+\infty&ifR_{i}\cap R_{j}\neq\emptyset\end{cases}$ $None$

we have proved the following elementary but fundamental theorem

###### Theorem 6.1

The partition function $Z_{\Lambda}(\beta)$ defined in (6.21) can be written
as

$Z_{\Lambda}(\beta)~{}=~{}\sum_{k\geq 0}{1\over
k!}\\!\sum_{R_{1},\dots,R_{k}\atop|R_{i}|\geq
2}\\!\prod_{j=1}^{k}\rho_{\beta}(R_{j})e^{-\sum_{1\leq i<j\leq
n}U(R_{i},R_{j})}$ $None$

where

$\rho_{\beta}(R)~{}=~{}\int\prod_{x\in
R}d\mu(\phi_{x})\sum_{g\in{G}_{R}}\prod_{X\in
g}(e^{-\beta\Psi(X,\phi_{X})}-1)$ $None$

In conclusion theorem 6.1 states that the partition function of discrete spin
systems can be rewritten as the grand canonical partition function of a
polymer gas in which the polymers $R$ are finite subsets of $\mathbb{V}$ with
cardinality greater or equal than 2 and a polymer $R$ has activity $\rho(R)$.
Such way to rewrite the partition function of a discrete system is called the
high temperature polymer gas expansion of the system.

Note that all the information about the original system is now contained in
the structure of the activity $\rho(R)$ of a given polymer, given by (6.25).

Note also that $\rho_{\beta}(R)\to 0$ as $\beta\to 0$. Hence the activity of
polymers is small when $\beta$ is small. So it is reasonable to expect the
convergence of this polymer gas when the parameter $\beta$ is sufficiently
small. This is the reason why such an expansion is called “high temperature
expansion”.

We want to outline here that we meet again in the structure of polymer
activity (6.25), the combinatorial problems we have seen in section 3.1.3.
Here such problems are even worst because the factors
$(e^{-\beta\Psi(X,\phi_{X})}-1)$ in (6.25) when dealing with many-body
interactions and hypergraphs. As far as discrete spin systems with purely pair
interactions are concerned, we will get rid of this challenging combinatorial
problem by means of the previously seen tree graph identities.

Many body interaction may also be treated, with a completely different
technique, as we will see in the section 6.3.

#####  6.2.1 General spin systems interacting via pair potentials

General many body interactions are usually quite difficult be treated, in
particular in the case in which the single spin probability space $\Omega$ is
non compact. Hereafter we will consider only pair interaction, i.e. we will
assume that $\Psi(X,\phi_{X})~{}=~{}0$ whenever $|X|\neq 2$. Hence our
interactions will be always pair interactions of the form
$\Psi(\\{x,y\\},\phi_{x},\phi_{y})$ with $x$ and $y$ varying in
$\mathbb{Z}^{d}$. This function $\Psi(x,y,\phi_{x},\phi_{y})$ is supposed to
satisfies some properties. In particular we will demand that there exist a
positive symmetric function $J(\\{x,y\\})$ (i.e. $J_{yx}=J_{xy}$) such that

$|\Psi(\\{x,y\\},\phi_{x},\phi_{y})|\leq J_{xy}\|\phi_{x}\|\|\phi_{y}\|$
$None$

and

$\sup_{x\in\mathbb{Z}^{d}}\sum_{y\in\mathbb{Z}^{d}\atop y\neq
x}J_{xy}~{}\doteq~{}J<\infty$ $None$

It is crucial to point out that conditions (6.26) and (6.27) automatically
implies that the potential $\Psi(\\{x,y\\},\phi_{x},\phi_{y})$ is stable in
the following sense. Given any finite set $R\subset\mathbb{Z}^{d}$ (with
cardinality $|R|\geq 2$), it holds that

$\sum_{\\{x,y\\}\subset R}\Psi(\\{x,y\\},\phi_{x},\phi_{y})\geq-{J\over
2}\sum_{x\in R}\|\phi_{x}\|^{2}$ $None$

Indeed,

$\displaystyle\sum_{\\{x,y\\}\subset R}\Psi(\\{x,y\\},\phi_{x},\phi_{y})$
$\displaystyle\geq-\sum_{\\{x,y\\}\subset R}|J_{xy}\|\phi_{x}\|\|\phi_{y}\|$
$\displaystyle\geq~{}-\sum_{\\{x,y\\}\subset R}J_{xy}{1\over
2}\Big{[}\|\phi_{x}\|^{2}+\|\phi_{y}\|^{2}\Big{]}$
$\displaystyle=-\sum_{\\{x,y\\}\subset R}J_{xy}\|\phi_{x}\|^{2}$
$\displaystyle={1\over 2}\sum_{x\in R}\|\phi_{x}\|^{2}\sum_{y\in R\atop y\neq
x}J_{xy}$ $\displaystyle\geq-{J\over 2}\sum_{x\in R}\|\phi_{x}\|^{2}$

Once a function $\Psi$ with the properties (6.26) and (6.27) above has been
given, we can construct the Hamiltonian of the system. To do this we first fix
a spin configuration $\phi_{\Lambda^{c}}$ for the spins outside the box
$\Lambda$ (i.e. in the set $\Lambda^{c}~{}=~{}\mathbb{Z}^{d}-\Lambda$). This
correspond to choose the boundary conditions for the system restricted to
$\Lambda$. Such choice is totally free in the case of bounded spin system,
while, for consistency reasons, we will see in a moment that there are
limitations for such choice in the case of unbounded spin systems.

Once the boundary condition $\phi_{\Lambda^{c}}$ has been fixed outside
$\Lambda$, then, for any configuration $\phi_{\Lambda}$ inside $\Lambda$, the
Hamiltonian of the system is given by

$H^{\phi_{\Lambda^{c}}}(\phi_{\Lambda})~{}=~{}\sum_{\\{x,y\\}\cap\Lambda\neq\emptyset}\Psi(\\{x,y\\},\phi_{x},\phi_{y})$
$None$

The number $H(\phi_{\Lambda})$ represents the energy associated to the
configuration $\phi_{\Lambda}$ of the spin system. In order to make things
easier we will consider hereafter just free boundary condition (no influence
of the exterior on the system confined in the volume $\Lambda$). Namely we
will suppose that our Hamiltonian is

$H(\phi_{\Lambda})~{}=~{}\sum_{\\{x,y\\}\subset\Lambda}\Psi(\\{x,y\\},\phi_{x},\phi_{y})$
$None$

Now that we have defined the spin systems (and its single spin a priori
distribution), the space of configurations, the Hamiltonian (i.e. the energy
associated to each system configuration) and the boundary conditions we are
ready to study the statitic/thermodynamic behavior of the system. The last
step is to define a probability associate to each configuration
$\phi_{\Lambda}$ of the system in $\Lambda$. So we define the probability to
find the system in a given configuration $\\{\phi_{x}\\}_{x\in\Lambda}$ of
spins, with energy $H(\phi_{\Lambda})$ as

${\rm Prob}(\phi_{\Lambda})~{}=~{}{e^{-\beta
H(\phi_{\Lambda})}d\mu(\phi_{\Lambda})\over\int d\mu(\phi_{\Lambda})e^{-\beta
H}}$ $None$

the parameter $\beta^{-1}$ performs the link with thermodynamics and it is
interpreted as the temperature of the system. The constant
$Z_{\Lambda}(\beta)~{}=~{}\int d\mu(\phi_{\Lambda})e^{-\beta H}$ in (6.31) is
the partition function of the system

$Z_{\Lambda}(\beta)~{}=~{}\int
d\mu(\phi_{\Lambda})\exp\left[-\beta\sum_{\\{x,y\\}\subset\Lambda}\Psi(\\{x,y\\},\phi_{x},\phi_{y})\right]$
$None$

The partition function (6.32) is the key quantity in order to study
thermodynamic properties of the system. E.g. the thermodynamic free energy of
the system is defined as

$f(\beta)~{}=~{}\lim_{\Lambda\to\infty}{1\over|\Lambda|}\log
Z_{\Lambda}(\beta)$ $None$

where of course the limit $\Lambda\to\infty$ has to be interpreted in the
sense of Fisher or Van Hove or even in a more restrictive fashion, e.g. cubic
boxes with size $N\to\infty$.

In particular, the question about the analyticity of $f(\beta)$ as a function
of $\beta$ is very important. Accordingly to the physics belief: No phase
transition can occur in the system wherever $f(\beta)$ analytic. Concerning
this question, we could say that there is a sort of “meta-theorem” stating
that any reasonable spin system has free energy analytic if $\beta$ is
sufficiently small. Note that $\beta$ sufficiently small means substantially
that the system is weakly interacting or in other words spin are nearly
independent.

Now we will now show that $\log Z_{\Lambda}(\beta)$ can be written always in
term of a formal series expansion which converges absolutely if $\beta$ is
sufficiently small. Such an expansion is called the high temperature expansion
of the lattice system. We will also study directly (via cluster expansion
methods) this series expansion in the next section.

#####  6.2.2 Polymer representation of $Z_{\Lambda}(\beta)$ for the pair
potential case

Again, in order to make things easier we will consider just free boundary
condition. This means that the partition function of our spin system is given
by

$Z_{\Lambda}(\beta)~{}=~{}\int
d\mu(\phi_{\Lambda})\exp\left[-\beta\sum_{\\{x,y\\}\subset\Lambda}\Psi(\\{x,y\\},\phi_{x},\phi_{y})\right]$
$None$

Let us now use the “Mayer expansion trick” and rewrite our partition function
in the following way

$Z_{\Lambda}(\beta)~{}=~{}\int
d\mu(\phi_{\Lambda})\exp\left[-\beta\sum_{\\{x,y\\}\subset\Lambda}\Psi(\\{x,y\\},\phi_{x},\phi_{y})\right]~{}=~{}$
$~{}=~{}\int
d\mu(\phi_{\Lambda})\prod_{\\{x,y\\}\subset\Lambda}[(e^{-\beta\Psi(\\{x,y\\},\phi_{x},\phi_{y})})-1)+1]~{}=~{}$
$~{}=~{}1+\int d\mu(\phi_{\Lambda})\sum_{g\in{\cal
G}_{\Lambda}}\prod_{\\{x,y\\}\in
E_{g}}(e^{-\beta\Psi(\\{x,y\\},\phi_{x},\phi_{y})}-1)~{}=~{}$

where recall that ${\cal G}_{\Lambda}$ denotes the set of all graphs $g$ with
vertex set $\Lambda$ (either connected or not connected). Note also that the
graph with empty edge set has been separated and yields the contribution 1 in
equation above, since by definition $\int d\mu(\phi_{\Lambda})1~{}=~{}1$. As
we did in Section 3.1.2, we can group together connected components in
$\sum_{g\in{\cal G}_{\Lambda}}$ namely

$\sum_{g\in{\cal G}_{\Lambda}}~{}=~{}\sum_{\\{R_{1},\dots
R_{k}\\}\in\Lambda\atop R_{i}\cap R_{j}~{}=~{}\emptyset,~{}|R_{i}|\geq
2}\prod_{i~{}=~{}1}^{k}\sum_{g\in{{G}_{R_{i}}}}$

where $\\{R_{1},\dots R_{k}\\}$ is a non ordered $k$-ple of mutually non
intersecting subsets of $\Lambda$ with cardinality greater than 2. Note in
fact that the sets with cardinality 1 yields contribution equal to 1. Now,
recalling that $d\mu(\phi_{\Lambda})$ is a product measure, we get

$\int d\mu(\phi_{\Lambda})\sum_{g\in{\cal G}_{\Lambda}}\prod_{X\in
g}(e^{-\Psi(\\{x,y\\},\phi_{x},\phi_{y})}-1)~{}=~{}\sum_{\\{R_{1},\dots
R_{k}\\}\in\Lambda\atop R_{i}\cap R_{j}~{}=~{}\emptyset,~{}|R_{i}|\geq
2}\rho_{\beta}(R_{1})\dots\rho_{\beta}(R_{k})$

with

$\rho_{\beta}(R)~{}=~{}\int\prod_{x\in
R}d\mu(\phi_{x})\sum_{g\in{G}_{R}}\prod_{X\in
g}(e^{-\beta\Psi(\\{x,y\\},\phi_{x},\phi_{y})}-1)$

and evidently

$\sum_{{\\{R_{1},\dots R_{k}\\}\in\Lambda\atop R_{i}\cap
R_{j}=\emptyset}\atop|R_{i}|\geq
2}~{}~{}\rho_{\beta}(R_{1})\dots\rho_{\beta}(R_{k})=~{}~{}~{}~{}\sum_{k\geq
1}{1\over k!}\sum_{{R_{1},\dots,R_{k}\subset\Lambda\atop R_{i}\cap
R_{j}=\emptyset}\atop|R_{i}|\geq
2}~{}~{}~{}~{}\rho_{\beta}(R_{1})\dots\rho_{\beta}(R_{k})$

Namely, if we define for any pair of sets $R_{i},R_{j}$

$V(R_{i},R_{j})~{}=~{}\begin{cases}0&{\rm if}~{}R_{i}\cap
R_{j}~{}=~{}\emptyset\\\ +\infty&{\rm if}~{}R_{i}\cap
R_{j}\neq\emptyset\end{cases}$ $None$

We can rewrite the grand-canonical partition function (3.16) as

$Z_{\Lambda}(\beta)~{}=~{}1+\sum_{k\geq 1}{1\over
k!}\\!\sum_{R_{1},\dots,R_{k}\atop|R_{i}|\geq
2}\\!\rho_{\beta}(R_{1})\dots\rho_{\beta}(R_{k})e^{-\sum_{1\leq i<j\leq
n}V(R_{i},R_{j})}$ $None$

where

$\rho_{\beta}(R)~{}=~{}\int\prod_{x\in
R}d\mu(\phi_{x})\sum_{g\in{G}_{R}}\prod_{X\in
g}(e^{-\beta\Psi(\\{x,y\\},\phi_{x},\phi_{y})}-1)$ $None$

So we have shown that the partition function $Z_{\Lambda}(\beta)$ defined in
(6.21) can always be written as the grand canonical partition function of a
gas of “polymers” in which polymers are subsets $R$ of $\Lambda$ with
cardinality greater or equal 2, with activity $\rho_{\beta}(R)$ given by
(6.25), and with pair interaction given by (6.23), i.e., they cannot overlap.

Threfore, according to Theorem 4.2, whenever the condition

$\sup_{x\in Z^{d}}\sum\limits_{\gamma\in\mathbb{P}\atop x\in
R}|\rho_{\beta}(R)|~{}e^{a|R|}\leq e^{a}-1$ $None$

is satisfied, the free energy is analytic in $\beta$.

Note that the information about the original system is contained in the
structure of the activity $\rho_{\beta}(R)$ of a given polymer $R$. Note also
that $\rho_{\beta}(R)\to 0$ as $\beta\to 0$. Hence the activity of polymers is
small when $\beta$ is small. So it is reasonable to expect for convergence of
this polymer gas when the parameter $\beta$ is sufficiently small. This is the
reason why such an expansion is called the high temperature expansion of the
spin system.

#####  6.2.3 The bounded spin system case

Let us suppose that our spin sysytem is bounded, namely it holds that

$\|\phi_{x}\|\leq c<+\infty$ $None$

then we can state the following theorem.

###### Theorem 6.2

Consider a spin system at inverse temperature $\beta$ with partition function
$Z_{\Lambda}(\beta)$ given by (6.32), interacting via a pair potential
satisfying (6.26) and (6.27). Suppose that (6.39) holds and let $\beta$ such
that

$\beta\leq{0,058\over c^{2}J}$

Then $|\Lambda|^{-1}\log Z_{\Lambda}(\beta)$ is an absolutely convergent
series uniformly in $\Lambda$.

Remark. (6.26) and (6.27) are satisfied for pratically all studied bounded
spin system (e.g. Ising-type models, Potts model, Heisenberg model).

Proof. All we need to do in order to prove the theorem is to check condition
(6.38). First note that (6.38) is satisfied if

$\sum_{n=2}^{\infty}e^{an}\sup_{x\in\mathbb{Z}^{d}}\sum\limits_{R:\in\mathbb{P}\atop|R|=n,\;x\in
R}|\rho_{\beta}(R)|\leq e^{a}-1$ $None$

For that we need need an upper bound on the modulous of the activity
$\rho_{\beta}(R)$ given in (6.25) when $|R|=n$. To bound $|\rho_{\beta}(R)|$
we use Theorem 3.7 and in particular the tree graph inequality (3.69).
Supposing that $R$ with cardinality $|R|=n$ is the unordered set
$\\{x_{i},\dots,x_{n}\\}\subset\mathbb{Z}^{d}$ and setting, for
$\\{x_{i},x_{j}\\}\subset R$,

$\Psi(\\{x_{i},x_{y}\\},\phi_{x_{i}},\phi_{x_{j}})=\Psi_{ij}$

we have

$\displaystyle|\rho_{\beta}(R)|$ $\displaystyle=\Bigg{|}\int\prod_{x\in
R}d\mu(\phi_{x})\sum_{g\in{G}_{R}}\prod_{X\in
g}(e^{-\beta\Psi(\\{x,y\\},\phi_{x},\phi_{y})}-1)\Bigg{|}$
$\displaystyle=\bigg{|}\int\prod_{i=1}^{n}d\mu(\phi_{x_{i}})\sum_{g\in
G_{n}}\prod_{\\{i,j\\}\in E_{g}}(e^{-\beta\Psi_{ij}}-1)\bigg{|}$
$\displaystyle\leq\int\prod_{i=1}^{n}d\mu(\phi_{x_{i}})\bigg{|}\sum_{g\in
G_{n}}\prod_{\\{i,j\\}\in E_{g}}(e^{-\beta\Psi_{ij}}-1)\bigg{|}$

Now observe that in force of (6.28) and (6.39) we have that

$\sum_{\\{x,y\\}\subset R}\Psi(\\{x,y\\},\phi_{x},\phi_{y})\geq-{c^{2}J\over
2}|R|$

i.e., if we pose $R=\\{x_{i},\dots,x_{n}\\}$ and
$\Psi(\\{x_{i},x_{y}\\},\phi_{x_{i}},\phi_{x_{j}})=\Psi_{ij}$, we have that
the real numbers $\\{\Psi_{ij}\\}_{\\{i,j\\}\in E_{n}}$ are such that

$|\Psi_{ij}|\leq c^{2}J_{x_{i}x_{j}}$

and

$\sum_{1\leq i<j\leq n}\Psi_{ij}\geq-{c^{2}J\over 2}n$

Therefore, according to Theorem 3.7

$\bigg{|}\sum_{\tau\in T_{n}}\prod_{\\{i,j\\}\in
E_{g}}(e^{-\beta\Psi_{ij}}-1)\bigg{|}\leq e^{{c^{2}\beta J\over
2}n}\sum_{\tau\in T_{n}}\prod_{\\{i,j\\}\in
E_{\tau}}(1-e^{-\beta|\Psi_{ij}|})$

Therefore, since $\int d\mu(\phi_{x_{i}})=1$, we have

$\displaystyle|\rho_{\beta}(R)|$ $\displaystyle\leq e^{{c^{2}\beta J\over
2}n}\sum_{\tau\in T_{n}}\prod_{\\{i,j\\}\in E_{\tau}}(1-e^{-\beta
c^{2}J_{x_{i}x_{j}}})$ $\displaystyle\leq e^{{c^{2}\beta J\over 2}n}(\beta
c^{2})^{n-1}\sum_{\tau\in T_{n}}\prod_{\\{i,j\\}\in E_{\tau}}J_{x_{i}x_{j}}$

and thus

$\displaystyle\sup_{x\in\mathbb{Z}^{d}}\sum\limits_{R:\in\mathbb{P}\atop|R|=n,\;x\in
R}|\rho_{\beta}(R)|$ $\displaystyle\leq e^{{c^{2}\beta J\over 2}n}(\beta
c^{2})^{n-1}\sup_{x\in\mathbb{Z}^{d}}\sum\limits_{\\{x_{1},\dots,x_{n}\\}\subset\mathbb{Z}^{d}\atop
x_{1}=x}\sum_{\tau\in T_{n}}\prod_{\\{i,j\\}\in E_{\tau}}J_{x_{i}x_{j}}$
$\displaystyle=e^{{c^{2}\beta J\over 2}n}(\beta
c^{2})^{n-1}\sup_{x\in\mathbb{Z}^{d}}{1\over(n-1)!}\sum_{(x_{i},\dots,x_{n})\in\mathbb{Z}^{dn}\atop
x_{1}=x,\,x_{i}\neq x_{j}}\sum_{\tau\in T_{n}}\prod_{\\{i,j\\}\in
E_{\tau}}J_{x_{i}x_{j}}$ $\displaystyle\leq e^{{c^{2}\beta J\over 2}n}{(\beta
c^{2})^{n-1}\over(n-1)!}\sum_{\tau\in
T_{n}}\sup_{x\in\mathbb{Z}^{d}}\sum_{(x_{i},\dots,x_{n})\in\mathbb{Z}^{dn}\atop
x_{1}=x,\,x_{i}\neq x_{j}}\prod_{\\{i,j\\}\in E_{\tau}}J_{x_{i}x_{j}}$

and since (recall Proposition 3.1 and (6.27)), for any $\tau\in T_{n}$

$\sup_{x\in\mathbb{Z}^{d}}\sum_{(x_{i},\dots,x_{n})\in\mathbb{Z}^{dn}\atop
x_{1}=x,\,x_{i}\neq x_{j}}\prod_{\\{i,j\\}\in E_{\tau}}J_{x_{i}x_{j}}\leq
J^{n-1}$ $None$

we get in the end that (recall: $\sum_{\tau\in T_{n}}1=n^{n-2}$)

$\sup_{x\in\mathbb{Z}^{d}}\sum\limits_{R:\in\mathbb{P}\atop|R|=n,\;x\in
R}|\rho_{\beta}(R)|\leq e^{{c^{2}\beta J\over 2}n}(c^{2}\beta
J)^{n-1}{n^{n-2}\over(n-1)!}$

Therefore condition (6.40) is satisfied if (as we did earlier, we choose for
simplicity $a=\ln 2$ and we bound ${n^{n-2}\over(n-1)!}\leq e^{n-1}$)

$\sum_{n=2}^{\infty}\Big{(}2e^{{c^{2}\beta J\over 2}}\Big{)}^{n}(c^{2}e\beta
J)^{n-1}\leq 1$

i.e., setting $x=c^{2}\beta J$

$\sum_{n=2}^{\infty}\Big{(}2e^{{x\over 2}}\Big{)}^{n}(ex)^{n-1}\leq 1$

i.e.

$2e^{{x\over 2}}\sum_{n=2}^{\infty}\Big{(}2e^{{x\over 2}+1}x\Big{)}^{n-1}\leq
1$

i.e

${4e^{{x}+1}x\over 1-2e^{{x\over 2}+1}x}\leq 1$

which is satisfied if $x\leq 0,058$.

In conclusion (6.40) is satisfied for all $\beta$ such that

$\beta J\leq{0,058\over c^{2}}$

which ends the proof of Theorem 6.2. $\Box$

#####  6.2.4 The unbounded spin system case

Concerning unbounded spin system we will prove the following theorem

###### Theorem 6.3

Consider a spin system at inverse temperature $\beta$ with partition function
$Z_{\Lambda}(\beta)$ given by (6.32), interacting via a pair potential
satisfying (6.26) and (6.27). Suppose that the norm
$\|\cdot\|:\Omega\to[0,+\infty)$ is a function such that, for $n\in\mathbb{N}$

$\int e^{+{\beta J\over 2}\|\phi\|^{2}}\|\phi\|^{n}d\mu(\phi)\leq n![C(\beta
J)]^{n}$ $None$

and let $\beta$ such that

$\beta JC^{2}(\beta J)\leq{e-1\over 4e^{2}}$ $None$

Then $|\Lambda|^{-1}\log Z_{\Lambda}(\beta)$ is an absolutely convergent
series uniformly in $\Lambda$.

Proof.

Again, we just need to check the condition (6.38). Thus, given
$R\subset\mathbb{Z}^{d}$ such that $|R|\geq 2$, let us first estimate the
modulus of the polymer activity is given by (6.25). We have as before

$|\rho_{\beta}(R)|~{}\leq~{}\int\prod_{x\in R}d\mu(\phi_{x})\Big{|}\sum_{g\in
G_{R}}\prod_{\\{x,y\\}\in
g}(e^{-\beta\Psi(\\{x,y\\},\phi_{x},\phi_{y})}-1)\Big{|}$

As before let $R$ such that $|R|=n$ ; so $R$ is the unordered set
$\\{x_{i},\dots,x_{n}\\}\subset\mathbb{Z}^{d}$ and let set, for
$\\{x_{i},x_{j}\\}\subset R$,

$\Psi(\\{x_{i},x_{y}\\},\phi_{x_{i}},\phi_{x_{j}})=\Psi_{ij}$

Thus, as before, we have

$|\rho_{\beta}(R)|\leq\int\prod_{i=1}^{n}d\mu(\phi_{x_{i}})\bigg{|}\sum_{g\in
G_{n}}\prod_{\\{i,j\\}\in E_{g}}(e^{-\beta\Psi_{ij}}-1)\bigg{|}$

Now, by (6.26) and (6.28), with $R=\\{x_{1},\dots,x_{n}\\}$ and
$\Psi(\\{x_{i},x_{y}\\},\phi_{x_{i}},\phi_{x_{j}})=\Psi_{ij}$, we have that
the real numbers $\\{\Psi_{ij}\\}_{\\{i,j\\}\in E_{n}}$ are such that

$|\Psi_{ij}|\leq J_{x_{i}x_{j}}\|\phi_{x_{i}}\|\|\phi_{x_{j}}\|$

and, for any $S\subset[n]$,

$\sum_{\\{i,j\\}\subset S}\Psi_{ij}\geq-{J\over 2}\sum_{i\in
S}\|\phi_{x_{i}}\|^{2}$

Therefore, according to Theorem 3.7

$\bigg{|}\sum_{g\in G_{n}}\prod_{\\{i,j\\}\in
E_{g}}(e^{-\beta\Psi_{ij}}-1)\bigg{|}\leq e^{{\beta J\over
2}\sum_{i=1}^{n}\|\phi_{x_{i}}\|^{2}}\sum_{\tau\in T_{n}}\prod_{\\{i,j\\}\in
E_{\tau}}(1-e^{-\beta J_{x_{i}x_{j}}\|\phi_{x_{i}}\|\|\phi_{x_{j}}\|})$

Now we get

$\displaystyle|\rho_{\beta}(R)|$
$\displaystyle\leq\int\prod_{i=1}^{n}d\mu(\phi_{x_{i}})e^{{\beta J\over
2}\|\phi_{x_{i}}\|^{2}}\sum_{\tau\in T_{n}}\prod_{\\{i,j\\}\in
E_{\tau}}(1-e^{-\beta J_{x_{i}x_{j}}\|\phi_{x_{i}}\|\|\phi_{x_{j}}\|})$
$\displaystyle\leq\beta^{n-1}\sum_{\tau\in
T_{n}}\int\prod_{i=1}^{n}d\mu(\phi_{x_{i}})e^{{\beta J\over
2}\|\phi_{x_{i}}\|^{2}}\prod_{\\{i,j\\}\in
E_{\tau}}J_{x_{i}x_{j}}\|\phi_{x_{i}}\|\|\phi_{x_{j}}\|$

Then observe that if $\tau\in T_{n}$ has degrees $d_{1},\dots d_{n}$ at
vertices $1,2,\dots,n$ respectively, then

$\prod_{\\{i,j\\}\in
E_{\tau}}\|\phi_{x_{i}}\|\|\phi_{x_{j}}\|=\prod_{i=1}^{n}\|\phi_{i}\|^{d_{i}}$

and thus

$|\rho_{\beta}(R)|\leq b^{n-1}\\!\\!\\!\\!\sum_{d_{1},\dots,d_{n}:\;d_{i}\geq
1\atop d_{1}+\dots+d_{n}=2n-2}\prod_{i=1}^{n}\int d\mu(\phi_{x_{i}})e^{{\beta
J\over
2}\|\phi_{x_{i}}\|^{2}}\|\phi_{x_{i}}\|^{d_{i}}\\!\\!\\!\\!\sum_{\tau\in
T_{n}\atop d_{1},\dots,d_{n}~{}{\rm fixed}}\prod_{\\{i,j\\}\in
E_{\tau}}J_{x_{i}x_{j}}$

whence, using again (6.41), and recalling also Cayley formula (3.5), we get

$\displaystyle\sup_{x\in\mathbb{Z}^{d}}\sum\limits_{R:\in\mathbb{P}\atop|R|=n,\;x\in
R}|\rho_{\beta}(R)|$ $\displaystyle\leq{(\beta J)^{n-1}\over
n-1}\sum_{d_{1},\dots,d_{n}:\;d_{i}\geq 1\atop
d_{1}+\dots+d_{n}=2n-2}\prod_{i=1}^{n}\int
d\mu(\phi_{x_{i}}){\|\phi_{x_{i}}\|^{d_{i}}e^{{\beta J\over
2}\|\phi_{x_{i}}\|^{2}}\over(d_{1}-1)!}$ $\displaystyle\leq{(\beta
J)^{n-1}\over n-1}\sum_{d_{1},\dots,d_{n}:\;d_{i}\geq 1\atop
d_{1}+\dots+d_{n}=2n-2}\prod_{i=1}^{n}d_{i}C(\beta J)^{d_{i}}$
$\displaystyle\leq{(\beta J)^{n-1}\over n-1}[C(\beta
J)]^{2n-2}\sum_{d_{1},\dots,d_{n}:\;d_{i}\geq 1\atop
d_{1}+\dots+d_{n}=2n-2}\prod_{i=1}^{n}d_{i}$

where in the second inequality we have used hypothesis (6.42). Observe now
that

$\displaystyle\sum_{d_{1},\dots,d_{n}:\;d_{i}\geq 1\atop
d_{1}+\dots+d_{n}=2n-2}\prod_{i=1}^{n}d_{i}$
$\displaystyle=\sum_{s_{1},\dots,s_{n}:\;s_{i}\geq 0\atop
s_{1}+\dots+s_{n}=n-2}\prod_{i=1}^{n}(1+s_{i})$
$\displaystyle\leq\sum_{s_{1},\dots,s_{n}:\;s_{i}\geq 0\atop
s_{1}+\dots+s_{n}=n-2}\prod_{i=1}^{n}e^{s_{i}}$
$\displaystyle=e^{n-2}\sum_{d_{1},\dots,d_{n}:\;d_{i}\geq 1\atop
d_{1}+\dots+d_{n}=2n-2}1$ $\displaystyle=e^{n-2}{2n-3\choose n-1}$
$\displaystyle\leq(4e)^{n-1}$

Therefore we get

$\sup_{x\in\mathbb{Z}^{d}}\sum\limits_{R:\in\mathbb{P}\atop|R|=n,\;x\in
R}|\rho_{\beta}(R)|\leq{(4e\beta J)^{n-1}\over n-1}[C(\beta J)]^{2n-2}$

In conclusion we have that condition (6.38) is satisfied if (as we did
earlier, we choose for simplicity $a=\ln 2$)

$\sum_{n=2}^{\infty}{(4e\beta JC^{2}(\beta J))^{n-1}\over n-1}\leq 1$

i.e. if

$-\ln[1-4e\beta JC^{2}(\beta J)]\leq 1$

i.e. if

$\beta JC^{2}(\beta J)\leq{e-1\over 4e^{2}}$

This concludes the proof of Theorem 6.3. $\Box$

####  6.3 Discrete Spin systems with many-body interactions

We have found a general condition, the equation (4.82), that guarantees the
absolute convergence uniformly in $\Lambda$ of $\Sigma^{\Lambda}_{x}$, and
hence of $\log\Xi_{\Lambda}(\rho)$. The ways we found so far to check that the
polymer activities satisfy (4.82) may be used only for pair interaction. We
have now to face a new problem: starting from the structure of the polymer
activity $\rho_{\gamma}$ in the case of general many body interactions, which
is given by

$\rho_{\gamma}~{}=~{}\int\prod_{x\in\gamma}d\mu(\phi_{x})\sum_{g\in{G}_{\gamma}}\prod_{X\in
g}(e^{-\beta\Psi(X,\phi_{X})}-1)$ $None$

we want to give general criteria in order to satisfy the condition (4.82),
that in this case take the form

$\sum\limits_{\gamma\in\mathbb{P}_{\mathbb{V}}\atop
x\in\gamma}|\rho_{\gamma}|~{}e^{a|\gamma|}\leq e^{a}-1$ $None$

Here we want to present a simple technical device, introduced in 1981 by M.
Cassandro and E. Olivieri, which exploits the fact that for bounded spin
systems on the lattice the sum over the Mayer graphs can be directly bounded,
for $N$ body potentials absolutely summable in an appropriate sense. This
method has two main advantages: it is relatively simple, since it avoids the
combinatorial complexity arising in the tree identities presented above, and
it may be applied to a large class of discrete systems. In particular, for
systems with many-body interactions it is difficult to obtain better estimates
of the interactions $\beta\Psi(X,\phi_{X})$ for which (6.45) is satisfied.

The method can be also applied to systems with two body interactions but the
results are obviously far from optimal, as we shall see below.

######  Theorem 6.4 (Cassandro, Olivieri)

Consider the polymer activity (6.44). If it is possible to give for any
connected graph $g$ such that ${\rm supp}\ g=n$ the following bound

$\int\prod_{x\in{\rm supp}\ g}d\mu(\phi_{x})\prod_{X\in
g}\left|e^{-\beta\Psi(X,\phi)}-1\right|\leq{\bar{\sigma}}^{n}\prod_{X\in
g}W(X)$ $None$

with

$\sum_{X\ni x}W(X)\leq K,\qquad\qquad{\bar{\sigma}}<1$ $None$

and the following relation between ${\bar{\sigma}}$ and $K$ holds

$K<{|\log\bar{\sigma}|\over 4}$ $None$

then the condition (6.45) is satisfied for some $a>0$.

Remark

It is important, in order to be able to apply this theorem to concrete models,
to understand the structure of the estimate needed in order to satisfy (6.45).
Given a graph, the contribution of that graph to the polymer activity have to
be bounded by the product of a vertex contribution $\bar{\sigma}<1$, meaning
that the bigger ${\rm supp}\ g$ the smaller its contribution, and an edge
contribution $J(X)$, whose estimate has to be suitably related to
$\bar{\sigma}$.

Proof.

We will bound directly l.h.s. of (6.45)

$\sum_{\gamma\ni x}|\rho_{\gamma}|e^{a|\gamma|}=\sum_{\gamma\ni
x}e^{a|\gamma|}\left|\int\prod_{x\in\gamma}d\mu(\phi_{x})\sum_{g\in{\cal
G}_{\gamma}}\prod_{X\in g}[e^{-\beta\Psi(X,\phi)}-1]\right|\leq$
$\leq\sum_{\gamma\ni x}\sum_{g\in{\cal
G}_{\gamma}}{\bar{\sigma}}^{|\gamma|}e^{a|\gamma|}\prod_{X\in
g}W(X)\leq\sum_{\gamma\ni x}\sum_{g\in{\cal
G}_{\gamma}}{\sigma}^{|\gamma|}\prod_{X\in g}W(X)$ $None$

Hence the theorem follows if we are able to show the inequality

$F(\sigma)=\sum_{\gamma\ni x}\sum_{g\in{\cal
G}_{\gamma}}{\sigma}^{|\gamma|}\prod_{X\in g}W(X)<e^{a}-1$ $None$

where

$\sigma={\bar{\sigma}}e^{a}$

We now rewrite the sum over connected graphs $g$ passing through the lattice
point $x$ using Cassandro- Olivieri hierarchy. The latter is constructed
observing that, given a connected graph $g$, its links can be always ordered,
for some positive integer $t$, in the following way

$g=\\{C_{0},C^{1}_{1},\dots,C^{1}_{k_{1}},\dots,C^{t}_{1},\dots,C^{t}_{k_{t}}\\}$

where $C_{0}$ is a link of $g$ such that $x\in C_{0}$. For $1\leq s\leq t$,
the links $C^{s}_{1},\dots,C^{s}_{k_{s}}$ represent the $s^{\rm th}$ hierarchy
and we have the following relations between different hierarchies:

$C^{s}_{i}\cap\left[\cup_{i=1}^{k_{s-1}}C^{s-1}_{i}\right]\neq\emptyset$
$C^{s}_{i}\cap\left[\cup_{l=0}^{s-2}\cup_{i=1}^{k_{l}}C^{l}_{i}\right]=\emptyset$
$None$

Thus the first hierarchy $C^{1}_{1},\dots,C^{1}_{k_{1}}$ of links of $g$ is
the collection of all links of $g$ which have a non empty intersection with
$C_{0}$; the second hierarchy $C^{2}_{1},\dots,C^{2}_{k_{2}}$ is the
collection of links of $g$ which have a non empty intersection with
$\cup_{i}C^{1}_{i}$, but has an empty intersection with the set $C_{0}$, and
so on.

We will also denote, for $s\geq 1$

$\Delta_{s}=\cup_{i=1}^{k_{s}}C^{s}_{i}\backslash\left[\left(\cup_{i=1}^{k_{s}}C^{s}_{i}\right)\cap\left(\cup_{i=1}^{k_{s-1}}C^{s-1}_{i}\right)\right]$
$None$

while $\Delta_{0}=C_{0}$. The set $\Delta_{s}$ represents therefore the set of
the new points reached by the $s$-th hierarchy. By definition $|\Delta_{s}|>0$
for $s<t$, because the bonds of the $(s+1)^{\rm th}$ hierarchy have
intersection only with the set $\Delta_{s}$, else they would belong to an
earlier hierarchy. Note that $\\{\Delta_{0},\Delta_{1},\dots,\Delta_{t}\\}$ is
a partition of the set ${\rm supp}\ g$ and thus

$|\Delta_{0}|+|\Delta_{1}|+\cdots+|\Delta_{t}|=|{\rm supp}\ g|$ $None$

thus, for a given
$g=\\{C_{0},C^{1}_{1},\dots,C^{1}_{k_{1}},\dots,C^{t}_{1},\dots,C^{t}_{k_{t}}\\}$,
we can always write

$\sigma^{|{\rm supp}\
g|}\leq\sigma^{|\Delta_{0}|}\sigma^{|\Delta_{1}|}\cdots\sigma^{|\Delta_{t-1}|}$

Hence we can get the estimate

$F(\sigma)\leq\sum_{C_{0}\ni 0}W(C_{0})\sigma^{|C_{0}|}+\sum_{C_{0}\ni
0}W(C_{0})\sigma^{|\Delta_{0}|}\sum_{t=1}^{\infty}\sum_{k_{1}=1}^{\infty}\sum_{C^{1}_{1},\dots
C^{1}_{k_{1}}}\prod_{j=1}^{k_{1}}W(C^{1}_{j})\sigma^{|\Delta_{1}|}\cdots$
$\cdots\sum_{k_{t-1}=1}^{\infty}\sum_{C^{t-1}_{1},\dots
C^{t-1}_{k_{t-1}}}\prod_{j=1}^{k_{t-1}}W(C^{t-1}_{j})\sigma^{|\Delta_{t-1}|}\sum_{k_{t}=1}^{\infty}\sum_{C^{t}_{1},\dots
C^{t}_{k_{t}}}\prod_{j=1}^{k_{t}}W(C^{t}_{j})$ $None$

Let us start to perform the last sum in r.h.s. of (6.54), i.e. the sum over
the $t^{\rm th}$ hierarchy, supposing to keep fixed all sets concerning the
previous hierarchies. We obtain

$\displaystyle\sum_{k_{t}=1}^{\infty}\sum_{C^{t}_{1},\dots
C^{t}_{k_{t}}}\prod_{j=1}^{k_{t}}W(C^{t}_{j})$
$\displaystyle\leq\sum_{k_{t}=1}^{\infty}{1\over
k_{t}!}\left(\sum_{C:\,C\cap\Delta_{t-1}\neq\emptyset}W(C)\right)^{k_{t}}$
$\displaystyle\leq\sum_{k_{t}=1}^{\infty}{1\over
k_{t}!}\left(|\Delta_{t-1}|K\right)^{k_{t}}$ $\displaystyle\leq
e^{K|\Delta_{t-1}|}-1$

where the sum $\sum_{C:\,C\cap\Delta_{t-1}\neq\emptyset}W(C)$ is bounded by
fixing one point in $\Delta_{t-1}$ (and this gives the factor
$|\Delta_{t-1}|$) and then by summing over all $C$ passing for such point.
Hence

$F(\sigma)\leq\sum_{C_{0}\ni 0}W(C_{0})\sigma^{|C_{0}|}+\sum_{C_{0}\ni
0}W(C_{0})\sigma^{|\Delta_{0}|}\sum_{t=1}^{\infty}\sum_{k_{1}=1}^{\infty}\sum_{C^{1}_{1},\dots
C^{1}_{k_{1}}}\prod_{j=1}^{k_{1}}W(C^{1}_{j})\sigma^{|\Delta_{1}|}\cdots$
$\cdots\sum_{k_{t-1}=1}^{\infty}\sum_{C^{t-1}_{1},\dots
C^{t-1}_{k_{t-1}}}\prod_{j=1}^{k_{t-1}}W(C^{t-1}_{j})\sigma^{|\Delta_{t-1}|}(e^{K|\Delta_{t-1}|}-1)$

Note now that $e^{K}\sigma<1$ certainly holds if (6.48) does and $a$ is
choosen small enough, namely $a<(3/4)|\log{\bar{\sigma}}|$. To find a bound of
the maximum of the factor $\sigma^{|\Delta_{t-1}|}(e^{K|\Delta_{t-1}|}-1)$ it
is convenient to write

$\sigma^{|\Delta_{t-1}|}(e^{K|\Delta_{t-1}|}-1)=e^{(-|\log\sigma|+K)|\Delta_{t-1}|)}(1-e^{-K|\Delta_{t-1}|})$

The maximum in $|\Delta_{t-1}|$ of this function is assumed for
$|\Delta_{t-1}|=x$ such that $e^{-Kx}={|\log\sigma|-k\over|\log\sigma|}$, and
using $e^{K}\sigma<1$ and hence $e^{(-|\log\sigma|+K)|\Delta_{t-1}|)}<1$, this
gives easily

$\sigma^{|\Delta_{t-1}|}(e^{K|\Delta_{t-1}|}-1)\leq{K\over|\log\sigma|}$

We sum now the next hierarchy $t-1$ as before keeping fixed everything
concerning the previous hierarchies and obtain

$\sum_{k_{t-1}=1}^{\infty}\sum_{C^{t-1}_{1},\dots
C^{t-1}_{k_{t}}}\prod_{j=1}^{k_{t-1}}W(C^{t-1}_{j})\sigma^{|\Delta_{t-1}|}(e^{K|\Delta_{t-1}|}-1)$
$\leq{K\over|\log\sigma|}\sum_{k_{t-1}=1}^{\infty}{1\over
k_{t-1}!}\left(\sum_{C:\,C\cap\Delta_{t-2}\neq\emptyset}W(C)\right)^{k_{t}}\leq{K\over|\log\sigma|}(e^{K|\Delta_{t-2}|}-1)$

Iterating until $t=1$, we obtain

$F(\sigma)\leq\sum_{C_{0}\ni 0}W(C_{0})\sigma^{|C_{0}|}+\sum_{C_{0}\ni
0}W(C_{0})\sum_{t=1}^{\infty}\left({K\over|\log\sigma|}\right)^{t}\leq
K\left[\sigma+{K\over|\log\sigma|-K}\right]$

hence we get (6.45), and then the proof of the theorem, if the following
condition is satisfied

$K\left[\sigma+{K\over|\log\sigma|-K}\right]<e^{a}-1$ $None$

The relation (6.55) give a condition between $K$, $a$ and $\bar{\sigma}$. If
for instance one suppose to know the value of $\bar{\sigma}$, the problem is
to find the value of $a$ which maximizes the constant $K$. This problem can be
solved numerically. A rough estimate, once the value of $\bar{\sigma}$ is
fixed, is to choose e.g. $a=(1/2)|\log{\bar{\sigma}}|$, which corresponds to
the choice $\sigma=\sqrt{\bar{\sigma}}$ (i.e. $|\log\sigma|={1\over
2}|\log{\bar{\sigma}}|$), obtaining easily

$K<{|\log\bar{\sigma}|\over 4}$ $None$

$\Box$

####  6.4 Some applications of theorem 6.4

#####  6.4.1 The Ising model

A rough but fast estimate of the analyticity region in $\beta$ (high
temperature regime) of the partition function of the Ising model can be
obtained using theorem 6.4.

In order to apply theorem 6.4 to the partition function (6.7) we have to find
$\bar{\sigma}$ and $J$ such that

$\int\prod_{x\in{\rm supp}\ g}d\mu(\sigma_{x})\prod_{\\{x,y\\}\in g\atop
x,y~{}{\rm near.~{}neigh.}}\left|e^{\beta
J\sigma_{x}\sigma_{y}}-1\right|\leq{\bar{\sigma}}^{n}\prod_{\\{x,y\\}\in g}W$
$None$

This can be done observing that, for $\beta<1$, $\left|e^{\beta
J\sigma_{x}\sigma_{y}}-1\right|<2\beta J$. The l.h.s. of (6.57) can be
estimated then as $(2\beta J)^{|E_{g}|}$. Since $|E_{g}|\geq n-1$ and $n\geq
2$, we can choose $\bar{\sigma}=(\beta J)^{1/3}$, $J=2(\beta J)^{1/3}$ and
then, having each site $2d$ nearest neighbors, $K=4d(\beta J)^{1/3}$. The
condition (6.48) becomes then

$4d(\beta J)^{1/3}<{|\log(\beta J)|\over 12}\Rightarrow{(\beta
J)^{1/3}\over|\log(\beta J)|}<{1\over 48d}$ $None$

which is satisfied for $(\beta J)<1$ small enough. By a rapid computation
(6.58) is satisfied by $\beta\approx 4\times 10^{-5}$. However, we will see in
the next chapter that this estimate is far from optimal.

#####  6.4.2 Direct proof of the Israel theorem

Discrete spin systems with many body interactions have been studied since the
early ’70. After the pioneering work by Gallavotti and Miracle-Sole, which
proved analiticity of the logarithm of the partition function when the spin
space state has only a finite number of values, Israel proved in [I] a general
condition, valid for all compact spins, to guarantee that uniformly in the
volume the partition function stays different from zero, and hence indirectly
the analyticity of its logarithm. This analyticity result can be found easily
and directly with a proof based only on combinatorial arguments by means of
theorem 6.4.

Consider a lattice system described by the partition function (6.44) with
many-body potentials such that

$\beta|\Psi(X,\phi_{X})|\leq w(X),\quad{\rm\ for\ all}\ X$ $None$

with $w(X)$ positive function defined on the sets of $Z^{d}$ such that, for
some $a>0$

$\sum_{X\ni 0}w(X)e^{a|X|}=I(a)<\infty$ $None$

which is a condition analogous to [I]. We also define

${\bar{I}}=\sup_{X\subset\Lambda}w(X)$

Then we have

###### Theorem 6.5

The logarithm of the partition function (6.44) of a lattice system with $\Psi$
satisfying (6.59) and (6.60), converges absolutely uniformely in the volume,
provided the following relation between $I(a)$ and $a$ is satisfied

$I(a)<{e^{-{\bar{I}}}\over 4}\ a$ $None$

Proof.

For any $\gamma\subset Z^{d}$ and for any graph $g\in{\cal G}_{\gamma}$ we
have

$\displaystyle\left|\int\prod_{x\in\gamma}d\mu(\phi_{x})\prod_{X\in
g}[e^{-\beta\Psi(X,\phi)}-1]\right|$ $\displaystyle\leq\prod_{X\in
g}w(X)e^{w(X)}$ $\displaystyle\leq e^{-a|R|}\prod_{X\in
g}e^{a|X|}w(X)e^{\bar{I}}$

Hence we can apply theorem 6.4 posing $\bar{\sigma}=e^{-a}$ and
$W(X)=e^{a|X|}w(X)e^{\bar{I}}$. $\Box$

#####  6.4.3 The lattice gas with $N$ body interactions

The lattice gas interacting through many body interaction can be also treated
using theorem 6.4. We will prove the following theorem.

###### Theorem 6.6

Consider a lattice gas at temperature $\beta^{-1}$ and activity $z$, described
by the grand canonical partition function

$\Xi_{\Lambda}(\beta,z)=1+z|\Lambda|+\sum_{n\geq
2}z^{n}\sum_{X\subset\Lambda\atop|X|=n}e^{-\beta\sum_{Y\subset X\atop|Y|\geq
2}V(Y)}$ $None$

with

$\sum_{X\ni 0}|V(X)|=J<\infty,$ $None$

then $|\Lambda|^{-1}\log\Xi_{\Lambda}(\beta,z)$ can be written as a series
absolutely convergent uniformely in $\Lambda$, for all $\beta$ and $z$
satisfying the following condition

$z(\exp[4\beta Je^{\beta J}]-1)<1$ $None$

Proof.

We can trasform $\Xi_{\Lambda}(\beta,z)$ in a form suitable for polymer
expansion. We can write

$\Xi_{\Lambda}(\beta,z)=\sum_{\sigma_{\Lambda}}z^{\sum_{x\in\Lambda}\sigma_{x}}\exp\\{-\beta\sum_{X\subset\Lambda\atop|X|\geq
2}V(X)\prod_{x\in X}\sigma_{x}\\}$ $None$

where the ”spin” variables $\sigma_{x}$ at the site $x\in\Lambda$ can take the
two values $\sigma_{x}=0,1$, and $\sum_{\sigma_{\Lambda}}$ means the sum over
all possible configurations of the valus of spins in $\Lambda$. Then, by Mayer
expansion on the $\exp$

$\Xi_{\Lambda}(\beta,z)=\sum_{\gamma_{1},\gamma_{2},\dots,\gamma_{k}\in\pi(\Lambda)}\rho_{\gamma_{1}}\rho_{\gamma_{2}}\cdots\rho_{\gamma_{k}}$

where $\pi(\Lambda)$ is the set of the partitions of $\Lambda$ and

$\rho_{\gamma}=\begin{cases}1+z&if|\gamma|=1\\\
\sum_{\sigma_{\gamma}}z^{\sum_{x\in\gamma}\sigma_{x}}\left[\sum_{g\in\cal
G_{\gamma}}\prod_{X\in g}\left(e^{-\beta V(X)\prod_{x\in
X}\sigma_{x}}-1\right)\right]&if|\gamma|\geq 2\end{cases}$

Now note that, for any $g\in\cal G_{\gamma}$,

$\prod_{X\in g}\left(e^{-\beta V(X)\prod_{x\in
X}\sigma_{x}}-1\right)=\begin{cases}\prod_{X\in g}\left(e^{-\beta
V(X)}-1\right)&if\sigma_{x}=1\,\,\forall x\in\gamma\\\ 0&otherwise\end{cases}$

hence defining

$\zeta(\gamma)=\begin{cases}1&if|\gamma|=1\\\ \left({z\over
1+z}\right)^{|\gamma|}\left[\sum_{g\in\cal G_{\gamma}}\prod_{X\in
g}\left(e^{-\beta V(X)}-1\right)\right]&if|\gamma|\geq 2\end{cases}$

we can write

$\Xi_{\Lambda}(\beta,z)=(1+z)^{|\Lambda|}\sum_{\\{\gamma_{1},\dots,\gamma_{k}\\}\atop|\gamma_{i}|\geq
2\,\gamma_{i}\cap\gamma_{j}=\emptyset}\zeta(\gamma_{1})\zeta(\gamma_{2})\cdots\zeta(\gamma_{k})$

and thus also

$\log\Xi_{\Lambda}(\beta,z)=|\Lambda|\log(1+z)+\sum_{n\geq 1}{1\over
n!}\sum_{\gamma_{1},\dots,\gamma_{k}\atop|\gamma_{i}|\geq
2}\phi^{T}(\gamma_{1},\dots,\gamma_{k})\zeta(\gamma_{1})\zeta(\gamma_{2})\cdots\zeta(\gamma_{k})$

Hence the lattice gas pressure is analytic when the infinite sum in r.h.s. of
equation above is absolutely convergent uniformely in $\Lambda$. We have

$\left|\left({z\over 1+z}\right)^{n}\prod_{X\in g}[e^{-\beta
V(X)}-1]\right|\leq$ $\left({z\over 1+z}\right)^{n}\prod_{X\in
g}\beta|V(X)|e^{\beta J}\leq\left({z\over 1+z}\right)^{n}\prod_{X\in
g}|{\tilde{V}}(X)|$

where ${\tilde{V}}(X)=\beta|V(X)|e^{\beta J}$. Hence we can apply theorem 2
with $K=\beta Je^{\beta J}$ and

${\bar{\sigma}}=\left({z\over 1+z}\right)$

and the convergence occurs when

$\beta Je^{\beta J}<{1\over 4}\left|\log\left({z\over 1+z}\right)\right|$

which is the condition (6.64). $\Box$

Note that the relation between $z$ and $\beta$ in (6.64), namely the fact that
$z$ can be taken large if $\beta$ is sufficently small, is of the form needed
to apply the particle-hole symmetry pointed out in [16], enlarging the
analiticity region.

###  Chapter 7 Ising model

The Ising model is a lattice system enclosed in a box
$\Lambda\subset\mathbb{Z}^{d}$. The box $\Lambda$ is a finite set, generally a
square of size $L$ which contains $|\Lambda|~{}=~{}L^{d}$ sites of the lattice
$\mathbb{Z}^{d}$. In each site $x\in\Lambda$ there is a random variable
$\sigma_{x}$ which can take one of the two possible value
$\sigma_{x}~{}=~{}\pm 1$. A configuration ${\bf\sigma}_{\Lambda}$ of the
system is given when one declares the value of the spin $\sigma_{x}$ for each
site $x\in\Lambda$. Hence ${\bf\sigma}_{\Lambda}$ is a set
${\bf\sigma}_{\Lambda}~{}=~{}(\sigma_{x_{1}},\dots,\sigma_{x_{|\Lambda|}})$ of
$|\Lambda|$ numbers $\pm 1$. Equivalently one can say that a configuration
$\sigma_{\Lambda}$ of the system is a function
$\sigma_{\Lambda}:\Lambda\to\\{+1,-1\\}:x\mapsto\sigma_{x}$. Observe that the
total number of configurations of the system in $\Lambda$ is $2^{|\Lambda|}$.
We denote by $\Omega_{\Lambda}$ the set of all possible spin configurations in
$\Lambda$.

Given a configurations ${\bf\sigma}_{\Lambda}$ of the system in $\Lambda$, the
energy of such configuration is

$H_{\Lambda}({\bf\sigma}_{\Lambda})~{}=~{}-J\sum_{|x-y|=1\atop\\{x,y\\}\subset\Lambda}\sigma_{x}\sigma_{y}-h\sum_{x\in\Lambda}\sigma_{x}+{\cal
B}(\sigma_{\Lambda})$ $None$

where $h$ is an “external magnetic field”, $J>0$ is a positive constant and
${\cal B}(\sigma_{\Lambda})$ represent the interaction of the spins inside the
box $\Lambda$ with the world outside. Of course ${\cal B}(\sigma_{\Lambda})$
is rather arbitrary. We will list in a moment some typical boundary
conditions. Let us just remark here that ${\cal B}(\sigma_{\Lambda})$ has to
be in any case a “surface term”, i.e.

$\lim_{\Lambda\to\infty}{\max_{\sigma_{\Lambda}}|{\cal
B}(\sigma_{\Lambda})|\over|\Lambda|}~{}=~{}0$ $None$

The statistical mechanics is obtained by assigning to any configuration
$\sigma_{\Lambda}$ a probability to occur $P(\sigma_{\Lambda})$. This
probability is given by

$P(\sigma_{\Lambda})~{}=~{}{e^{-\beta H_{\Lambda}({\bf\sigma}_{\Lambda})}\over
Z_{\Lambda}(\beta,h)}$ $None$

where

$Z_{\Lambda}(\beta,h)~{}=~{}\sum_{{\bf\sigma}_{\Lambda}\in\Omega_{\Lambda}}e^{-\beta
H_{\Lambda}({\bf\sigma}_{\Lambda})}$

is the partition function in the grand canonical ensemble. If
$f(\sigma_{\Lambda})$ is a function depending on the configuration of the
system, its mean value in the grand canonical ensemble is

$\langle
f(\sigma_{\Lambda})\rangle_{\Lambda}~{}=~{}\sum_{\sigma_{\Lambda}\in\Omega_{\Lambda}}P(\sigma_{\Lambda})f(\sigma_{\Lambda})~{}=~{}\sum_{\sigma_{\Lambda}\in\Omega_{\Lambda}}{e^{-\beta
H_{\Lambda}({\bf\sigma}_{\Lambda})}f(\sigma_{\Lambda})\over
Z_{\Lambda}(\beta,h)}$

The finite volume free energy of the model is given by

$f_{\Lambda}(\beta,h)~{}=~{}{1\over|\Lambda|}\ln Z_{\Lambda}(\beta,h)$ $None$

Note that, at finite volume, this function is analytic in $\beta,h$ for all
$\beta>0$ and for all $h\in(-\infty,+\infty)$.

Thermodynamic is recovered evaluating the following limit

$f(\beta,h)~{}=~{}\lim_{\Lambda\to\infty}{1\over|\Lambda|}\ln
Z_{\Lambda}(\beta,h)$ $None$

The function $f(\beta,h)$ is called the infinite volume free energy of the
system. It is easy to show that the limit exists, but in general is not
expected to be analytic in the whole physical region. It is also worth to
stress that (7.2) ensures that the limit above in independent on the boundary
conditions $\cal B$. As a matter of fact, let ${\cal B}(\sigma_{\Lambda})$ and
${\cal B}^{\prime}(\sigma_{\Lambda})$ two different boundary conditions and
let us denote

$H^{\rm
bulk}_{\Lambda}({\bf\sigma}_{\Lambda})~{}=~{}-J\sum_{|x-y|=1\atop\\{x,y\\}\subset\Lambda}\sigma_{x}\sigma_{y}-h\sum_{x\in\Lambda}\sigma_{x}$

then we have

${Z_{\Lambda,{\cal B}}(\beta,h)\over Z_{\Lambda,{\cal
B}^{\prime}}(\beta,h)}~{}=~{}{\sum_{{\bf\sigma}_{\Lambda}}e^{-\beta H^{\rm
bulk}_{\Lambda}({\bf\sigma}_{\Lambda})}e^{-\beta{\cal
B}(\sigma_{\Lambda})}\over\sum_{{\bf\sigma}_{\Lambda}}e^{-\beta H^{\rm
bulk}_{\Lambda}({\bf\sigma}_{\Lambda})}e^{-\beta{\cal
B}^{\prime}(\sigma_{\Lambda})}}\leq\max_{\sigma_{\Lambda}}e^{\beta|{\cal
B}(\sigma_{\Lambda})|+\beta|{\cal B}^{\prime}(\sigma_{\Lambda})|}$

i.e

${Z_{\Lambda,{\cal B}}(\beta,h)\over Z_{\Lambda,{\cal
B}^{\prime}}(\beta,h)}\leq e^{\beta\max_{\sigma_{\Lambda}}(|{\cal
B}(\sigma_{\Lambda})|+|{\cal B}^{\prime}(\sigma_{\Lambda})|)}$

analogously one can get

$e^{-\beta\max_{\sigma_{\Lambda}}(|{\cal B}(\sigma_{\Lambda})|+|{\cal
B}^{\prime}(\sigma_{\Lambda})|)}\leq{Z_{\Lambda,{\cal B}}(\beta,h)\over
Z_{\Lambda,{\cal B}^{\prime}}(\beta,h)}$

whence

${1\over|\Lambda|}|\ln Z_{\Lambda,{\cal B}}(\beta,h)-\ln Z_{\Lambda,{\cal
B}^{\prime}}(\beta,h)|\leq{1\over|\Lambda|}\beta\max_{\sigma_{\Lambda}}(|{\cal
B}(\sigma_{\Lambda})|+|{\cal B}^{\prime}(\sigma_{\Lambda})|)$ $None$

and taking the limit $\Lambda\to\infty$ and using (7.2) we get the result.

We list below some typical boundary conditions.

1) Open (or free) boundary conditions. This is the case

${\cal B}_{0}(\sigma_{\Lambda})~{}=~{}0$

2) Periodic boundary conditions. This is the case in which $\Lambda$ is a
Torus, i.e. spins of opposite faces interact via the constant $-J$, as if they
were nearest neighbors. Clearly this can be obtained by a suitable choice of
${\cal B}_{p}(\sigma_{\Lambda})$ 3) $+$ boundary conditions. This is the
following case. Let
$\partial\Lambda=\\{x\in\mathbb{Z}^{2}\backslash\Lambda:|x-y|=1~{}~{}{\rm
for~{}some}~{}y\in\Lambda\\}$ be the “external boundary” of $\Lambda$ and put
$\sigma_{x}~{}=~{}+1$ for all $x\in\partial\Lambda$. Then

$B_{+}(\sigma_{\Lambda})~{}=~{}-J\sum_{x\in\partial\Lambda}\sum_{y\in\Lambda\atop|x-y|=1}\sigma_{y}$

Physically this means to fix the spins outside $\Lambda$ (those who can
interact with spin inside!) to the value $\sigma_{x}=+1$.

The fact that the free energy is independent from boundary conditions does not
mean that the system is stable respect to boundary conditions. Precisely the
instability of the system respect to boundary conditions is an evidence for
phase transition. Even if the free energy is independent of boundary
conditions it still can occur a discontinuity of some derivative of the free
energy at some point in the $(\beta,h)$ region of the physical parameters. We
will show below that this indeed happens.

We will consider ahead the magnetization of the system, which measures if spin
are mostly oriented up or down. This quantity is the partial derivative of the
free energy respect to the magnetic field.

$M_{\Lambda}^{{\cal
B}}(\beta,h)~{}=~{}{1\over|\Lambda|}\sum_{x\in\Lambda}\langle\sigma_{x}\rangle_{\Lambda,{\cal
B}}~{}=~{}\beta^{-1}{\partial\over\partial h}f_{\Lambda}(\beta,h)$

We will see that the magnetization in the thermodynamic limit can take
different values according to different boundary conditions.

#####  7.0.1 High temperature expansion

We now develop a (high temperature) polymer expansion for the partition
function of the two dimensional Ising model with zero magnetic field (i.e. we
set $h=0$ in (7.1)). We suppose from now on that $\Lambda$ is a square of size
$L$ which thus contains $L^{2}$ lattice sites. We also choose free boundary
conditions (i.e. we set ${\cal B}(\sigma_{\Lambda})=0$ in (7.1)). The
Hamiltonian of the zero field, free boundary conditions Ising model is thus

$H_{\Lambda}({\bf\sigma}_{\Lambda})~{}=~{}-J\sum_{|x-y|=1\atop\\{x,y\\}\subset\Lambda}\sigma_{x}\sigma_{y}~{}=~{}-J\sum_{b\in
B(\Lambda)}\tilde{\sigma}_{b}$

where $B(\Lambda)$ is the set of all pairs $b=\\{x,y\\}\subset\Lambda$ such
that $|x-y|~{}=~{}1$ (nearest neighbor pairs) and for $b=\\{x,y\\}$ we put
$\tilde{\sigma}_{b}=\sigma_{x}\sigma_{y}$. Recall that, since $\Lambda$ is a
$L\times L$ box in $\mathbb{Z}^{2}$, then $|B(\Lambda)|~{}=~{}2L(L-1)$

The partition function of the Ising model at zero magnetic field is thus

$Z_{\Lambda}(\beta)~{}=~{}\sum_{{\bf\sigma}_{\Lambda}\in\Omega_{\Lambda}}\prod_{b\in
B(\Lambda)}e^{+\beta J\tilde{\sigma}_{b}}$

We want to prove that the free energy

$f_{\Lambda}(\beta)={1\over|\Lambda|}\ln Z_{\Lambda}(\beta)$ $None$

is an analytic function of $\beta$ is $\beta$ is sufficiently small (i.e. in
the high temperature regime) uniforlmly in the volume $\Lambda$. Observe that
$\tilde{\sigma}_{b}~{}=~{}\pm 1$, hence

$e^{+\beta J\tilde{\sigma}_{b}}~{}=~{}\cosh(\beta
J\tilde{\sigma}_{b})+\sinh(\beta J\tilde{\sigma}_{b})~{}=~{}\cosh(\beta
J)+\tilde{\sigma}_{b}\sinh(\beta J)~{}=$ $=~{}\cosh(\beta
J)[1+\tilde{\sigma}_{b}\tanh(\beta J)]$

hence

$Z_{\Lambda}(\beta)~{}=~{}[\cosh(\beta
J)]^{2L(L-1)}\sum_{{\bf\sigma}_{\Lambda}\in\Omega_{\Lambda}}\prod_{b\in
B(\Lambda)}[1+\tilde{\sigma}_{b}\tanh(\beta J)]$

Developing the product $\prod_{b\in
B(\Lambda)}[1+\tilde{\sigma}_{b}\tanh(\beta J)]$ we get terms of the type

$[\tanh(\beta J)]^{k}\tilde{\sigma}_{b_{1}}\dots\tilde{\sigma}_{b_{k}}$

which has a clear geometric interpretation. The set of bonds
${b_{1}},\dots,{b_{k}}$ form a graph (connected or not) in $\Lambda$ whose
links are nearest neighbors.

When one perform the sum over ${\bf\sigma}_{\Lambda}$ we get that

$\sum_{{\bf\sigma}_{\Lambda}}\tilde{\sigma}_{b_{1}}\dots\tilde{\sigma}_{b_{k}}$

is zero whenever there is a not paired spin. See Figures 11 and 12.

Figure 11. A non vanishing graph with seven connected components

The only graphs which yield a non vanishing contribution to the partition
function are those whose vertices have incidence number zero, two or four
(Figure 11), while the contribution of all other graphs is zero once the sum
over configurations $\sigma_{\Lambda}$ has been done (Figure 12). If the graph
$\tilde{\sigma}_{b_{1}}\dots\tilde{\sigma}_{b_{k}}$ is non vanishing then

$\sum_{{\bf\sigma}_{\Lambda}}\tilde{\sigma}_{b_{1}}\dots\tilde{\sigma}_{b_{k}}~{}=~{}2^{L^{2}}$

We can naturally split a non vanishing graph in non intersecting connected
components which we will call lattice animals. For example in figure 11 it is
drawn a non vanishing graph formed by seven non intersecting lattice animals.
A lattice animal $\gamma$ is thus nothing but a graph $g$ with edge set
$E_{\gamma}~{}=~{}\\{b_{1},\dots,b_{k}\\}\subset B(\Lambda)$ formed by nearest
neighbor links $b~{}=~{}\\{x,y\\}$ and with vertex set
$V_{g}=\cup_{i=1}^{k}b_{i}\subset\Lambda$ which is a connected graph in
$V_{\gamma}$ (in usual sense). The allowed lattice animals are only those
$\gamma$ with incidence number at the vertices equal to two or four. Let us
denote by ${\cal L}$ the set of possible lattice animals in $\mathbb{Z}^{d}$
and by ${\cal L}_{\Lambda}$ the set of all possible lattice animals in
$\Lambda$.

Two lattice animals $\gamma$ and $\gamma^{\prime}$ are non overlapping (i.e.
compatible), and we write $\gamma\sim\gamma^{\prime}$ if and only if
$V_{\gamma}\cap V_{\gamma^{\prime}}^{\prime}=\emptyset$. We will denote
shortly $|\gamma|=|E_{\gamma}|$ (i.e. $|\gamma|$ is the number of nearest
neighbor bonds which constitute $\gamma$, i.e. if
$\gamma=\\{b_{1},\dots,b_{k}\\}$ then $|\gamma|=k$. Note that only such
lattice animals (i.e. just those with incidence number at the vertices equal
to 2 or to 4) survive because we are using free boundary conditions. Note also
that lattice animal $c\in{\cal L}$ with incidence number equal to 2 in anyone
of its vertices is a simple cycle. For a cycle $c$ we have that $|V_{c}|=|c|$,
while for a lattice lattice animal $\gamma$ which is not a cycle we have
$|V_{\gamma}|<|\gamma|$.

Figure 12. A vanishing graph.

In conclusion we can write

$Z_{\Lambda}(\beta)~{}=~{}[\cosh(\beta
J)]^{2L(L-1)}2^{L^{2}}\;\Xi_{\Lambda}(\beta)$ $None$

where

$\Xi_{\Lambda}(\beta)~{}=~{}1+\sum_{n\geq
1}\sum_{\\{\gamma_{1},\dots\gamma_{n}\\}\subset{\cal
L}_{\Lambda}\atop|\gamma_{i}|\geq
4,\;\gamma_{i}\sim\gamma_{j}}\xi(\gamma_{1})\dots\xi(\gamma_{n})$ $None$

where $\gamma$ denote a allowed lattice animal (whence the condition
$|\gamma|\geq 4$) with activity

$\xi(\gamma)~{}=~{}[\tanh(\beta J)]^{|\gamma|}$ $None$

Thus the partition function of the Ising model (7.8) can be written, modulo a
constant (the factor $[\cosh(\beta J)]^{2L(L-1)}2^{L^{2}}$), as the partition
function of a hard core polymer gas, i.e the term (7.9). In this case polymers
are lattice animals, i.e. elements of ${\cal L}$ with the incompatibility
relation $\gamma\nsim\gamma^{\prime}$ if and only if $V_{\gamma}\cap
V_{\gamma^{\prime}}\neq\emptyset$.

Let us apply to this polymer gas the convergence criterion (4.48) of Theorem
4.1 to the polymers system with partition function $\Xi_{\Lambda}(\beta)$
given by (7.9) above. Namely we need to find numbers
$\mu(\gamma)\in(0,+\infty)$ such that

$\xi(\gamma)\leq{\mu(\gamma)\over\Xi_{{\cal L}_{\gamma}}(\bm{\mu})}$ $None$

with ${\cal L}_{\gamma}=\\{\gamma^{\prime}\in{\cal
L}:\gamma^{\prime}\nsim\gamma\\}$.

As we did previoulsy we set $\mu(\gamma)=\xi(\gamma)e^{a{|\gamma|}}$ so that
condition becomes

$\Xi_{{\cal L}_{\gamma}}(\bm{\mu})\leq e^{a{|\gamma|}}$ $None$

where

$\Xi_{{\cal L}_{\gamma}}(\bm{\mu})=1+\sum_{n=1}^{|V_{\gamma}|}{1\over
n!}\sum_{(\gamma_{1},\dots,\gamma_{n})\in{\cal
L}^{n}_{\gamma}\atop\gamma_{i}\sim\gamma_{j}}\prod_{i=1}^{n}|\xi(\gamma_{i})|e^{a|\gamma_{i}|}$

Consider now the factor

$F_{n}(\gamma,a)\doteq\sum_{(\gamma_{1},\dots,\gamma_{n})\in{\cal
L}^{n}_{\gamma}\atop\gamma_{i}\sim\gamma_{j}}\prod_{i=1}^{n}|\xi(\gamma_{i})|e^{a|\gamma_{i}|}$
$None$

We have thus to choose $n$ lattice animals $\gamma_{1},\dots\gamma_{n}$ all
incompatible with a given lattice animal $\gamma$ and all pairwise compatible.
We recall that two lattice animals are incompatible of they share a vertex os
$\mathbb{Z}^{2}$. The factor (7.13) is zero whenever $n>|V_{\gamma}|$,
$\gamma_{i}\nsim\gamma$ since $\gamma$ has $|V_{\gamma}|$ vertices and thus we
can arrange at most $|V_{\gamma}|$ lattice animals pairwise compatible each
one sharing a different vertex of $V_{\gamma}$. Therefore, when the factor
above is not zero, i.e. for $n\leq|V_{\gamma}|$, it can be bounded at least by
(again a very rough bound)

$\displaystyle F_{n}(\gamma,a)$
$\displaystyle\leq|V_{\gamma}|(|V_{\gamma}|-1)\cdots(|V_{\gamma}|-n+1)\Bigg{[}\sup_{x\in\mathbb{Z}^{2}}\,\sum_{\gamma\in{\cal
L}\atop x\in\gamma}|{\xi(\gamma)}|e^{a|\gamma|}\Bigg{]}^{n}$
$\displaystyle={|V_{\gamma}|\choose
n}n!\Bigg{[}\sup_{x\in\mathbb{Z}^{2}}\,\sum_{\gamma\in{\cal L}\atop
x\in\gamma}|{\xi(\gamma)}|e^{a|\gamma|}\Bigg{]}^{n}$

Thus

$\displaystyle\Xi_{{\cal L}_{\gamma}}(\bm{\mu})$ $\displaystyle\leq
1+\sum_{n=1}^{|V_{\gamma}|}{|V_{\gamma}|\choose
n}\Bigg{[}\sup_{x\in\mathbb{Z}^{2}}\,\sum_{\gamma\in{\cal L}\atop
x\in\gamma}|{\rho(\gamma)}|e^{a|\gamma|}\Bigg{]}^{n}$
$\displaystyle=\Bigg{[}1+\sup_{x\in\mathbb{Z}^{2}}\,\sum_{\gamma\in{\cal
L}\atop x\in\gamma}|{\xi(\gamma)}|e^{a|\gamma|}\Bigg{]}^{|V_{\gamma}|}$

Thus (7.12) can be written as

$\Bigg{[}1+\sup_{x\in\mathbb{Z}^{2}}\,\sum_{\gamma\in{\cal L}\atop
x\in\gamma}|{\xi(\gamma)}|e^{a|\gamma|}\Bigg{]}^{|V_{\gamma}|}\leq
e^{a|\gamma|}$ $None$

and since, $|V_{\gamma}|\leq|\gamma|$ for any $\gamma\in{\cal L}$ (the
equality holding only if $\gamma$ is a cycle) we have that (7.14) is surely
satisfied if

$\sup_{x\in\mathbb{Z}^{2}}\,\sum_{\gamma\in{\cal L}\atop
x\in\gamma}|{\xi(\gamma)}|e^{a|\gamma|}\leq e^{a}-1$ $None$

Observe finally that, due to the symmetry of the problem the function

$f(x)=\sum_{\gamma\in{\cal L}\atop x\in\gamma}|{\xi(\gamma)}|e^{a|\gamma|}$

is constant as $x$ varies in $\mathbb{Z}^{2}$. Therefore (7.15) is equivalent
to the condition

$\sum_{\gamma\in{\cal L}\atop 0\in\gamma}|{\xi(\gamma)}|e^{a|\gamma|}\leq
e^{a}-1$ $None$

where $0$ is the origin in $\mathbb{Z}^{2}$.

The condition (7.16) is a (high temperature) convergence condition for the
analyticity of the free energy of the Ising model at zero magnetic field and
free boundary conditions. Now recalling (7.10) and due to the symmetry of the
problem

$\sum_{\gamma\in{\cal L}\atop 0\in\gamma}[\tanh(\beta
J)]^{|\gamma|}e^{a|\gamma|}=\sum_{n\geq 4}[\tanh(\beta
J)]^{n}e^{an}\sum_{\gamma\in{\cal L}\atop
0\in\gamma,\;|\gamma|=n}1=\sum_{n\geq 4}[\tanh(\beta J)]^{n}e^{an}C_{n}$

with

$C_{n}=\sum_{\gamma\in{\cal L}\atop 0\in\gamma,\;|\gamma|=n}1$

being the number of lattice animals in ${\cal L}$ made by $n$ nearest neighbor
bonds containing the origin. I.e. we need to count all lattice animals
$\gamma$ with a given cardinality $|\gamma|=n$ that pass through the origin.
To do this just observe that the nearest neighbor bonds of a lattice animal
form a graph with degree 2 or 4, i.e. a graph with even degree vertices. It is
long known that any graph with even degree at its vertices admits an Eulerian
cycle (i.e. a graph cycle that crosses each edge exactly once). Therefore all
lattice animals containing the origin can be formed by performing a cycle
starting at the origin. To form a cycle starting at the origin in
$\mathbb{Z}^{2}$ we can take each time 3 directions (also at the beginning
since the cycle can be traveled in two direction) This immediately implies
that

$C_{n}=\sum_{\gamma\in{\cal L}\atop 0\in\gamma,\;|\gamma|=n}1\leq 3^{n}$

Therefore condition (7.16) is surely satisfied if

$\sum_{n\geq 4}[3\tanh(\beta J)]^{n}e^{an}\leq e^{a}-1$

i.e. posing $x=|3\tanh(\beta J)|$ as soon as

$e^{4a}[3\tanh(\beta J)]^{4}+(e^{2a}-e^{a})[3\tan(\beta J)]-(e^{a}-1)\leq 0$

which yields (for $a=0.21$)

$\tanh(\beta J)\leq 0.1525$

Hence convergence of the free energy, uniformly in the volume $\Lambda$ occurs
as soon as

$\beta\leq\beta_{0}\doteq{1\over J}\tanh^{-1}(0.1525)\approx{0.151\over J}$
$None$

#####  7.0.2 Low temperature expansion

We consider now the zero magnetic field Ising model with $+$ boundary
conditions in a box $\Lambda$ which is assumed to be a $L\times L$ square with
$L^{2}$ sites. Thus, given a configuration $\sigma_{\Lambda}$ of the spins
inside $\Lambda$, the Hamiltonian is

$H^{+}_{\Lambda}({\bf\sigma}_{\Lambda})~{}=~{}-J\sum_{\\{x,y\\}\subset\Lambda\atop|x-y|~{}=~{}1}\sigma_{x}\sigma_{y}-J\sum_{y\in\partial\Lambda}\sum_{x\in\Lambda,\atop|x-y|~{}=~{}1}\sigma_{x}$

Another way to write the Hamiltonian of the zero field + boundary condition
Ising model is as follows is

$H^{+}_{\Lambda}({\bf\sigma}_{\Lambda})~{}=~{}-J\sum_{\\{x,y\\}\cap\Lambda\neq\emptyset\atop|x-y|~{}=~{}1}\sigma_{x}\sigma_{y}$

recalling that $\sigma_{y}=+1$ whenever $y\in\partial\Lambda$.

We now rewrite the partition function $Z^{+}_{\Lambda}(\beta)$ via a contour
gas in the following way. For a fixed configuration $\sigma_{\Lambda}$ draw a
unit segment perpendicular to the center of each bond $b$ of nearest neighbors
having opposite spins at its extremes (in three dimensions this segment
becomes a unite square surface). Such unit segments have extremities which are
in a lattice shifted respect to the original lattice by a factor $1/2$ along
both $x$ and $y$ axis. This lattice is called the dual lattice and denoted by
$\mathbb{Z^{*}}^{2}$. Hence this set of unit segments forms a graph of nearest
neighbor links in $\Lambda^{*}\subset\mathbb{Z^{*}}^{2}$ where $\Lambda^{*}$
is a $(L+1)\times(L+1)$ square in the dual lattice. With the particular choice
of + boundary conditions such graphs are exactly the same of Figure 11, with
the only difference that now they live in $\Lambda^{*}$, i.e. in a square with
$(L+1)\times(L+1)$ sites in the dual lattice. See Figure 13.

Figure 13. The grey square is $\Lambda^{*}$. The outer spins are those fixed
by the boundary conditions.

To any configuration $\sigma_{\Lambda}$ in $\Lambda$ we can thus associate a
graph $\Gamma$ in the dual lattice made by nearest neighbor links with
incidence number equal either 0, 2 or 4. The graph $\Gamma$ splits into its
connected components in the usual manner. Hence
$\Gamma=\\{\gamma_{1},\gamma_{2},\dots,\gamma_{k}\\}$ with
$\gamma_{i}\cap\gamma_{j}=\emptyset$. In this case (low temperature
expansion), such connected components $\gamma$ are called contours and
$|\gamma|$ is now called perimeter of the contour. Note that now a contour
configuration correspond to a spin configuration. I.e., given the spin
configuration $\sigma_{\Lambda}$ in $\Lambda$ we can recover the contour
configuration $\Gamma=\\{\gamma_{1},\gamma_{2},\dots,\gamma_{k}\\}$ and
conversely, given the contour configuration
$\Gamma=\\{\gamma_{1},\gamma_{2},\dots,\gamma_{k}\\}$ we can recover the spin
configuration $\sigma_{\Lambda}$ which originated it.

I.e. there is a one to one correspondence
$\sigma_{\Lambda}\to\\{\gamma_{1},\dots,\gamma_{k}\\}$. We can express the
Hamiltonian, which is a function of $\sigma_{\Lambda}$, as a function of the
contour configuration $\Gamma=\\{\gamma_{1},\gamma_{2},\dots,\gamma_{k}\\}$
uniquely determined by $\sigma_{\Lambda}$. This is done very easily, just
observing that, for a fixed configuration $\sigma_{\Lambda}$ we have

$H^{+}_{\Lambda}({\bf\sigma}_{\Lambda})~{}=~{}-J\sum_{\\{x,y\\}\cap\Lambda\neq\emptyset\atop|x-y|~{}=~{}1}\sigma_{x}\sigma_{y}~{}=~{}-J\left[B^{+}-B^{-}\right]$

where

$B^{+}~{}=~{}\sharp~{}{\rm
of\;nearest\;neighbor\;pairs}~{}\sigma_{x}\sigma_{y}~{}{\rm
with}~{}\sigma_{x}~{}=~{}\sigma_{y}$ $B^{-}~{}=~{}\sharp~{}{\rm
of\;nearest\;neighbor\;pairs}~{}\sigma_{x}\sigma_{y}~{}{\rm
with}~{}\sigma_{x}~{}=~{}-\sigma_{y}$

If now $\\{\gamma_{1},\dots,\gamma_{k}\\}$ is the contour configurations
corresponding to $\sigma_{\Lambda}$ we have that the number of nearest
neighbor pairs with opposite signal is by construction equal to
$\sum_{i=1}^{k}|\gamma_{i}|$, where recall that $|\gamma_{i}|$ is the
perimeter of $\gamma$

$B^{-}~{}=~{}\sum_{i=1}^{k}|\gamma_{i}|$

Denoting with

$\tilde{B}_{\Lambda}~{}=~{}2L(L-1)+4L~{}=~{}2L(L+1)$

the number of nearest neighbor pairs with non void intersection with
$\Lambda$, we also have

$B^{+}~{}=~{}\tilde{B}_{\Lambda}-B^{-}$

hence

$H^{+}_{\Lambda}({\bf\sigma}_{\Lambda})~{}=~{}H_{\Lambda^{*}}(\gamma_{1},\dots,\gamma_{k})~{}=~{}-J\tilde{B}_{\Lambda}+2J\sum_{i=1}^{k}|\gamma_{i}|$

Hence the partition function of the Ising model with $h=0$ and with
$+$-boundary conditions can be written as

$Z^{+}_{\Lambda}(\beta)~{}=~{}e^{+\beta
J\tilde{B}_{\Lambda}}\left[1+\sum_{n\geq
1}\sum_{\\{\gamma_{1},\dots,\gamma_{n}\\}\subset{\cal
L}_{\Lambda^{*}}\atop|\gamma_{i}|\geq 4,~{}\gamma_{i}\sim\gamma_{j}}e^{-2\beta
J\sum_{i=1}^{n}|\gamma_{i}|}\right]$ $None$

the factor 1 inside the bracket correspond to the situation in which no
contour is present or equivalently when all spin are +1. As usual
$\gamma_{i}\sim\gamma_{j}$ means that $V_{\gamma_{i}}\cap
V_{\gamma_{j}}~{}=~{}\emptyset$ where recall that $V_{\gamma_{i}}$ and
$V_{\gamma_{j}}$ are subsets of $\Lambda^{*}$.

The expression in square bracket in (7.18) it is the grand canonical partition
function of a hard core lattice polymer gas in which the polymers are as
before contours $\gamma_{i}$ but now with activity $\exp[-2\beta
J|\gamma_{i}|]$, i.e.

$\Xi_{\Lambda^{*}}(\beta)~{}=~{}1+\sum_{n\geq
1}\sum_{\\{\gamma_{1},\dots,\gamma_{n}\\}\subset{\cal
L}_{\Lambda^{*}}\atop|\gamma_{i}|\geq 4,~{}\gamma_{i}\sim\gamma_{j}}e^{-2\beta
J\sum_{i=1}^{n}|\gamma_{i}|}~{}=~{}$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}1+\sum_{n\geq 1}{1\over
n!}\sum_{(\gamma_{1},\dots,\gamma_{n})\in{\cal
L}^{n}_{\Lambda^{*}}\atop|\gamma_{i}|\geq
4,~{}\gamma_{i}\sim\gamma_{j}}\rho(\gamma_{1})\dots\rho(\gamma_{n})$

where now the activity of contours is

$\rho(\gamma)~{}=~{}e^{-2\beta J|\gamma|}$ $None$

Note that the activity of a contour goes to zero if the perimeter $|\gamma|$
of the contour increases, so that big contours are depressed, but also the
activity of a contour goes to zero when $\beta$ goes to infinity, i.e all
contours tend to be depressed when the temperature is lower and lower. Hence
at very low temperature one is expected to see a sea of $+$ with very small
and very few contours which surround islands of -. This means that the in
Ising model at zero magnetic field and very low temperature spins are
predominantly oriented +, and this fact occurs independently on the size
$\Lambda$ where the system is confined. This give a quite clear a detailed
picture of the low temperature phase of the Ising model with + boundary
condition and $h~{}=~{}0$.

We can now formulate a condition for the analyticity of the free energy of the
Ising model at $h~{}=~{}0$ with +boundary conditions in a completely analogous
way as we did for the case of high temperature. The only thing that changes is
that now the activity of contours (which in the high temperature case where
called lattice animals, but they are the same objects) is defined by (7.19).
Again we have to check a formula identical to (7.16) where now in place of
$\xi(\gamma)$ we have to put $\rho(\gamma)$ defined in (7.19). Hence we reduce
ourselves to check for which $\beta$ the following inequality holds

$\sum_{n\geq 4}[3e^{a}\,e^{-2\beta J}]^{n}\leq e^{a}-1$

which is satisfied, for $a=0.21$, if

$3e^{-2\beta J}\leq 0.4577$

i.e.

$\beta\geq\beta_{1}~{}\doteq~{}{1\over 2J}\ln\left[3\over
0.4577\right]\approx{0.94\over J}$ $None$

we have that the finite volume free energy $f_{\Lambda}(\beta)$ is analytic in
$\beta$ for all $\beta\geq\beta_{1}={0.94\over J}$ uniformly in $\Lambda$.
Note that in this case $f_{\Lambda}(\beta)$ is calculated with + boundary
condition, but the limit
$f(\beta)~{}=~{}\lim_{\Lambda\to\infty}f_{\Lambda}(\beta)$ which is also
analytic in $\beta$ for all $\beta>\beta_{1}$, does not depend on boundary
conditions.

In conclusion we have proved the following theorem

Theorem. The free energy of the Ising model at zero magnetic field

$f(\beta,h=0)~{}=~{}\lim_{\Lambda\to\infty}{1\over|\Lambda|}\ln
Z_{\Lambda}(\beta,h=0)$

is analytic in $\beta$ for all $\beta\in(0,\beta_{0}]\cup[\beta_{1},+\infty)$
where $\beta_{0}$ is defined in (7.17) and $\beta_{1}$ is defined in (7.20).
We now show that there is at least a point in the interval
$(\beta_{0},\beta_{1})$ where $f(\beta,h=0)$ is non analytic.

#####  7.0.3 Existence of phase transitions

First recall the definition of the magnetization of the system as

$M_{\Lambda}(\beta)~{}=~{}{1\over|\Lambda|}\sum_{x\in\Lambda}\langle\sigma_{x}\rangle_{\Lambda}$
$None$

The quantity $M_{\Lambda}(\beta)$ measures the mean orientation of spins of
the finite system confined in $\Lambda$. If $M_{\Lambda}(\beta)>0$ then it
means that spin + are predominant while if $M_{\Lambda}(\beta)<0$ this means
that spin - are predominant. It is just a simple calculus to show the identity

$M_{\Lambda}(\beta)~{}=~{}\beta^{-1}\left.{\partial
f_{\Lambda}(\beta,h)\over\partial h}\right|_{h~{}=~{}0}$

which tells us that the (finite volume) magnetization is, modulo a factor
$\beta^{-1}$, the first derivative respect to $h$ of the finite volume free
energy of the system.

By a Theorem called “Lee Yang theorem” is possible to show that $f(\beta,h)$
is analytic for all $\beta>0$ when $h\neq 0$ hence the only singularity of
this function can occur when $h=0$.

Hence in order to show existence of transition phase, our strategy will be to
show that, depending on boundary condition the infinite volume magnetization
$M(\beta)~{}=~{}\lim_{\Lambda\to\infty}M_{\Lambda}(\beta)$ is not stable for
variations of the boundary conditions.

Thus we will show first that the magnetization $M^{+}_{\Lambda}(\beta)$ of the
Ising model with + boundary conditions (the superscript “+” remember us that
we are using + boundary conditions) is a number arbitrarily near 1 for $\beta$
sufficiently big. The result will also immediately imply that the
magnetization $M^{-}_{\Lambda}(\beta)$ of the Ising model with - boundary
conditions is a number arbitrarily near -1 for $\beta$ sufficiently big.

Let us denote $\langle\cdot\rangle_{\Lambda}^{+}$ the mean values when the
Hamiltonian is taken with $h=0$ and with + boundary condition. It is easy to
see that, for any $x\in\Lambda$

$\langle\sigma_{x}\rangle_{\Lambda}^{+}~{}=~{}1-2\Big{[}{\rm
Prob}_{\Lambda}^{+}(\sigma_{x}=-1)\Big{]}$ $None$

where ${\rm Prob}_{\Lambda}^{+}(\sigma_{x}=-1)$ denotes the probability that
the spin $\sigma_{x}$ at the site $x$ is equal to $-1$ for the Ising model
with zero magnetic field and + boundary conditions. As a matter of fact, by
definition

$\displaystyle\langle\sigma_{x}\rangle_{\Lambda}^{+}$
$\displaystyle=~{}(+1)~{}\Big{[}{\rm
Prob}_{\Lambda}^{+}(\sigma_{x}=+1)\Big{]}~{}~{}+~{}~{}(-1)~{}\Big{[}{\rm
Prob}_{\Lambda}^{+}(\sigma_{x}=-1)\Big{]}$ $None$
$\displaystyle=~{}1\;-\,2\Big{[}{\rm
Prob}_{\Lambda}^{+}(\sigma_{x}=-1)\Big{]}$

Thus in order to evaluate $\langle\sigma_{x}\rangle_{\Lambda}^{+}$ it is
sufficient to evaluate ${\rm Prob}_{\Lambda}^{+}(\sigma_{x}=-1)$.

We have seen before that the Ising model with + boundary condition and zero
magnetic field can be mapped in a contour gas, and that there is a one to one
correspondence between spin configurations $\sigma_{\Lambda}$ and contour
configurations $\gamma_{1},\dots\gamma_{k}$. Now a $\sigma_{\Lambda}$ such
that the spin at $x$ is $-1$ is such that the site $x$ is the “interior” of at
least one contour associated to $\sigma_{\Lambda}$, i.e. $x$ is surrounded at
least by a contour (actually by an odd number of contours). Hence, if we
denote ${\rm Prob}_{\Lambda}^{+}(\exists\gamma\odot x)$ the probability that
at least one contour surrounds $x$, we have surely

${\rm Prob}_{\Lambda}^{+}(\sigma_{x}=-1)\leq{\rm
Prob}_{\Lambda}^{+}(\exists\gamma\odot x)$

But now

${\rm Prob}_{\Lambda}^{+}(\exists\gamma\odot
x)~{}=~{}{\sum_{\\{\gamma_{1},\dots,\gamma_{n}\\}:\exists\gamma_{i}\odot
x\atop|\gamma_{i}|\geq 4,~{}\gamma_{i}\sim\gamma_{j}}e^{-2\beta
J\sum_{i=1}^{n}|\gamma_{i}|}\over\sum_{\\{\gamma_{1},\dots,\gamma_{n}\\}\atop|\gamma_{i}|\geq
4,~{}\gamma_{i}\sim\gamma_{j}}e^{-2\beta J\sum_{i=1}^{n}|\gamma_{i}|}}$

where the denominator is the grand canonical partition function of the contour
gas and in the sum in the denominator is also included the empty graph which
contribute with the factor 1. Now we can write

$\displaystyle{\rm Prob}_{\Lambda}^{+}(\exists\gamma\odot x)$
$\displaystyle=~{}{\sum_{\\{\gamma_{1},\dots,\gamma_{n}\\}:\exists\gamma_{i}\odot
x\atop|\gamma_{i}|\geq 4,~{}\gamma_{i}\sim\gamma_{j}}e^{-2\beta
J\sum_{i=1}^{n}|\gamma_{i}|}\over\sum_{\\{\gamma_{1},\dots,\gamma_{n}\\}\atop|\gamma_{i}|\geq
4,~{}\gamma_{i}\sim\gamma_{j}}e^{-2\beta J\sum_{i=1}^{n}|\gamma_{i}|}}$
$\displaystyle=~{}{\sum_{\gamma\odot x}e^{-2\beta
J|\gamma|}\sum_{\\{\gamma_{1},\dots,\gamma_{n}\\}:\gamma_{i}\sim\gamma\atop|\gamma_{i}|\geq
4,~{}\gamma_{i}\sim\gamma_{j}}e^{-2\beta
J\sum_{i=1}^{n}|\gamma_{i}|}\over\sum_{\\{\gamma_{1},\dots,\gamma_{n}\\}\atop|\gamma_{i}|\geq
4,~{}\gamma_{i}\sim\gamma_{j}}e^{-2\beta J\sum_{i=1}^{n}|\gamma_{i}|}}$
$\displaystyle\leq\sum_{\gamma\odot x}e^{-2\beta J|\gamma|}$

I.e. in conclusion we get

${\rm Prob}_{\Lambda}^{+}(\exists\gamma\odot x)\leq\sum_{\gamma\odot
x}e^{-2\beta J|\gamma|}\leq\sum_{n\geq 4}e^{-2\beta Jn}\sum_{\gamma\odot x}1$

It is now easy to bound $\sum_{\gamma\odot x}$ with $n3^{n}$. As a matter of
fact, let us denote with $x^{\prime}$ the point of intersection of a contour
$\gamma$ (such that $|\gamma|~{}=~{}n)$which surrounds $x$ with the horizontal
axis passing through $x$. Let us ask ourselves how many possible $x^{\prime}$
we can get. Obviously not more than $n$. Now the possible contours with fixed
perimeter $|\gamma|~{}=~{}n$ which pass through a fixed $x^{\prime}$ are at
most $3^{n}$, so $\sum_{\gamma\odot x}1\leq n3^{n}$. Hence

${\rm Prob}_{\Lambda}^{+}(\exists\gamma\odot x)\leq\sum_{n\geq 4}e^{-2\beta
Jn}n3^{n}$

Setting

$x_{\beta}=3e^{-2\beta J},$

the sum in l.h.s. converges to if $x_{\beta}<1$ and in this case we have

${\rm Prob}_{\Lambda}^{+}(\exists\gamma\odot x)\leq\sum_{n\geq
4}nx_{\beta}^{n}={x_{\beta}^{4}(4-3x_{\beta})\over(1-x_{\beta}))^{2}}~{}\doteq~{}g(\beta)$
$None$

Recalling (7.23), in order to guarantee that
$\langle\sigma_{x}\rangle^{+}_{\Lambda}>0$ uniformly in $\Lambda$ we need to
impose that

$g(\beta)<{1\over 2}$

which holds if

$x_{\beta}<0.4795$

i.e. if

$\beta>\beta^{\prime}_{1}\doteq~{}{0.917\over J}$

Note that $g(\beta)$ defined in (7.24) above goes to zero as $\beta\to\infty$.
Thus when $\beta>\beta^{\prime}_{1}$ we have that

$\langle\sigma_{x}\rangle_{\Lambda}^{+}~{}\geq~{}1-2g(\beta)$

and recalling (7.21)

$M_{\Lambda}(\beta,h=0)\geq 1-2g(\beta)$

and therefore

$\lim_{\Lambda\to\infty}M_{\Lambda}(\beta,h=0)\geq 1-2g(\beta)$

Note that $\langle\sigma_{x}\rangle_{\Lambda}^{+}$ tends to 1 as
$\beta\to\infty$ and this estimate is uniform in $\Lambda$.

On the other side, if we bound $\langle\sigma_{x}\rangle_{\Lambda}^{-}$ (i.e.
the mean value of the spin at a site $x$ with - boundary conditions) we have

$\langle\sigma_{x}\rangle_{\Lambda}^{-}~{}=~{}(+1)~{}\Big{[}{\rm
Prob}_{\Lambda}^{-}(\sigma_{x}=+1)\Big{]}~{}~{}+~{}~{}(-1)~{}\Big{[}{\rm
Prob}_{\Lambda}^{-}(\sigma_{x}=-1)\Big{]}~{}=~{}$ $=~{}2~{}\Big{[}{\rm
Prob}_{\Lambda}^{-}(\sigma_{x}=+1)\Big{]}~{}-~{}1$

but now we have obviously that ${\rm
Prob}_{\Lambda}^{-}(\sigma_{x}=+1)~{}=~{}{\rm
Prob}_{\Lambda}^{+}(\sigma_{x}=-1)$ and hence

$\langle\sigma_{x}\rangle_{\Lambda}^{-}~{}\leq~{}2g(\beta)-1$

In this case $\langle\sigma_{x}\rangle_{\Lambda}^{-}$ tends to -1 as
$\beta\to\infty$.

In conclusion the system show spontaneous magnetization for $\beta$
sufficiently high uniformly in the volume $\Lambda$. I.e. we have shown that
e.g.

$\lim_{\Lambda\to\infty}\langle\sigma_{x}\rangle_{\Lambda}^{+}\neq
0~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm if}~{}~{}\beta>{0.917\over J}$

In other words the system is not stable even at the infinite volume limit to
boundary conditions.

As a last computation we show that when $\beta$ is small there is no such
instability. As a matter of fact, we can express
$\langle\sigma_{x}\rangle_{\Lambda}^{+}$ in term of high temperature lattice
animals. Recall that the Hamiltonian on the Ising model with + boundary
conditions and zero magnetic field is

$H^{+}_{\Lambda}({\bf\sigma}_{\Lambda})~{}=~{}-J\sum_{\\{x,y\\}\cap\Lambda\neq\emptyset\atop|x-y|~{}=~{}1}\sigma_{x}\sigma_{y}$

with $\sigma_{y}~{}=~{}+1$ whenever $y\in\partial\Lambda$. And the partition
function is

$Z^{+}_{\Lambda}(\beta)~{}=~{}\sum_{\sigma_{\Lambda}}e^{+\beta
J\sum_{\\{x,y\\}\cap\Lambda\neq\emptyset\atop|x-y|~{}=~{}1}\sigma_{x}\sigma_{y}}~{}=~{}\sum_{\sigma_{\Lambda}}\prod_{\\{x,y\\}\cap\Lambda\neq\emptyset\atop|x-y|~{}=~{}1}e^{+\beta
J\sigma_{x}\sigma_{y}}~{}=~{}$ $=~{}\sum_{\sigma_{\Lambda}}\prod_{b\atop
b\cap\Lambda\neq\emptyset}e^{+\beta J\tilde{\sigma}_{b}}$

where now the nearest neighbor pair $b$ are those strictly contained in
$\Lambda$ plus the nearest neighbor pairs $\\{x,y\\}$ for which $x\in\Lambda$
and $y\in\partial\Lambda$. In this last case, since we are using + boundary
conditions, $\tilde{\sigma}_{b}=\sigma_{x}\sigma_{y}=+\sigma_{x}$. As before
we can write

$Z^{+}_{\Lambda}(\beta)~{}=~{}\sum_{\sigma_{\Lambda}}\prod_{b\atop
b\cap\Lambda\neq\emptyset}\cosh\beta J\left[1+\tilde{\sigma}_{b}\tanh\beta
J\right]$

and supposing as usual that $\Lambda$ is a square of size $L$ we have that the
number of $b$ such that $b\cap\Lambda\neq\emptyset$ is
$2L(L-1)+4L~{}=~{}2L(L+1)$, hence

$Z^{+}_{\Lambda}(\beta)~{}=~{}[\cosh\beta
J]^{2L(L+1)}\sum_{\sigma_{\Lambda}}\prod_{b\atop
b\cap\Lambda\neq\emptyset}\left[1+\tilde{\sigma}_{b}\tanh\beta J\right]$

As before the development of the product

$\prod_{b\atop b\cap\Lambda\neq\emptyset}\left[1+\tilde{\sigma}_{b}\tanh\beta
J\right]$ $None$

gives rise to terms of the form

$(\tanh\beta J)^{k}\tilde{\sigma}_{1}\dots\tilde{\sigma}_{k}$

which can be associated to graphs in $\Lambda\cup\partial\Lambda$. It is
important to stress that when $b\subset\Lambda$ then we call $b$ internal
bond, i.e. $b~{}=~{}\\{x,y\\}$ with $x\in\Lambda$ and $y\in\Lambda$ and
$\tilde{\sigma}_{b}~{}=~{}\sigma_{x}\sigma_{y}$ where both $\sigma_{x}$ and
$\sigma_{y}$ are summed in the summation $\sum_{\sigma_{\Lambda}}$ over spin
configurations. On the other hand when $b\cap\partial\Lambda\neq\emptyset$,
then we call $b$ boundary bond i.e. $b~{}=~{}\\{x,y\\}$ with $x\in\Lambda$ and
$y\in\partial\Lambda$ and $\tilde{\sigma}_{b}~{}=~{}\sigma_{x}\sigma_{y}$
where only $\sigma_{x}$ is summed in the summation $\sum_{\sigma_{\Lambda}}$,
while $\sigma_{y}~{}=~{}1$

So this time graphs which will not vanish after the sum over spin
configuration are not only closed polygons type lattice animals as before.
There are also new type of lattice animals. New non vanishing lattice animals
are also those which start in a site $y\in\partial\Lambda$ and end in another
site $y^{\prime}\in\partial\Lambda$. See Figure 14 below.

Figure 14. A non vanishing configuration of lattice animals in
$Z_{\Lambda}^{+}(\beta)$.

Any non vanish factor
$(\tanh\beta)^{k}J\tilde{\sigma}_{1}\dots\tilde{\sigma}_{k}$, independently of
the nature of the bonds $b$ (i.e. internal bonds or boundary bonds) becomes,
once the sum over configuration $\sigma_{\Lambda}$ is preformed,
$2^{L^{2}}(\tanh\beta J)^{k}$. Note that all non vanishing factors
$(\tanh\beta)^{k}J\tilde{\sigma}_{1}\dots\tilde{\sigma}_{k}$ are all positive
even if we were imposing $-$ boundary conditions (so that now for a boundary
bond $b=\\{x,y\\}$ with $y\in\partial\Lambda$ we should have $\sigma_{y}=-1$),
because any new type (connected) lattice animals has always an even number of
paths ending at the boundary so that it has only an even number of such points
where $\sigma_{y}=-1$. So that negative values of spins at the boundary
cancels. In conclusion we have again

$Z^{+}_{\Lambda}(\beta)~{}=~{}[\cosh(\beta
J)]^{2L(L-1)}2^{L^{2}}\;\Xi^{+}_{\Lambda}(\beta)$ $None$

where

$\Xi^{+}_{\Lambda}(\beta)~{}=~{}1+\sum_{n\geq
1}\sum_{\\{\gamma_{1},\dots\gamma_{n}\\}:\;|\gamma_{i}|\geq
4\atop\gamma_{i}\sim\gamma_{j}}\xi(\gamma_{1})\dots\xi(\gamma_{n})$ $None$

where again $\gamma$ denote a allowed lattice animal (just recall that there
are in this case more new lattice animals allowed respect to the case of free
boundary conditions) with activity

$\xi(\gamma)~{}=~{}[\tanh(\beta J)]^{|\gamma|}$

Let us now express the mean value $\langle\sigma_{x}\rangle^{+}_{\Lambda}$
(where $x\in\Lambda$) in terms of lattice animals. By definition we have

$\langle\sigma_{x}\rangle^{+}_{\Lambda}~{}=~{}{\sum_{\sigma_{\Lambda}}\sigma_{x}e^{+\beta
J\sum_{\\{x,y\\}\cap\Lambda\neq\emptyset\atop|x-y|~{}=~{}1}\sigma_{x}\sigma_{y}}\over
Z^{+}_{\Lambda}(\beta)}$ $None$

The numerator of the expression above can be easily rewritten in term of
lattice animals as

$\sum_{\sigma_{\Lambda}}\sigma_{x}e^{+\beta
J\sum_{\\{x,y\\}\cap\Lambda\neq\emptyset\atop|x-y|~{}=~{}1}\sigma_{x}\sigma_{y}}~{}=~{}$
$=~{}[\cosh(\beta J)]^{2L(L-1)}2^{L^{2}}\sum_{n\geq
1}\sum_{\\{\gamma_{1},\dots\gamma_{n}\\}:\atop\gamma_{i}\sim\gamma_{j},\;\exists\gamma_{x}}\xi(\gamma_{1})\dots\xi(\gamma_{n})$
$None$

where now in $\sum$ the notation $\exists\gamma_{x}$ means that among lattice
animals $\gamma_{1},\dots\gamma_{n}$ at least one of them starts at $x$ and
end at a boundary site, see Figure 15 for such kind of lattice animals
(actually are also allowed lattice animals which have incidence number $3$ in
$x$). Let as indicate with ${\cal L}_{x}$ the set os all this lattice animals

Figure 15. A non vanishing configuration of lattice animals in the numerator
of $\langle\sigma_{x}\rangle^{+}_{\Lambda}$.

Note also that the factor $1$ (i.e that corresponding to the empty lattice
animal) is no more present in the numerator (7.29).

Thus we can write

$\displaystyle\langle\sigma_{x}\rangle^{+}_{\Lambda}$
$\displaystyle=~{}{\sum_{\\{\gamma_{1},\dots\gamma_{n}\\}:\atop\gamma_{i}\sim\gamma_{j},~{}\exists\gamma_{x}}\xi(\gamma_{1})\dots\xi(\gamma_{n})\over
1+\sum_{\\{\gamma_{1},\dots\gamma_{n}\\}:\;|\gamma_{i}|\geq
4\atop\gamma_{i}\sim\gamma_{j}}\xi(\gamma_{1})\dots\xi(\gamma_{n})}$
$\displaystyle\leq~{}{\sum\limits_{\gamma\in{\cal L}_{x}\atop
V_{\gamma}\cap\partial\Lambda\neq\emptyset}\xi(\gamma)\sum_{\\{\gamma_{1},\dots\gamma_{n}\\}:\;|\gamma_{i}|\geq
4\atop\gamma_{i}\sim\gamma_{j},\;{\gamma_{i}\sim\gamma}}\xi(\gamma_{1})\dots\xi(\gamma_{n})\over
1+\sum_{\\{\gamma_{1},\dots\gamma_{n}\\}:\;|\gamma_{i}|\geq
4\atop\gamma_{i}\sim\gamma_{j}}\xi(\gamma_{1})\dots\xi(\gamma_{n})}$
$\displaystyle\leq~{}\sum_{\gamma\in{\cal L}_{x}\atop
V_{\gamma}\cap\partial\Lambda\neq\emptyset}\xi(\gamma)$

In conclusion we have that

$\langle\sigma_{x}\rangle^{+}_{\Lambda}~{}\leq~{}\sum_{\gamma\in{\cal
L}_{x}\atop V_{\gamma}\cap\partial\Lambda\neq\emptyset}\xi(\gamma)$ $None$

Observe now that if ${\cal L}^{1}_{x}$ denotes the set of paths starting at
$x$ (i.e. lattice animals having degree 1 at $x$) we can reorganize the sum
above as follows

$\sum_{\gamma\in{\cal L}_{x}\atop
V_{\gamma}\cap\partial\Lambda\neq\emptyset}\xi(\gamma)\leq\sum_{\gamma\in{\cal
L}^{1}_{x}\atop
V_{\gamma}\cap\partial\Lambda\neq\emptyset}\xi(\gamma)\left[1+\left(\sum_{{\tilde{\gamma}}\in{\cal
L}^{1}_{x}\atop
V_{\tilde{\gamma}}\cap\partial\Lambda\neq\emptyset}\xi({\tilde{\gamma}})\right)^{2}\right]$

If we denote by $d(x,\partial\Lambda)$ the minimum distance (in nearest
neighbor bonds) between $x$ and the boundary $\partial\Lambda$, we have
clearly that a $\gamma$ in the sum must have at leat
$|\gamma|~{}=~{}d(x,\partial\Lambda)$ bonds. So

$\sum_{\gamma\in{\cal L}^{1}_{x}\atop
V_{\gamma}\cap\partial\Lambda\neq\emptyset}\xi(\gamma)~{}~{}\leq~{}~{}\sum_{n\geq
d(x,\partial\Lambda)}(\tanh\beta J)^{n}\sum_{\gamma\in{\cal
L}^{1}_{x}:\;|\gamma|=n\atop
V_{\gamma}\cap\partial\Lambda\neq\emptyset}1~{}\leq~{}{4\over 3}\sum_{n\geq
d(x,\partial\Lambda)}(3\tanh\beta J)^{n}$

where we have estimated

$\sum_{\gamma\in{\cal L}^{1}_{x}:\;|\gamma|=n\atop
V_{\gamma}\cap\partial\Lambda\neq\emptyset}\leq 4\cdot 3^{n-1}$

The series $\sum_{n\geq d(x,\partial\Lambda)}(3\tanh\beta J)^{n}$ converges
for $3\tanh\beta J<1$ i.e for

$\beta<\beta_{0}^{\prime}\doteq{1\over J}\tanh^{-1}(1/3)\approx{0.34\over J}$
$None$

and in conclusion

$\sum_{\gamma\in{\cal L}^{1}_{x}\atop
V_{\gamma}\cap\partial\Lambda\neq\emptyset}\xi(\gamma)~{}\leq~{}{4\over
3}{(3\tanh\beta J)^{d(x,\partial\Lambda)}\over 1-3\tanh\beta J}$

Therfore we get finally, for $\beta$ as small as in condition (7.31)

$\langle\sigma_{x}\rangle^{+}_{\Lambda}~{}~{}\leq~{}~{}{4\over 3}{(3\tanh\beta
J)^{d(x,\partial\Lambda)}\over 1-3\tanh\beta J}\left[1+\left({4\over
3}{(3\tanh\beta J)^{d(x,\partial\Lambda)}\over 1-3\tanh\beta
J}\right)^{2}\right]~{}\leq$ $\leq~{}{100\over 3}{(3\tanh\beta
J)^{d(x,\partial\Lambda)}\over(1-3\tanh\beta J)^{3}}$

where in the las line we have used that $3\tanh\beta J<1$. Hence the
magnetization at zero magnetic field and at finite volume is bounded by

$\displaystyle M^{+}_{\Lambda}(\beta,h=0)$
$\displaystyle=~{}{1\over|\Lambda|}\sum_{x\in\Lambda}\langle\sigma_{x}\rangle^{+}_{\Lambda}$
$\displaystyle\leq~{}{100\over 3(1-3\tanh\beta
J)^{3}}{1\over|\Lambda|}\sum_{x\in\Lambda}[3\tanh\beta
J]^{d(x,\partial\Lambda)}$ $\displaystyle\leq{100\over 3(1-3\tanh\beta
J)^{3}}{1\over|\Lambda|}\sum_{n=1}^{L/2}[3\tanh\beta
J]^{n}\sum_{x\in\Lambda\atop d(x,\partial\Lambda)=n}1$
$\displaystyle\leq{300\tanh\beta J\over 3(1-3\tanh\beta
J)^{4}}{|\partial\Lambda|\over|\Lambda|}$

hence we get, for $\beta<\beta_{0}^{\prime}$

$\lim_{\Lambda\to\infty}M^{+}_{\Lambda}(\beta,h=0)~{}=~{}~{}0$

Note also, for any $\Lambda$

$\lim_{\beta\to 0}M^{+}_{\Lambda}(\beta,h=0)~{}=~{}~{}0$

Thus when the temperature is sufficiently high the magnetization of the
system, even with + boundary conditions, tends to be zero in the thermodynamic
limit. This in contrast with the result that we have shown for low
temperature, where $M^{+}_{\Lambda}(\beta,h=0)$ is definitely away from zero
and is near to one.

This result say to us that the Ising model does not present the phenomenon of
the spontaneous magnetization when the temperature is sufficiently high, or in
other words the bulk system is not sensible to change of boundary conditions
in the thermodynamic limit. On the contrary, when the temperature is very low,
the system shows indeed spontaneous magnetization and its bulk becomes
sensible to boundary conditions even in the thermodynamic limit.

The interpretation of this fact is that the Ising model has a phase transition
at zero magnetic field and at some critical value $\beta_{c}$ of the inverse
temperature. Below $\beta_{c}$ the system behaves like in the high temperature
regime and above $\beta_{c}$ the system behave like the low temperature
regime.

Note also that $\lim_{\beta\to 0}M^{+}_{\Lambda}(\beta,h=0)$ and
$\lim_{\beta\to 0}M^{-}_{\Lambda}(\beta,h=0)$ has different values, the first
near 1 and the second near -1, and therefore the thermodynamic limit
$M(\beta,h=0)$ of the magnetization (i.e. the derivative of the free energy
$f(\beta,h)$) does not have a definite value. This is an evidence that
$f(\beta,h)$ has discontinous derivative respect to $h$ at $h=0$ when the
inverse temperature is greater than $\beta>\beta_{1}$. On the other hand we
have seen that $\lim_{\beta\to 0}M^{+}_{\Lambda}(\beta,h=0)=\lim_{\beta\to
0}M^{-}_{\Lambda}(\beta,h=0)=0$ when $\beta<\beta^{\prime}_{0}$, therefore
there is a value $\beta_{c}$ such that for $\beta<\beta_{c}$ the infinite
volume magnetization is zero and for $\beta>\beta_{c}$ the magnetization is
different from zero. It can be shown that the free energy is non analytic in
$\beta=\beta_{c}$ as a function of $\beta$.

#####  7.0.4 The critical temperature

As a very last exercise we show that if there is a unique non analytic point
of the free energy $f(\beta)$ at $h~{}=~{}0$ in the interval $(0,\infty)$ then
this point can stay only on a well defined $\beta_{c}$.

Consider the partition function of the Ising model at zero magnetic field
assuming without loss of generality that $J=1$ to simplify things. The low
temperature expansion in terms of closed closed contours of this partition
function in a box $\Lambda$ which is a square of size $L-1$ where $L$ with +
boundary conditions is such that the contours live in the square $\Lambda^{*}$
of size $L$ and the partition function is

$Z^{+}_{(L-1)\times(L-1)}(\beta)~{}=~{}e^{+\beta
2L(L-1)}\sum_{\gamma_{1},\dots,\gamma_{n}:~{}\gamma_{i}\sim\gamma_{j}\atop\gamma_{i}\in
L\times L}\prod_{i=1}^{n}e^{-2\beta|\gamma_{i}|}$ $None$

On the other hand, via high temperature expansion, the partition function of
the zero magnetic field, free boundary conditions Ising model in a box of size
$L$ can be written as

$Z^{\rm open}_{L\times
L}(\beta)~{}=~{}\cosh(\beta)^{2L(L-1)}2^{L^{2}}\sum_{\gamma_{1},\dots,\gamma_{n}:~{}\gamma_{i}\sim\gamma_{j}\atop\gamma\in
L\times L}\prod_{i=1}^{n}[\tanh{\beta}]^{|\gamma_{i}|}$ $None$

Compare now (7.32) and (7.33) and note that the sums on closed polymers in
both equations are identical.

Let $\varphi(\beta)$ be the function of $\beta$ defined by

$\varphi(\beta)~{}=~{}-{\ln[\tanh{\beta}]\over 2}$ $None$

so that

$e^{-2\varphi(\beta)}~{}=~{}\tanh{\beta}$

Then we can write, by (7.32) and (7.33)

${Z^{+}_{(L-1)\times(L-1)}(\varphi(\beta))\over
e^{+2\varphi(\beta)(L-1)(L-2)}}~{}=~{}{Z^{\rm open}_{L\times
L}(\beta)\over\cosh(\beta J)^{2L(L-1)}2^{L^{2}}}$

and taking the logarithm on both sides

$\ln
Z^{+}_{(L-1)\times(L-1)}(\varphi(\beta)){-2\varphi(\beta)(L-1)(L-2)}~{}=~{}$
$=~{}\ln Z^{\rm open}_{L\times L}(\beta)-{2L(L-1)}\ln\cosh(\beta)-L^{2}\ln 2$

Dividing by $L^{2}$ and taking the limit $L\to\infty$

$\lim_{L\to\infty}\Bigg{\\{}{(L-1)^{2}\over L^{2}}{1\over(L-1)^{2})}\ln
Z^{+}_{(L-1)\times(L-1)}(\beta^{*})~{}-~{}2\beta^{*}J{(L-1)(L-2)\over
L^{2}}\Bigg{\\}}~{}=~{}$ $~{}=~{}\lim_{L\to\infty}\left\\{{1\over L^{2}}\ln
Z^{\rm open}_{L\times L}(\beta)-{2L(L-1)\over L^{2}}\ln\cosh(\beta J)-\ln
2\right\\}$

i.e., since free energy $f(\beta)~{}=~{}\lim_{\Lambda\to\infty}\ln
Z^{\tau}_{\Lambda}(\beta)$ does not depend on boundary conditions

$f(\varphi(\beta))-2\varphi(\beta)~{}=~{}f(\beta)-\ln[2\cosh^{2}(\beta)]$

hence we get

$f(\beta)~{}=~{}f(\varphi(\beta))+\ln[2\cosh^{2}(\beta)]-2\varphi(\beta)$

i.e.

$f(\beta)~{}=~{}f(\varphi(\beta))+\ln[2\cosh^{2}(\beta)]+\ln[\tanh{\beta}]$
$None$

We now suppose that $f(\beta)$ can have at most one singularity, and we
conclude via (7.35) that this singularity, if exists, can occur only at
$\beta_{c}$ where $\beta_{c}$ is the solution of the equation
$\beta=\varphi(\beta)$.

Note that $\varphi(\beta)$ (see (7.34)) is analytic as a function of $\beta$
for all $\beta\in(0,\infty)$. Now, by (7.35), $f(\beta)-f(\varphi(\beta))$ is
analytic for all $\beta\in(0,\infty)$. Let now $\beta^{\prime}\in(0,\infty)$
such that $\beta^{\prime}\neq\beta_{c}$, then by (7.35) $f(\beta)$ is analytic
in $\beta~{}=~{}\beta^{\prime}$. In fact suppose, by absurd, that $f(\beta)$
is non analytic in $\beta~{}=~{}\beta^{\prime}$, then
$f(\varphi(\beta^{\prime}))$ must also be non analytic, since only in this way
$f(\beta^{\prime})-f(\varphi(\beta^{\prime}))=\ln
2\cosh(\beta^{\prime})+2\ln[\tanh{\beta^{\prime}}]$ can be non singular. But
if $\beta^{\prime}\neq\beta_{c}$ then
$f(\varphi(\beta^{\prime}))=~{}f(\beta^{\prime\prime})$ with
$\beta^{\prime\prime}\neq\beta^{\prime}$. Hence $f(\beta)$ would have two
singularities which is in contradiction with the hypothesis. This equation is
saying that if $f(\beta)$ has a singularity point in the domain
$\beta\in(0,\infty)$, this point can occur only when $\beta=\beta_{c}$, i.e.
where $\beta_{c}$ is the solution of

$e^{-2\beta}~{}=~{}\tanh{\beta}$

which gives $\beta_{c}={1\over 2}\ln(1+\sqrt{2})$ which is the correct value
calculated via the Onsager solution.

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### Index

  * activity §1.9
  * boundary condition Chapter 7
  * boundary conditions §2.10
  * Cayley formulas §3.1.1
  * ensembles §1.2
    * canonical §1.7
    * grand canonical §1.9, §2.1
    * Micro canonical §1.3
  * entropy §1.3
    * entropy of the ideal gas §1.5, §1.7
  * ergodic hypothesis §1.2, §1.3
  * Gibbs measure §1.2, §1.2
    * configurational §2.1
    * grand canonical §1.9
  * Gibbs paradox §1.6
  * graph §3.1.1
    * connected §3.1.1
    * tree graph §3.1.1
  * Hamiltonian §1.1, §6.2.1
  * ideal gas §1.10, §1.5
  * Ising model §6.1, Chapter 7
  * lattice gas Chapter 6
  * Mayer series §3.1.2
  * pair potential §2.1
    * hard core §2.3.1
    * Lennard-Jones §2.5
    * positive type §2.5
    * regularity §2.1
    * repulsive §2.5
    * stability §2.1
    * stability constant §2.1
  * partition function §1.3, §6.2.1
    * canonical §1.7
    * grand canonical §1.9, §2.1, §5.1.1
    * microcanonical §1.3
    * polymer gas representation §6.2, §6.2.2
  * phase space §1.1
  * phase transition §2.10, §7.0.3
  * Plank constant §1.3
  * polymer gas Part II
  * pressure §1.9, §2.8
    * analyticity §2.10
    * continuity §2.9
  * spin system Chapter 6
  * thermodynamic limit §2.6