathrm{o}$ is any fixed vertex. The property (28) follows if the
cluster size $|C_{\mathrm{o}}|$ has exponentially bounded tails, i.e. if

(29) $\mathsf{P}(|C_{\mathrm{o}}|>n)\leq A\cdot e^{-t_{0}n}/\sqrt{n}$

for some $A$ and $t_{0}>0$. We can easily see that (29) implies (28) using the
identity

$\mathsf{E}\left(e^{sX}\right)=\int_{0}^{\infty}se^{sx}\mathsf{P}(X>x)\,dx.$

Moreover, it is well-known that (29) holds (see Panagiotis [28] Theorem 1.2.1;
Aizenman and Newman [2], Proposition 5.1) for $\beta<\beta_{c}^{1}(\alpha)$.
By stochastic domination we have $\beta_{c}^{1}(\alpha)\leq\beta_{c}(\alpha)$
and we can thus infer that

(30) $\beta_{c}^{1}(\alpha)\leq\beta_{*}\leq\beta_{c}(\alpha)$

as claimed in discussion following Theorem 1. From now on we assume that
$\beta<\beta_{*}$ and thus that (28) holds.

A major part of our argument depends on the following lemma stating that the
moment generating function (MGF) $\mathsf{E}(e^{sQ})$ of $Q$ from (10) is
finite for all $s$.

######  Lemma 2.

If $\nu(\gamma_{-})$ satisfies (28) then

(31)
$\int\exp\left(sQ(\gamma_{-},{\gamma_{0}})\right)\,d\tilde{\eta}({\gamma_{0}})\,d\nu(\gamma_{-})<\infty,$

for every $s>0$.

####  2.3.3. Proof of Lemma 2

The edge-indicators ${\gamma_{0}}(ij)$ distributed according to $\tilde{\eta}$
are independent with Bernoulli distribution $\mathrm{Be}(\tilde{p}(ij))$ where
$\tilde{p}(ij)\leq p(ij)=1-e^{-J(ij)}$. Since
$\mathrm{Be}(1-e^{-J})\prec\mathrm{Po}(J)$, we have

$\tilde{\eta}({\gamma_{0}})\prec\eta({\gamma_{0}})\prec\bigotimes_{ij}\mathrm{Po}(J(ij)).$

We assume an underlying probability space $(\Omega,\mathscr{F},\mathsf{P})$,
carrying the processes $(\gamma_{-},\eta)\sim\nu_{-}\otimes\tilde{\eta}$ as in
(31). In addition, we assume a discrete Poisson process $C\mapsto X(C)$,
$C\subset V_{-}$, specified by

$X(C):=\sum_{i\in C}\sum_{j\in
V_{+}}X(ij)\sim\mathrm{Po}(\lambda(C)),\quad\lambda(C)=\sum_{i\in C}\sum_{j\in
V_{+}}J(ij),$

where $D(C)=\sum_{i\in C}\sum_{j}{\gamma_{0}}(ij)\leq X(C)$.

Let $Y(C)={(X(C)-1)}_{+}$. To prove Lemma 2, it is enough to show that

(32)
$\mathsf{E}\left(\exp\left(s\cdot\sum_{C\in\mathscr{C}(\gamma_{-})}Y(C)\right)\right)<\infty,$

for $s>0$.

Choose $m_{0}\geq 2$ so that

(33)
$t=\frac{e^{s}\beta}{\alpha^{\prime}{(m_{0}-1)}^{\alpha^{\prime}}}<t_{0}.$

where $t_{0}=t_{0}(\beta)$ is the radius of convergence from (28). Let
$S=\\{-1,-2,\dots,-m_{0}\\}$ and let $\mathscr{C}^{\prime}=\\{C\setminus S\mid
C\in\mathscr{C}(\gamma)\\}$. Note that, for every $C\subset V_{-}$, we have

$Y(C)\leq X(S\cap C)+Y(C\setminus S).$

Hence, it follows that

$\sum_{C\in\mathscr{C}(\gamma_{-})}Y(C)\leq
X(S)+\sum_{C\in\mathscr{C}^{\prime}}Y(C).$

Since $\sum_{C\in\mathscr{C}^{\prime}}Y(C)$ is independent of
$X(S)\sim\mathrm{Po}(\lambda(S))$ by disjointness, it is enough to show that
$\mathsf{E}\left(e^{s\sum_{C\in\mathscr{C}^{\prime}}Y(C)}\right)<\infty$. This
amounts to show that,

(34)
$K_{3}:=\mathsf{E}\left(\prod_{C\in\mathscr{C}^{\prime}}\Psi(C)\right)<\infty,$

where

$\displaystyle\Psi(C)$
$\displaystyle=\mathsf{E}\left(e^{s\,{(X(C)-1)}_{+}}\mid C\right)$
$\displaystyle=\sum_{k=0}^{\infty}e^{-\lambda}\cdot\frac{\lambda^{k}}{k!}\cdot
e^{s\,{(k-1)}_{+}}$ (35) $\displaystyle=e^{-\lambda}+\lambda
e^{-\lambda}+e^{-\lambda-s}\cdot\sum_{k=2}^{\infty}\frac{{(e^{s}\lambda)}^{k}}{k!},\quad\text{with
$\lambda=\lambda(C)$.}$

For $i\geq 2$, an elementary integral estimate of
$\lambda(\\{i\\})=\sum_{j\in\mathbb{N}}J(ij)$ gives that

$\lambda(\\{i\\})\leq\frac{\beta}{\alpha^{\prime}\cdot{(|i|-1)}^{\alpha^{\prime}}},$

where $\alpha^{\prime}=\alpha-1$. Hence, for any $C\subset V_{-}$

(36)
$\lambda(C)\leq\frac{\beta}{\alpha^{\prime}\cdot{(J(C)-1)}^{\alpha^{\prime}}}\cdot|C|,$

where $J(C)=\min\\{|i|:i\in C\\}$ is the rightmost (first) element of $C$.

Since $e^{-\lambda}+\lambda e^{-\lambda}<1$ and $e^{-\lambda-s}<1$, the
expression (35) and the bound (36) implies that

(37) $\Psi(C)\leq
1+\sum_{k=2}^{\infty}{\left(\frac{e^{s}\beta}{\alpha^{\prime}{(J(C)-1)}^{\alpha^{\prime}}}\right)}^{k}\cdot\frac{|C|^{k}}{k!}$

Since $J(C)\geq m_{0}$, for all $C\in\mathscr{C}^{\prime}$, we obtain

(38) $\Psi(C)\leq 1+w(J)\cdot\Theta(|C|)$

where

$w(J)=\frac{{(m_{0}-1)}^{2\alpha^{\prime}}}{{(J-1)}^{2\alpha^{\prime}}}\quad\text{and}\quad\Theta(N)=\sum_{k=2}^{\infty}t^{k}\cdot\frac{N^{k}}{k!}<\infty,$

with $t<t_{0}$ as in (33).

Order the elements in $\mathscr{C}^{\prime}=\\{C_{1},C_{2},\dots\\}$ so that

$m_{0}+1=J(C_{1})<J(C_{2})<\cdots.$

Note that $J(C_{k})=\min\\{|i|:i\not\in S\cup C_{1}\cup\dots\cup C_{k-1}\\}$,
i.e., we can determine $J(C_{k})$ from the preceding clusters. By induction on
(17), we see that

$\mathsf{P}(|C_{k}|\mid C_{1},\dots,C_{k-1})\prec\mathsf{P}(|C_{\mathrm{o}}|)$

and, hence, with $\Theta(N)$ as in (38), we have, for all $k$,

(39) $\mathsf{E}\left(\Theta(|C_{k}|)\mid
C_{1},C_{2},\dots,C_{k-1}\right)\leq\mathsf{E}\left(\Theta(|C_{\mathrm{o}}|)\right)=:\Theta_{0}.$

which, since $t<t_{0}$, is less than $\infty$ by (28).

To complete the proof, we take conditional expectations in (34) and obtain
from (38) and (39) that

$\displaystyle K_{3}$
$\displaystyle\leq\mathsf{E}\left(\prod_{k=1}^{\infty}\mathsf{E}\left(1+w(J(C_{k}))\cdot\Theta(|C_{k}|)\mid
C_{1},C_{2},\dots,C_{k-1}\right)\right)$
$\displaystyle\leq\mathsf{E}\left(\prod_{k=1}^{\infty}\left(1+w(J(C_{k}))\cdot\Theta_{0}\right)\right)$
$\displaystyle\leq\exp\left(\Theta_{0}\cdot\text{const}\cdot\sum_{k=m_{0}}^{\infty}\frac{1}{{(m_{0}+k-1)}^{2\alpha^{\prime}}}\right)<\infty,$

since $J(C_{k})\geq m_{0}+k$ and $w(k)$ is decreasing in $k$ and the sum is
finite due to $2\alpha^{\prime}>1$.

∎

###  2.4. The proof of the theorem

Recall that our aim is to show that the sequence of the local likelihood
ratios

$h_{n}(x)=\frac{\mu({[x]}_{n})}{\nu({[x]}_{n})}$

in (4) is a Cauchy sequence with respect to the supremum norm. That is, we aim
to show that

(40) $\|h_{n}(x)-h_{m}(x)\|_{\infty}\to 0\quad\text{as $n,m\to\infty$}$

which means that the limit $h(x)$ is a continuous function bounded away from
$0$ and $\infty$.

We will refer to previous relations concerning representations and cylinder
probabilities such as (24), (22), (20), etc. which use notation for a more
general setting. We can now specialise to the case where we consider cylinders
at $F=[0,n-1]=\\{0,1,\dots,n-1\\}$ where $n\to\infty$. For notational
simplicity, we use subscript $n$ instead of $F$ when referring to cylinders,
cluster counts and measures, etc. i.e. ${[x]}_{n}$ stands for the cylinder
${[x]}_{[0,n-1]}$ and we write $\omega_{n}$ for $\omega_{[0,n-1]}$, $\nu^{n}$
for $\nu^{F}$, $R_{n}$ for $R_{F}$, etc.

From (22) and (20), we have in this notation

$\mu({[x]}_{n})=k_{2}\cdot\int B_{n}(x,\gamma)\cdot
2^{R_{n}(\gamma)}\,d\left(\nu^{n}(\gamma_{+})\otimes\tilde{\eta}({\gamma_{0}})\otimes\,d\nu(\gamma_{-})\right)$

and

$\nu({[x]}_{n})=k_{1}\cdot\int B_{n}(x,\gamma_{+})\,d\nu^{n}(\gamma_{+}),$

where $k_{2}=\int 2^{-\omega_{n}(\gamma)}\,d\mu(\gamma)$ and $k_{1}=\int
2^{-\omega_{n}(\gamma_{+})}\,d\nu(\gamma_{+})$. Hence, by taking the ratio, we
have

(41) $h_{n}(x)=K_{n}\cdot\frac{1}{L_{n}}\cdot I_{n}(x)$

where $K_{n}$ and $L_{n}$ are

(42) $\displaystyle K_{n}$ $\displaystyle=\frac{k_{2}}{k_{1}}=\frac{\int
2^{-\omega_{n}(\gamma)}\,d\mu(\gamma)}{\int
2^{-\omega_{n}(\gamma_{+})}\,d\nu(\gamma_{+})}$ (43) $\displaystyle L_{n}$
$\displaystyle=\int
2^{R_{n}(\gamma)}\,d(\nu^{n}(\gamma_{+})\otimes\tilde{\eta}(\eta)\otimes\nu(\gamma_{-})).$

Only the integral $I_{n}(x)$ depends on $x\in X$ and we have

(44) $I_{n}(x)=\frac{\int B_{n}(x,\gamma)\cdot
2^{R_{n}(\gamma)}\cdot\,d(\nu^{n}(\gamma_{+})\otimes\tilde{\eta}({\gamma_{0}})\otimes\nu(\gamma_{-}))}{\int
B_{n}(x,\gamma_{+})\,d\nu^{n}(\gamma_{+})}.$

Note that $K_{n}$ and $L_{n}$ does not depend on $x$.

Let

$B^{\prime}_{n}(x,\gamma)=\begin{cases}1&B_{n}(x,\gamma_{+})=0\\\
B_{n}(x,\gamma)&\text{otherwise.}\end{cases}$

Thus $B^{\prime}_{n}(x,\gamma)$ is zero only if $B_{n}(x,\gamma_{+})=1$ and
their is some cluster $C$ in $\mathscr{C}(\gamma_{-})$ sends a pair of edges
in ${\gamma_{0}}$ that joins two clusters in $\gamma_{+}$ that intersect
$[1,n-1]$ at positions with opposite spins. We can now rewrite $I_{n}(x)$ as

(45) $I_{n}(x)=\int
B_{n}^{\prime}(x,\gamma)2^{R_{n}(\gamma)}\,\,d(\hat{\nu}_{n}(\gamma_{+})\otimes\tilde{\eta}({\gamma_{0}})\otimes\nu(\gamma_{-})),$

where $\hat{\nu}_{n}(\gamma_{+})$ is the probability measure given by

(46)
$\hat{\nu}_{n}=\frac{B_{n}(x,\gamma_{+})\cdot\,d\nu^{n}(\gamma_{+})}{\,\int
B_{n}(x,\gamma_{+})\,d\nu^{n}(\gamma_{+})\,}.$

In other words, it is the measure $\nu^{n}(\gamma_{+})$ conditioned on
$\gamma_{+}$ and ${[x]}_{n}$ being compatible.

We define the endpoint of the “last” irrelevant edge as

(47) $N=\max\\{j\in V_{+}\mid ij\in{\gamma_{0}},\ i\in
C\in\mathscr{C}(\gamma_{-}),\ D(C)\geq 2\\}.$

By Lemma 2 $\mathsf{P}(N<\infty)=1$. Let $A(x,\gamma_{-},{\gamma_{0}})$
indicate the event that no cluster $C\in\mathscr{C}(\gamma_{-})$ sends two
edges in ${\gamma_{0}}$ to opposite spins of $x$. We have

(48) $B^{\prime}_{n}(x,\gamma)=A(x,\gamma_{-},{\gamma_{0}})\quad\text{for all
$n\geq N$.}$

Moreover, by (12), we have for $n\geq N$

(49) $R_{n}(\gamma)=Q(\gamma_{-},{\gamma_{0}}).$

We note that the quantities $B^{\prime}_{n}$ and $R_{n}$ are conditionally
independent on the event $n\geq N$.

We can now start to establish the convergence of the quantities (42), (43) and
(44). From (49) it is clear that the integrals $L_{n}$ converge

$L_{n}\to\int
2^{Q({\gamma_{0}},\gamma_{-})}\,d(\tilde{\eta}({\gamma_{0}})\otimes\nu(\gamma_{-})),$

as $n\to\infty$. This is finite by Lemma 2.

It also follows from (48) and (49) that conditioning on
$(\gamma_{-},{\gamma_{0}})$ gives

$g_{n}(x):=\mathsf{E}(I_{n}(x)\mid\gamma_{-},{\gamma_{0}})=A(x,g_{-},{\gamma_{0}})\cdot
2^{Q(\gamma_{-},{\gamma_{0}})}>0$

on $n\geq N$ and hence $g_{n}(x)-g_{m}(x)$ are eventually equal to $0$. It
follows from dominated convergence that

$\|I_{n}(x)-I_{m}(x)\|_{\infty}\leq\|2^{Q}\|_{L^{1}}\cdot\mathsf{P}(N\geq\min(n,m)),$

which goes to zero as $n,m\to\infty$. Thus, the functions $\\{I_{n}(x)\\}$
constitute a Cauchy sequence with respect to the supremum norm and the limit
$I(x)=\lim I_{n}(x)$ is thus a continuous function and it is also clear that
$I_{n}(x)>0$ for all $x$.

In order to establish (40), we must also show that the limit of $K_{n}$ exists
as a value bounded away from zero and infinity. That is, we want to show that

(50) $\lim_{n\to\infty}\log K_{n}=\log K$

exists as a finite value. By the representation (24) of $\mu$, we can write

$\displaystyle\log K_{n}$ $\displaystyle=\log\int
2^{-\omega_{n}(\gamma)}\,d\mu(\gamma)-\log\int
2^{-\omega_{n}(\gamma_{+})}\,d\nu(\gamma_{+})$
$\displaystyle=\log\frac{\int\,2^{\,\omega_{n}(\gamma_{+})-\omega_{n}(\gamma)+R(\gamma)}\cdot
2^{-\omega_{n}(\gamma_{+})}\,d(\nu(\gamma_{+})\otimes\tilde{\eta}({\gamma_{0}})\otimes\nu(\gamma_{-}))}{\int
2^{-\omega_{n}(\gamma_{+})}\,d\nu(\gamma_{+})}$
$\displaystyle=\log\int\,2^{\,\omega_{n}(\gamma_{+})-\omega_{n}(\gamma)+R(\gamma)}\,d(\hat{\nu}_{n}(\gamma_{+})\otimes\tilde{\eta}({\gamma_{0}})\otimes\nu(\gamma_{-}))$

where $\hat{\nu}_{n}(\gamma_{+})$ is the probability distribution
$2^{-\omega_{n}(\gamma_{+})}\ltimes\nu(\gamma_{+})$.

But, each cluster $C$ in $\gamma_{-}$ can only contribute with at most
${(D(C)-1)}_{+}$ to the difference
$\omega_{n}(\gamma_{+})-\omega_{n}(\gamma)$, since each irrelevant edge from
$C$ can join at most two clusters intersecting $S=n$. It follows that

(51) $\omega_{n}(\gamma_{+})-\omega_{n}(\gamma)\leq
Q(\gamma_{-},{\gamma_{0}})$

Thus by (11), we have

$\omega_{n}(\gamma_{+})-\omega_{n}(\gamma)+R(\gamma)\leq 2\cdot
Q(\gamma_{-},{\gamma_{0}}),$

and, since $\int 2^{2Q}\,d\tilde{\eta}({\gamma_{0}})\,d\nu(\gamma_{-})<\infty$
by Lemma 2, the dominated convergence theorem implies (50). ∎

## References

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Anders Johansson, Department of Mathematics, University of Gävle, 801 76
Gävle, Sweden. Email-address<EMAIL_ADDRESS>

Anders Öberg, Department of Mathematics, Uppsala University, P.O. Box 480, 751
06 Uppsala, Sweden. E-mail-address<EMAIL_ADDRESS>

Mark Pollicott, Mathematics Institute, University of Warwick, Coventry, CV4
7AL, UK. Email-address<EMAIL_ADDRESS>