Let $V$ be a stable and tempered pair potential with Basuev stability constant
$\bar{B}$. Then the convergence radius ${\cal R}$ of the Virial series (3.77)
admits the following lower bound.

${\cal R}\geq{0.14477\over\tilde{C}(\beta)e^{\beta\bar{B}}}$ $None$

Proof. We follow the steps done by Lebowitz and Penrose in [25] to obtain, via
Lagrange inversion, a lower bound for ${\cal R}$.

We start by observing that, due to (3.76) and the fact that
$C_{1}(\beta,\Lambda)=1$, there exists a circle $C$ of some radius
$R<1/[\tilde{C}(\beta)e^{\beta B+1}]$ and center in the origin $\lambda=0$ of
the complex $\lambda$-plane such that $\rho_{\Lambda}(\beta,\lambda)$ has only
one zero in the disc ${D_{R}}=\\{\lambda\in\mathbb{C}:|\lambda|\leq R\\}$ and
this zero occurs precisely at $\lambda=0$. Let now $\rho\in\mathbb{C}$ be such
that

$|\rho|<\min_{\lambda\in C}|\rho_{\Lambda}(\beta,\lambda)|$ $None$

Then by Rouché’s theorem $\rho_{\Lambda}(\beta,\lambda)$ and
$\rho_{\Lambda}(\beta,\lambda)-\rho$ have the same number of zeros (i.e. one)
in the region $D_{R}=\\{\lambda\in\mathbb{C}:|\lambda|\leq R\\}$. In other
words, for any complex $\rho$ satisfying (3.87) there is only one $\lambda\in
D_{R}$ such that $\rho=\rho_{\Lambda}(\beta,\lambda)$ and therefore we can
invert the equation $\rho=\rho_{\Lambda}(\beta,\lambda)$ and write
$\lambda=\lambda_{\Lambda}(\beta,\rho)$. Thus, according to Cauchy’s argument
principle, we can write the pressure $\beta P_{\Lambda}(\beta,\lambda)$ as a
function of the density $\rho=\rho_{\Lambda}(\beta,\lambda)$ as

$P_{\Lambda}(\rho,\beta)={1\over 2\pi
i}\oint_{\gamma}P_{\Lambda}(\beta,\lambda){d\rho_{\Lambda}(\beta,\lambda)\over
d\lambda}{d\lambda\over\rho_{\Lambda}(\beta,\lambda)-\rho}$ $None$

where $\gamma$ can be any circle centered at the origin in the complex
$\lambda$-plane fully contained in the region $D_{R}$ and such that

$|\rho|<\min_{\lambda\in\gamma}|\rho_{\Lambda}(\beta,\lambda)|$ $None$

By standard complex analysis $P_{\Lambda}(\rho,\beta)$ is analytic in $\rho$
in the region (3.89). Indeed, once (3.89) is satisfied we can write

${1\over\rho_{\Lambda}(\beta,\lambda)-\rho}=\sum_{n=0}^{\infty}{\rho^{n}\over[\rho_{\Lambda}(\beta,\lambda)]^{n+1}}$
$None$

and inserting (3.90) in (3.88) we get

$P_{\Lambda}(\rho,\beta)=\sum_{n=1}^{\infty}c_{n}(\beta,\Lambda)\rho^{n}$
$None$

with

$\displaystyle c_{n}(\beta,\Lambda)$ $\displaystyle={1\over 2\pi
i}\oint_{\gamma}P_{\Lambda}(\beta,\lambda){d\rho_{\Lambda}(\beta,\lambda)\over
d\lambda}{d\lambda\over[\rho_{\Lambda}(\beta,\lambda)]^{n+1}}$
$\displaystyle=-{1\over 2\pi
in}\oint_{\gamma}P_{\Lambda}(\beta,\lambda){d\over
d\lambda}\left[{1\over[\rho_{\Lambda}(\beta,\lambda)]^{n}}\right]d\lambda$
$\displaystyle={1\over 2\pi in}\oint_{\gamma}{dP_{\Lambda}(\beta,\lambda)\over
d\lambda}{1\over[\rho_{\Lambda}(\beta,\lambda)]^{n}}d\lambda$
$\displaystyle=~{}{1\over 2\pi
in\beta}\oint_{\gamma}{1\over[\rho_{\Lambda}(\beta,\lambda)]^{n-1}}{d\lambda\over\lambda}$

Therefore

$|c_{n}(\beta,\Lambda)|\leq{1\over
n\beta}{1\over\left[\min_{\lambda\in\gamma}|\rho_{\Lambda}(\beta,\lambda)|\right]^{n-1}}$
$None$

Inequality (3.92) shows that the convergence radius ${\cal R}$ of the series
(3.91) (i.e. of the virial series (3.77)) is such that

${\cal R}\geq\min_{\lambda\in\gamma}|\rho_{\Lambda}(\beta,\lambda)|$ $None$

Therefore the game is to find an optimal circle $\gamma$ in the region $D_{R}$
which maximizes the r.h.s. of (3.93). We proceed as follows. Recalling (3.76)
we have, by the triangular inequality, that

$|\rho_{\Lambda}(\beta,\lambda)|\geq|\lambda|-\sum_{n=2}^{\infty}n|C_{n}(\beta,\Lambda)||\lambda|^{n}$
$None$

We now use estimate (3.84) to bound the sum in the r.h.s. of (3.94). Therefore
we get

$\displaystyle|\rho_{\Lambda}(\beta,\lambda)|{}$
$\displaystyle\geq~{}|\lambda|-\sum_{n=2}^{\infty}{n^{n-1}\over
n!}[\tilde{C}(\beta)e^{\beta\bar{B}}]^{n-1}|\lambda|^{n}$
$\displaystyle=2|\lambda|-{1\over\tilde{C}(\beta)e^{\beta\bar{B}}}\sum_{n=1}^{\infty}{n^{n-1}\over
n!}[\tilde{C}(\beta)e^{\beta\bar{B}}|\lambda|]^{n}$

Now following [25] let us set $w$ to be the first positive solution of

$we^{-w}=\tilde{C}(\beta)e^{\beta\bar{B}}|\lambda|$ $None$

If we take $|\lambda|<1/e^{\beta\bar{B}+1}\tilde{C}(\beta)$ (which is surely
inside the convergence region since $\bar{B}\geq B$) then
$\tilde{C}(\beta)e^{\beta\bar{B}}|\lambda|<1/e$ and since the function
$we^{-w}$ is increasing in the interval $[0,1]$ and takes the value $1/e$ at
$w=1$, there is a unique $w$ in the interval $[0,1]$ which solves (3.95). Now
we use the Euler’s formula

$w=\sum_{n=1}^{\infty}{n^{n-1}\over n!}(we^{-w})^{n}$

to get

$|\rho_{\Lambda}(\beta,\lambda)|\geq{w\over\tilde{C}(\beta)e^{\beta\bar{B}}}\left[2e^{-w}-1\right]$
$None$

Observe that the r.h.s. (3.96) is greater than zero when $w$ varies in the
interval $(0,\ln 2)$. Therefore, any circle $\gamma$ around the origin with
radius $R_{\gamma}$ between zero and $\ln
2/(2e^{\beta\bar{B}+1}\tilde{C}(\beta))$, which corresponds, in force of
(3.95), to any $w$ in the interval $(0,\ln 2)$, is surely in the region
$D_{R}$. So we have

$\max_{\gamma\subset
D_{R}}\min_{\lambda\in\gamma}|\rho_{\Lambda}(\beta,\lambda)|\geq{1\over\tilde{C}(\beta)e^{\beta\bar{B}}}\max_{w\in(0,\ln
2)}w\left[2e^{-w}-1\right]$ $None$

which, recalling (3.93) and since $\max_{w\in(0,\ln 2)}{w}[2e^{-w}-1]\geq
0.14477$, concludes the proof.

##  Part II Discrete systems

###  Chapter 4 The Abstract Polymer gas

####  4.1 Setting

The abstract polymer gas is a discrete model which plays a very important role
in many physical situations, such as spin systems on the lattice at low or
high temperature, or continuous or discrete field theories. The polymer gas
was first introduced by Gruber and Kunz [20] in 1970. In the original Gruber
and Kunz paper polymer were finite subsets of $\mathbb{Z}^{d}$ (the unit cubic
lattice). Later, Kotecyý and Preiss proposed the abstract model in which
polymers were abstract objects belonging to some set ${\cal P}$, the polymer
set, whose unique structure was given by mean of a symmetric and reflexive
relation in ${\cal P}$, that they called the incompatibility relation. As we
will see, this is equivalent to assume that the interaction between polymers
occurs via a hard core pair potential. In the present chapter we revisit the
abstract polymer gas.

We first specify a countable set ${\cal P}$ whose elements are all possible
polymers (i.e. ${\cal P}$ is the single particle state space). We then
associate to each polymer $\gamma\in{\cal P}$ a complex number $z_{\gamma}$ (a
positive number in physical situations) which is interpreted as the activity
of the polymer $\gamma$. We will denote
$\bm{z}=\\{z_{\gamma}\\}_{\gamma\in{\cal P}}$ and for any $\Lambda\subset{\cal
P}$, $\bm{z}_{\Lambda}=\\{z_{\gamma}\\}_{\gamma\in\Lambda}$.

In the general situation polymers interact through a pair potential. Namely,
the energy $E$ of a configuration $\gamma_{1},\dots,\gamma_{n}$ of $n$
polymers is given by

$E(\gamma_{1},\dots,\gamma_{n})=\sum_{1\leq i<j\leq
n}V(\gamma_{i},\gamma_{j})$ $None$

where pair potential $V(\gamma,\gamma^{\prime})$ is a symmetric function in
${\cal P}\times{\cal P}$ taking values in $\mathbb{R}\cup\\{+\infty\\}$. We
will make the assumption that the pair interaction $V(\gamma,\gamma^{\prime})$
is purely hard-core. Namely $V(\gamma,\gamma^{\prime})$ takes values in the
set $\\{0,+\infty\\}$. Observe that an hard-core pair potential
$V(\gamma,\gamma^{\prime})$ in ${\cal P}$ induces a relation $\mathcal{R}_{V}$
in ${\cal P}\times{\cal P}$ (i.e. $\mathcal{R}_{V}$ is a subset of ${\cal
P}\times{\cal P}$). Namely, we say that a pair $(\gamma,\gamma^{\prime})$
belongs to $\mathcal{R}_{V}$ if and only if
$V(\gamma,\gamma^{\prime})=+\infty$. Clearly $\mathcal{R}_{V}$ is symmetric
because by assumption $V$ is symmetric. When
$(\gamma,\gamma^{\prime})\in\mathcal{R}_{V}$ (i.e.
$V(\gamma,\gamma^{\prime})=+\infty$), we write $\gamma\not\sim\gamma^{\prime}$
and say that $\gamma$ and $\gamma^{\prime}$ are incompatible. Conversely, if
$(\gamma,\gamma^{\prime})\notin\mathcal{R}_{V}$ we say that the polymers
$\gamma$ and $\gamma^{\prime}$ are compatible and we write
$\gamma\sim\gamma^{\prime}$. Note that if $V$ is such that $\mathcal{R}_{V}$
is reflexive, then $\gamma\not\sim\gamma$ for all $\gamma\in{\cal P}$. The
relation ${\cal R}_{V}$ induced by the pair potential $V$ is called the
incompatibility relation.

Fix now a finite set $\Lambda\subset{\cal P}$ (the “volume” of the gas). Then,
for $z_{\gamma}\geq 0$, the probability to see the configuration
$(\gamma_{1},\dots,\gamma_{n})\in\Lambda^{n}$ is given by

$Prob(\gamma_{1},\dots,\gamma_{n})={1\over\Xi_{\Lambda}}{1\over
n!}{{z_{\gamma_{1}}}{z_{\gamma_{2}}}\dots z_{\gamma_{n}}e^{-\sum_{1\leq
i<j\leq n}V(\gamma_{i},\gamma_{j})}}$

where the normalization constant $\Xi_{\Lambda}$ is the grand-canonical
partition function in the volume $\Lambda$ and is given by

$\Xi_{\Lambda}(\bm{z}_{\Lambda})=1+\sum_{n\geq 1}{1\over
n!}\sum_{(\gamma_{1},\dots,\gamma_{n})\in\Lambda^{n}}{z_{\gamma_{1}}}{z_{\gamma_{2}}}\dots
z_{\gamma_{n}}e^{-\sum_{1\leq i<j\leq n}V(\gamma_{i},\gamma_{j})}$ $None$

Note that configurations $\gamma_{1},\dots,\gamma_{n}$ for which there exists
some incompatible pair $\gamma_{i},\gamma_{j}$, i.e. such that
$V(\gamma_{i},\gamma_{j})=+\infty$ have zero probability to occur, i.e. are
forbidden.

Remark. The fact that $V(\gamma,\gamma^{\prime})\geq 0$ immediately implies
that $\Xi_{\Lambda}$ is convergent and

$|\Xi_{\Lambda}(\bm{z}_{\Lambda})|\leq 1+\sum_{n\geq 1}{1\over
n!}\Bigg{[}\sum_{\gamma\subset\Lambda}|z_{\gamma}|\Bigg{]}^{n}\leq\exp\Big{\\{}\sum_{\gamma\in\Lambda}|z_{\gamma}|\Big{\\}}\leq\max_{\gamma\in\Lambda}e^{|z_{\gamma}||\Lambda|}$

Actually, $V(\gamma\,\gamma^{\prime})\geq 0$ implies that $\Xi_{\Lambda}(z)$
is analytic in the whole $\mathbb{C}^{|\Lambda|}$ ($|\Lambda|$ is the
cardinality of $\Lambda$). The “pressure” of this gas is defined via the
formula

$P_{\Lambda}(\bm{z}_{\Lambda})={1\over|\Lambda|}\log\Xi_{\Lambda}(\bm{z}_{\Lambda})$
$None$

While the partition function $\Xi_{\Lambda}(\bm{z}_{\Lambda})$ diverges as
$\Lambda\to{\cal P}$, the pressure $P_{\Lambda}(\bm{z}_{\Lambda})$ is supposed
to have a finite limit at least when $z$ varies in some finite polydisc
$|z_{\gamma}|\leq\rho_{\gamma}$ with
$\bm{\rho}=\\{\rho_{\gamma}\\}_{\gamma\in{\cal P}}$ being some positive
function $\bm{\rho}:{\cal P}\to\mathbb{R}^{+}:\gamma\mapsto\rho_{\gamma}$
defined on ${\cal P}$ independent on $\Lambda$. So that in principle it should
be possible to give an upper bound for $|P_{\Lambda}(\bm{z})|$ which is
uniform in $\Lambda$.

The pressure (4.3) can be written as a formal series using the Mayer trick in
the partition function (4.2) by writing the Gibbs factor as

$e^{-\\!\sum_{1\leq i<j\leq n}V(\gamma_{i},\gamma_{j})}\,=\,\prod_{1\leq
i<j\leq n}\\!\left[(e^{-V(\gamma_{i},\gamma_{j})}-1)+1\right]$

Proceeding exactly as in section 3.1.2 with the only difference if that now
$V(\gamma_{i},\gamma_{j})$ is in place of $V(x_{i}-x_{j})$ and
$\sum_{\gamma_{i}\in\Lambda}$ is in place of $\int_{\Lambda}dx_{i}$ we obtain

$\log\Xi_{\Lambda}(\bm{z}_{\Lambda})=\sum_{n=1}^{\infty}{1\over
n!}\sum_{(\gamma_{1},\dots,\gamma_{n})\in\Lambda^{n}}\phi^{T}(\gamma_{1},\dots,\gamma_{n})\,z_{\gamma_{1}}\dots\,z_{\gamma_{n}}$
$None$

with

$\phi^{T}(\gamma_{1},\dots,\gamma_{n})=\begin{cases}1&{\text{i}f}~{}n=1\\\
\sum\limits_{g\in G_{n}}\prod\limits_{\\{i,j\\}\in
E_{g}}(e^{-V(\gamma_{i},\gamma_{j})}-1)&{\text{i}f}~{}n\geq 2\end{cases}$
$None$

where $\sum_{g\in{G}_{n}}$ is the sum over all connected graphs between $[n]$.

Observe that, since $V(\gamma,\gamma^{\prime})\geq 0$, by Proposition 3.3 it
holds

$\phi^{T}(\gamma_{1},\dots,\gamma_{n})=(-1)^{n-1}|\phi^{T}(\gamma_{1},\dots,\gamma_{n})|$
$None$

The equation (4.4) makes sense only for those $\bm{z}\in\mathbb{C}^{\cal P}$
such that the formal series in the r.h.s. of (4.4) converge absolutely. To
study absolute convergence, we will consider, for any $\Lambda$, the positive
term series

$|\log\Xi|_{\Lambda}(\bm{\rho}_{\Lambda})=\sum_{n=1}^{\infty}{1\over
n!}\sum_{(\gamma_{1},\dots,\gamma_{n})\in\Lambda^{n}}|\phi^{T}(\gamma_{1},\dots,\gamma_{n})|\rho_{\gamma_{1}}\cdots{\rho_{\gamma_{n}}}$
$None$

for $\bm{\rho}\in(0,\infty)^{{\cal P}}$. Note that

$|\log\Xi_{\Lambda}(\bm{z}_{\Lambda})|\leq|\log\Xi|_{\Lambda}(\bm{\rho}_{\Lambda})$

for all $\bm{z}\in\mathbb{C}^{|\Lambda|}$ in the poly-disc
$\\{|z_{\gamma}|\leq\rho_{\gamma}\\}_{\gamma\in\Lambda}$, so that if we are
able to prove that the series (4.7) converges, for all $\Lambda$, at some
value $\bm{\rho}=\\{\rho_{\gamma}\\}_{\gamma\in{\cal P}}\in(0,\infty)^{{\cal
P}}$, then we have also proved that the series (4.4) converges absolutely, for
all $\Lambda$, whenever $\bm{z}$ is in the poly-disk
$\\{|z_{\gamma}|\leq\rho_{\gamma}\\}_{\gamma\in{\cal P}}$. We also observe
that, since $V(\gamma,\gamma^{\prime})\geq 0$, by (4.6)

$|\log\Xi|_{\Lambda}(\bm{\rho}_{\Lambda})=-\log\Xi_{\Lambda}(-\bm{\rho}_{\Lambda})$
$None$

To study the convergence of the pressure (4.3) it is convenient to consider
the quantity

$|\Pi|_{\gamma_{0}}(\bm{\rho})=\sum_{n=0}^{\infty}{1\over
n!}\sum_{(\gamma_{1},\gamma_{2},\dots,\gamma_{n})\in{\cal
P}^{n}}|\phi^{T}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})|\,\,\rho_{\gamma_{1}}\cdots{\rho_{\gamma_{n}}}$
$None$

Note that in (4.9) the sum over each polymer $\gamma_{i}$ is not anymore
restricted to the finite “volume” $\Lambda$ and it runs over the full polymer
space ${\cal P}$.

If we are able to show that $|\Pi|_{\gamma_{0}}(\bm{\rho})$ converges for some
(bounded) positive function $\bm{\rho}\in[0,\infty)^{\cal P}$, then also the
pressure $P_{\Lambda}(\bm{z}_{\Lambda})$ converges absolutely, for all
$\Lambda$, whenever $z$ is in the poly-disk
$\\{|z_{\gamma}|\leq\rho_{\gamma}\\}_{\gamma\in{\cal P}}$ and in this poly-
disk it uniformly in $\Lambda$. Indeed in the poly-disk
$\\{|z_{\gamma}|\leq\rho_{\gamma}\\}_{\gamma\in{\cal P}}$ we have

$|P_{\Lambda}(\bm{z}_{\Lambda})|={1\over|\Lambda|}|\log\Xi_{\Lambda}(\bm{z}_{\Lambda})|\leq{1\over|\Lambda|}|\log\Xi|_{\Lambda}(\bm{\rho}_{\Lambda})~{}=$
$={1\over|\Lambda|}\sum_{n=1}^{\infty}{1\over
n!}\sum_{(\gamma_{1},\dots,\gamma_{n})\in\Lambda^{n}}|\phi^{T}(\gamma_{1},\dots,\gamma_{n})|\rho_{\gamma_{1}}\cdots{\rho_{\gamma_{n}}}=$
$={1\over|\Lambda|}\sum_{\gamma_{0}\in\Lambda}\rho_{\gamma_{0}}\sum_{n=1}^{\infty}{1\over
n!}\sum_{(\gamma_{1},\dots,\gamma_{n-1})\in\Lambda^{n-1}}|\phi^{T}(\gamma_{0},\gamma_{1},\dots,\gamma_{n-1})|\rho_{\gamma_{1}}\cdots{\rho_{\gamma_{n-1}}}~{}=$
$\leq~{}{1\over|\Lambda|}\sum_{\gamma_{0}\in\Lambda}\rho_{\gamma_{0}}\sum_{m=0}^{\infty}{1\over(m+1)!}\sum_{(\gamma_{1},\dots,\gamma_{m})\in\Lambda^{m}}|\phi^{T}(\gamma_{0},\gamma_{1}\dots,\gamma_{m})|\rho_{\gamma_{1}}\cdots{\rho_{\gamma_{m}}}~{}\leq$
${1\over|\Lambda|}\sum_{\gamma_{0}\in\Lambda}\rho_{\gamma_{0}}\sum_{m=0}^{\infty}{1\over
m!}\sum_{(\gamma_{1},\dots,\gamma_{n})\in\Lambda^{m}}|\phi^{T}(\gamma_{0},\gamma_{1}\dots,\gamma_{m})|\rho_{\gamma_{1}}\cdots{\rho_{\gamma_{m}}}~{}=$
$=~{}{1\over|\Lambda|}\sum_{\gamma_{0}\in\Lambda}\rho_{\gamma_{0}}|\Pi|_{\gamma_{0}}(\bm{\rho})~{}\leq\sup_{\gamma_{0}\in\Lambda}\,\,\rho_{\gamma_{0}}\,|\Pi|_{\gamma_{0}}(\bm{\rho})~{}\leq~{}\sup_{\gamma_{0}\in{\cal
P}}\,\,\rho_{\gamma_{0}}\,|\Pi|_{\gamma_{0}}(\bm{\rho})$

In short, for all $\bm{z}$ in the polydisk
$\\{|z_{\gamma}|\leq\rho_{\gamma}\\}_{\gamma\in{\cal P}}$, it holds

$|P_{\Lambda}(\bm{z}_{\Lambda})|~{}\leq~{}\sup_{\gamma_{0}\in{\cal
P}}\,\,\rho_{\gamma_{0}}\,|\Pi|_{\gamma_{0}}(\bm{\rho})$ $None$

So in the next sections we will focus our attention on the formal series
$|\Pi|_{\gamma_{0}}(\bm{\rho})$ defined in equation (4.9). Indeed, when one is
able to prove that the series (4.9) then, by (4.10) he has also proved the
absolute convergence of the pressure of the polymer gas uniformly in the
volume $\Lambda$.

####  4.2 Convergence of the abstract polymer gas

As explained in the previous section, to study absolute convergence of the
pressure we will just need to prove that the positive term series defined in
(4.9)

$|\Pi|_{\gamma_{0}}(\bm{\rho})=\sum_{n=0}^{\infty}{1\over
n!}\sum_{(\gamma_{1},\gamma_{2},\dots,\gamma_{n})\in{\cal
P}^{n}}|\phi^{T}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})|\,\,\rho_{\gamma_{1}}\cdots{\rho_{\gamma_{n}}}$

is convergent for some $\bm{\rho}\in(0,+\infty)^{\cal P}$.

Since the interaction $V(\gamma,\gamma^{\prime})$ is purely hard core, we can
use the original Penrose identity to bound the factor Ursell factor
$|\phi^{T}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})|$. We define the map
$\bm{m}:T^{0}_{n}\to G^{0}_{n}$ as in Definition 3.6. By rooting trees in
$T_{n}^{0}$ in $0$, given $\tau\in T^{0}_{n}$, $d(i)$ denotes the depth of the
vertex $i$ (i.e. its edge distance from $0$) and that $i^{\prime}$ denotes the
parent of $i$. The map ${\rm\bf m}:T^{0}_{n}\to G^{0}_{n}$ is therefore the
map such that to each tree $\tau\in T^{0}_{n}$ associates the graph ${\rm\bf
m}(\tau)\in G^{0}_{n}$ formed by adding to $\tau$ all edges $\\{i,j\\}$ such
that either:

  * (p1)

$d_{\tau}(i)=d_{\tau}(j)$ (edges between vertices of the same generation), or

  * (p2)

$d_{\tau}(j)=d_{\tau}(i)-1$ and $j>i^{\prime}$ (edges between vertices with
generations differing by one).

As shown previously the map ${\rm\bf m}$ is a partition scheme and therefore
by Theorem 3.3 we have that

$\phi^{T}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})=\sum\limits_{g\in
G^{0}_{n}}\prod\limits_{\\{i,j\\}\in
E_{g}}(e^{-V(\gamma_{i},\gamma_{j})}-1)=\sum_{\tau\in
T^{0}_{n}}w_{\tau}(\gamma_{0}\,\gamma_{1},\dots,\gamma_{n})$

with

$w_{\tau}(\gamma_{0}\,\gamma_{1},\dots,\gamma_{n})=e^{-\sum_{\\{i,j\\}\in
E_{{\rm\bf m}(\tau)}\backslash
E_{\tau}}V(\gamma_{i},\gamma_{j})}\prod_{\\{i,j\\}\in\tau}\left(e^{-V(\gamma_{i},\gamma_{j})}-1\right)$

As shown in Section 3.4 (see there Theorem 3.5 and formula (3.55)) we have

$w_{\tau}(\gamma_{0}\,\gamma_{1},\dots,\gamma_{n})=(-1)^{n}\hbox
to0.0pt{\mbox{\small\rm 1}\hss}\kern
1.49994pt1_{P(\gamma_{0},\gamma_{1},\dots,\gamma_{n})}(\tau)$

where $P(\gamma_{0},\gamma_{1},\dots,\gamma_{n})$ is the set of Penrose trees
and $\hbox to0.0pt{\mbox{\small\rm 1}\hss}\kern 1.49994pt1_{\tau\in
P(\gamma_{0},\gamma_{1},\dots,\gamma_{n})}$ is the characteristic function of
the set $P(\gamma_{0},\gamma_{1},\dots,\gamma_{n})$ in $T^{0}_{n}$, i.e.

$\hbox to0.0pt{\mbox{\small\rm 1}\hss}\kern
1.49994pt1_{P(\gamma_{0},\gamma_{1},\dots,\gamma_{n})}(\tau)\;=\;\begin{cases}1&{\text{i}f}~{}\tau\in
P(\gamma_{0},\gamma_{1},\dots,\gamma_{n})\\\
0&{\text{i}fotherwise}\end{cases}$

We remind its definition below.

###### Definition 4.1

Given $(\gamma_{0},\gamma_{1},\dots,\gamma_{n})\in{\cal P}^{n+1}$, we define
the set of _Penrose trees_ $P(\gamma_{0},\gamma_{1},\dots,\gamma_{n})$, as the
set formed by the trees $\tau\in T^{0}_{n}$ such that

  1. (t0)

if $\\{i,j\\}\in E_{\tau}$ then $\gamma_{i}\nsim\gamma_{j}$

  2. (t1)

if two vertices $i$ and $j$ siblings then $\gamma_{i}\sim\gamma_{j}$;

  3. (t2)

if two vertices $i$ and $j$ are not siblings but $d_{\tau}(i)=d_{\tau}(j)$,
then $\gamma_{i}\sim\gamma_{j}$;

  4. (t3)

if two vertices $i$ and $j$ are s. t. $d_{\tau}(j)=d_{\tau}(i)-1$ and
$j>i^{\prime}$, then $\gamma_{i}\sim\gamma_{j}$.

Therefore we get

$\phi^{T}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})\;=\;(-1)^{n}\sum\limits_{\tau\in
T_{n}^{0}}\hbox to0.0pt{\mbox{\small\rm 1}\hss}\kern
1.49994pt1_{P(\gamma_{0},\gamma_{1},\dots,\gamma_{n})}(\tau)$ $None$

Using (4.11) we can now rewrite the formal series (4.9) as

$|\Pi|_{\gamma_{0}}(\bm{\rho})=\sum_{n=0}^{\infty}{1\over
n!}\sum_{(\gamma_{1},\gamma_{2},\dots,\gamma_{n})\in{\cal
P}^{n}}\sum\limits_{\tau\in T_{n}^{0}}\hbox to0.0pt{\mbox{\small\rm
1}\hss}\kern
1.49994pt1_{P_{G}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})}(\tau)\,\,{\rho_{\gamma_{1}}}\dots\rho_{\gamma_{n}}\;=$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\;\sum_{n=0}^{\infty}{1\over
n!}\sum\limits_{\tau\in
T_{n}^{0}}\sum_{(\gamma_{1},\gamma_{2},\dots,\gamma_{n})\in{\cal P}^{n}}\hbox
to0.0pt{\mbox{\small\rm 1}\hss}\kern
1.49994pt1_{P_{G}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})}(\tau)\,\,{\rho_{\gamma_{1}}}\dots\rho_{\gamma_{n}}\;=$
$=\;\sum_{n=0}^{\infty}{1\over n!}\sum\limits_{\tau\in
T_{n}^{0}}\phi_{\gamma_{0}}(\tau)\;\;\;\;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,~{}~{}~{}~{}~{}~{}~{}$
$None$

where

$\phi_{\gamma_{0}}(\tau)=\;\sum_{(\gamma_{1},\gamma_{2},\dots,\gamma_{n})\in{\cal
P}^{n}}\hbox to0.0pt{\mbox{\small\rm 1}\hss}\kern
1.49994pt1_{P_{G}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})}(\tau)\,\,{\rho_{\gamma_{1}}}\dots\rho_{\gamma_{n}}\;$
$None$

The structure of (4.12) is crucial. This equation shows that the formal series
$|\Pi|_{\gamma_{0}}(\rho)$ can be reaorganized as a sum over terms associated
to labelled trees.

We stress that here the factor $\phi_{\gamma_{0}}(\tau)$ depends on the
labelled tree $\tau$ because of the Penrose condition (t3) which indeed
depends on the labelling of the tree.

We will see below that efficient bounds on the factor
$\phi_{\gamma_{0}}(\tau)$ defined in (4.13) above can be obtained by choosing
a family of trees $\tilde{P}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})$ such
that
${P}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})\subset{\tilde{P}}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})$.
Below we will consider three possible choice of
$\tilde{P}{(\gamma_{0},\gamma_{1},\dots,\gamma_{n})}$ which are the choices
which yield the three known criteria for the convergence of cluster expansion
of the abstract polymer gas.

###### Definition 4.2

Given $(\gamma_{0},\gamma_{1},\dots,\gamma_{n})\in{\cal P}^{n+1}$, the set of
weakly the Penrose trees, which we denote by is defined as
${P}^{*}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})$, is formed by all trees
$\tau$ with vertex set $\\{0,1,\dots,n\\}$ and edge set $E_{\tau}$ such that

  1. $\rm(t0)\;\,$

if $\\{i,j\\}\in E_{\tau}$ then $\gamma_{i}\not\sim\gamma_{j}$

  2. $\rm(t1)^{*}$

if $i$ and $j$ are siblings then $\gamma_{i}\sim\gamma_{j}$;

###### Definition 4.3

Given $(\gamma_{0},\gamma_{1},\dots,\gamma_{n})\in{\cal P}^{n+1}$, the set of
the Dobrushin trees, which we denote by is defined as ${P}^{\rm
Dob}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})$, is the set the trees $\tau$
with vertex set $\\{0,1,\dots,n\\}$ and edge set $E_{\tau}$ such that

  1. $\rm(t0)\;\,$

if $\\{i,j\\}\in E_{\tau}$ then $\gamma_{i}\not\sim\gamma_{j}$

  2. $\rm(t1)^{D}$

if $i$ and $j$ are siblings then $\gamma_{i}\neq\gamma_{j}$;

###### Definition 4.4

Given $(\gamma_{0},\gamma_{1},\dots,\gamma_{n})\in{\cal P}^{n+1}$, the set of
Kotecký-Preiss trees, which we denote by ${P}^{\rm
KP}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})$ is formed by all trees $\tau$
with vertex set $\\{0,1,\dots,n\\}$ and edge set $E_{\tau}$ such that

  1. $\rm(t0)\;\,$

if $\\{i,j\\}\in E_{\tau}$ then $\gamma_{i}\not\sim\gamma_{j}$

Note that we have, by definition, that

$\displaystyle{P}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})$
$\displaystyle\subset\;{P}^{*}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})$ $None$
$\displaystyle\subset\;{P}^{\rm Dob}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})$
$\displaystyle\subset\;{P}^{\rm KP}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})$

So that, by (3.51), we have the bounds

$\displaystyle|\phi^{T}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})|$
$\displaystyle\leq\sum_{\tau\in T_{n}^{0}}\hbox to0.0pt{\mbox{\small\rm
1}\hss}\kern
1.49994pt1_{{P}^{*}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})}(\tau)$ $None$
$\displaystyle\leq\sum_{\tau\in T_{n}^{0}}\hbox to0.0pt{\mbox{\small\rm
1}\hss}\kern 1.49994pt1_{P^{\rm
Dob}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})}(\tau)$
$\displaystyle\leq\sum_{\tau\in T_{n}^{0}}\hbox to0.0pt{\mbox{\small\rm
1}\hss}\kern 1.49994pt1_{P^{\rm
KP}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})}(\tau)$

The latter (4.15), i.e. the worst among the three bound proposed, is known as
the Rota bound. This was the bound which was used by Cammarota [9], Brydges
[7] and Simon [50] to obtain a direct proof of the absolute convergence of the
pressure of a Polymer gas by directly estimating the Ursell coefficent.

Hence, we can bound the positive term series $|\Pi|_{\gamma_{0}}(\bm{\rho})$
defined in (4.9), using of course the first of the estimates (4.15) which is
the best among the three given above, as

$|\Pi|_{\gamma_{0}}(\bm{\rho})\;=\;\sum_{n=0}^{\infty}{1\over
n!}\sum_{(\gamma_{1},\gamma_{2},\dots,\gamma_{n})\in{\cal
P}^{n}}|\phi^{T}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})|{\rho_{\gamma_{1}}}\dots\rho_{\gamma_{n}}\;\leq$
$\leq\;\sum_{n=0}^{\infty}{1\over
n!}\sum_{(\gamma_{1},\gamma_{2},\dots,\gamma_{n})\in{\cal P}^{n}}\sum_{\tau\in
T_{n}^{0}}\hbox to0.0pt{\mbox{\small\rm 1}\hss}\kern
1.49994pt1_{{P}^{*}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})}(\tau){\rho_{\gamma_{1}}}\dots\rho_{\gamma_{n}}\;=$
$=\;\sum_{n=0}^{\infty}{1\over n!}\sum_{\tau\in
T_{n}^{0}}\sum_{(\gamma_{1},\gamma_{2},\dots,\gamma_{n})\in{\cal P}^{n}}\hbox
to0.0pt{\mbox{\small\rm 1}\hss}\kern
1.49994pt1_{{P}^{*}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})}(\tau){\rho_{\gamma_{1}}}\dots\rho_{\gamma_{n}}$

So we get

$|\Pi|_{\gamma_{0}}(\bm{\rho})\;\leq\;\Pi^{*}_{\gamma_{0}}(\bm{\rho})$ $None$

where

$\Pi^{*}_{\gamma_{0}}(\bm{\rho})\;=\;\sum_{n=0}^{\infty}{1\over
n!}\sum_{\tau\in T^{0}_{n}}\phi^{*}_{\gamma_{0}}(\tau,\bm{\rho})\;$ $None$

with

$\phi^{*}_{\gamma_{0}}(\tau,\bm{\rho})=\;\sum_{(\gamma_{1},\gamma_{2},\dots,\gamma_{n})\in{\cal
P}^{n}}\hbox to0.0pt{\mbox{\small\rm 1}\hss}\kern
1.49994pt1_{{P}^{*}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})}(\tau)\,\,{\rho_{\gamma_{1}}}\dots\rho_{\gamma_{n}}\;$
$None$

Analogously, using the second and third bounds (4.15) we can define two more
series which also majorize the series $|\Pi|_{\gamma_{0}}(\rho)$. Namely

$\Pi^{\rm Dob}_{\gamma_{0}}(\bm{\rho})\;=\;\sum_{n=0}^{\infty}{1\over
n!}\sum_{\tau\in T^{0}_{n}}\phi^{\rm Dob}_{\gamma_{0}}(\tau,\rho)\;$ $None$

and

$\Pi^{\rm KP}_{\gamma_{0}}(\bm{\rho})\;=\;\sum_{n=0}^{\infty}{1\over
n!}\sum_{\tau\in T^{0}_{n}}\phi^{\rm KP}_{\gamma_{0}}(\tau,\rho)\;$ $None$

with

$\phi^{\rm
Dob}_{\gamma_{0}}(\tau,\bm{\rho})=\;\sum_{(\gamma_{1},\gamma_{2},\dots,\gamma_{n})\in{\cal
P}^{n}}\hbox to0.0pt{\mbox{\small\rm 1}\hss}\kern 1.49994pt1_{P^{\rm
Dob}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})}(\tau)\,\,{\rho_{\gamma_{1}}}\dots\rho_{\gamma_{n}}\;$
$None$

and

$\phi^{\rm
KP}_{\gamma_{0}}(\tau,\bm{\rho})=\;\sum_{(\gamma_{1},\gamma_{2},\dots,\gamma_{n})\in{\cal
P}^{n}}\hbox to0.0pt{\mbox{\small\rm 1}\hss}\kern 1.49994pt1_{P^{\rm
KP}(\gamma_{0},\gamma_{1},\dots,\gamma_{n})}(\tau)\,\,{\rho_{\gamma_{1}}}\dots\rho_{\gamma_{n}}\;$
$None$

Here it is important to stress that, differently form the factor
$\phi_{\gamma_{0}}(\tau)$ defined in (4.13), the three factors $\phi^{*,\,\rm
Dob,\,KP}_{\gamma_{0}}(\tau)$ do not depend on the labels of the tree $\tau$,
but only on its topological structure. This means that the terms in the series
$\Pi^{*}_{\gamma_{0}}(\rho)$ can further be grouped together in terms of non
unlabelled rooted trees. We will make this concept precise in the next
section. We conclude this section by remarking that inequalities (4.15)
immediately imply

$|\Pi|_{\gamma_{0}}(\bm{\rho})\;\leq\;\Pi^{*}_{\gamma_{0}}(\bm{\rho})\;\leq\;\Pi^{\rm
Dob}_{\gamma_{0}}(\bm{\rho})\;\leq\;\Pi^{\rm KP}_{\gamma_{0}}(\bm{\rho})$
$None$

#####  4.2.1 Reorganization of the series $\Pi^{*}_{\gamma_{0}}(\bm{\rho})$

We now reorganize the sum over rooted labelled trees appearing in formula
(4.17) in terms of the plane rooted trees. Such reorganization is motivated by
the observation that the factors (4.18)-(4.22) do not depend on the labels
assigned to the vertices of $\tau$ but only its topological structure. As a
matter of fact, to each labelled ordered rooted tree $\tau\in T^{0}_{n}$ we
can associate a drawning in the plane known as the “plane rooted tree”
associated to $\tau$. The drawing of $\tau$ is obtained by putting parents at
the left of their children which are ordered in the top-to-bottom order
consistently with the order of their labels. For example the plane rooted
trees with $n+1=5$ vertices associated to the trees $\bf a$ with edge set
$\\{0,3\\},\\{1,3\\},\\{2,3\\},\\{1,4\\}$, $\bf b$ with edge set
$\\{0,2\\},\\{0,3\\},\\{1,2\\},\\{2,4\\}$ and $\bf c$ with edge set
$\\{0,2\\},\\{0,4\\},\\{4,3\\},\\{1,4\\}$ are drawn below

Figure

Observe that $\it b$ and $\it c$ are different plane rooted tree (because of
the rule of the ordering of the children from top-to-bottom).

In this way we have defined a map $m:\tau\mapsto m(\tau)$ which associate to
each labelled tree $\tau\in T^{0}_{n}$ a unique drawing $t=m(\tau)$ in the
plane, called the planar rooted tree associated to $\tau$. We denote by
${\mathcal{T}}^{0}_{n}=$ the set of all planar rooted trees with $n+1$
vertices and by $\mathcal{T}^{0,k}$ the set of planar rooted trees with
maximal generation number $k$; let also $\mathcal{T}^{0}=\cup_{n\geq
0}{\mathcal{T}}^{0}_{n}=\cup_{k\geq 0}\mathcal{T}^{0,k}$ be the set of all
planar rooted trees. An element $t\in\mathcal{T}^{0}_{n}$ can also be viewed
as an equivalence class of elements $\tau\in T_{n+1}$ with the equivalence
relation being that two elements $\tau$ and $\tau^{\prime}$ are equivalent if
the produce the same planar rooted tree. So when we write $\tau\in t$ with
$t\in\mathcal{T}^{0}_{n}$ we mean that $\tau$ is an element of the set of all
labelled trees in $T^{0}_{n}$ that produce the same plane ordered rooted tree.

We will use the following notations. Given a vertex $v\neq 0$ in a rooted tree
(with root $0$), we denote by $v^{\prime}$ is parent, we denote by $s_{v}$ the
number of its children and we denoted by $v^{1},\dots,v^{s_{v}}$ the children
of $v$. If $s_{v}=0$ we say that $v$ is an end-point or a leaf of $\tau$.

Note that the set of vertices of a labeled rooted tree $\tau\in T^{0}_{n}$ (a
plane rooted tree $t\in\mathcal{T}^{0}_{n}$) can be endowed with a total order
$\prec$ in a natural way in such way that for any $v$ the father $v^{\prime}$
of $v$ is such that $v^{\prime}\prec v$ and children of any vertex $v$ of a
labeled tree $\tau\in T^{0}_{n}$ (a plane rooted tree
$t\in\mathcal{T}^{0}_{n}$) are ordered following the order of their labels
(from high to low); namely, the ordering of $v^{1},\dots,v^{s_{v}}$ is such
that $v^{1}\prec v^{2}\prec\cdots\prec v^{s_{v}}$.

Clearly the map $\tau\mapsto m(\tau)=t$ is many-to-one and the cardinality of
the pre-image of a planar rooted tree $t$ (=number of ways of labelling the
$n$ non-root vertices of the tree with $n$ distinct labels consistently with
the rule “from high to low”) is given by

$\left|\\{\tau\in T^{0}_{n}:m(\tau)=t\\}\right|\;=\;{n!\over\prod_{v\succeq
0}s_{v}!}$ $None$

As a matter of fact, it is very easy to count how many labeled trees $\tau\in
T^{0}_{n}$ belong to the same equivalent class $t$, i.e. are associated to the
same plane rooted tree. One have just to count all permutations $\sigma$ of
$\\{0,1,2,\dots,n\\}$ which leaves the root unchanged and which respect the
order of the children in any vertex. Let $\tau\in T^{0}_{n}$ and let
$t=[\tau]$ the plane root tree associated to $\tau$ characterized by the
sequence $\\{s_{v}\\}_{v\succeq 0}$ then we have

$|[\tau]|={n!\over\prod_{v\succeq 0}s_{v}!}$

Indeed, $n!$ is the number of all permutations in the set $\\{1,2,\dots,n\\}$,
while for each vertex $v$, $s_{v}!$ are the permutation of the children. So
${n!/\prod_{v\succeq 0}s_{v}!}$ is the number of permutations of the vertices
of $\tau$ different from the root which do not change the order of the
children in every vertex.

Now observe that the factor (4.18) depend actually only on the plane rooted
tree associated to $\tau$. We have indeed

$\phi^{*}_{\gamma_{0}}(\tau,\bm{\rho})=\phi^{*}_{\gamma_{0}}([\tau],\bm{\rho})=\phi^{*}_{\gamma_{0}}(t,\bm{\rho})=\prod_{v\succeq
0}\Bigg{[}\sum_{(\gamma_{v^{1}},\dots,\gamma_{v^{s_{v}}})\in{\cal
P}^{s_{v}}\atop\gamma_{v^{i}}\not\sim\gamma_{v},\,\,\,\gamma_{v^{i}}\sim\gamma_{v^{j}}}\rho_{\gamma_{v^{1}}}\dots\rho_{\gamma_{v^{s_{v}}}}\Bigg{]}$
$None$

Note that, since in each vertex $v$ the sum over polymers
$\gamma_{v_{1}},\dots,\gamma_{v_{s_{v}}}$ associated to children of $v$
depends on the polymer $\gamma_{v}$ associated to $v$, in the expression above
the order of the product is relevant and it is organized in such way that
products corresponding to ancestors are at the left of products corresponding
to descendants. In the r.h.s. of (4.25) it also adopted the convention that
the sum in brackets is equal to 1 for a vertex $v$ such that $s_{v}=0$.

We now ready to reorganize the sum in the the r.h.s. of (4.17)

$\Pi^{*}_{\gamma_{0}}(\bm{\rho})=\sum_{n\geq 0}{1\over n!}\sum_{\tau\in
T^{0}_{n}}\phi^{*}_{\gamma_{0}}(\tau,\bm{\rho})\;=\sum_{n\geq 0}{1\over
n!}\sum\limits_{t\in\mathcal{T}^{0}_{n}}\sum_{\tau\in
t}\phi^{*}_{\gamma_{0}}(t,\rho)=$ $=\sum_{n\geq 0}{1\over
n!}\sum\limits_{t\in\mathcal{T}^{0}_{n}}\phi^{*}_{\gamma_{0}}(t,\rho)\sum_{\tau\in
t}1=\sum_{n\geq 0}{1\over
n!}\sum\limits_{t\in\mathcal{T}^{0}_{n}}\phi^{*}_{\gamma_{0}}(t,\rho)|t|=$
$=\sum_{n\geq 0}\sum\limits_{t\in\mathcal{T}^{0}_{n}}\Big{[}\prod_{v\succeq
0}{1\over s_{v}!}\Big{]}\phi^{*}_{\gamma_{0}}(t,\rho)=\sum_{n\geq
0}\sum\limits_{t\in\mathcal{T}^{0}_{n}}\prod_{v\succeq 0}\Bigg{[}{1\over
s_{v}!}\sum_{(\gamma_{v^{1}},\dots,\gamma_{v^{s_{v}}})\in{\cal
P}^{s_{v}}\atop\gamma_{v^{i}}\not\sim\gamma_{v},\,\,\,\gamma_{v^{i}}\sim\gamma_{v^{j}}}\rho_{\gamma_{v^{1}}}\dots\rho_{\gamma_{v^{s_{v}}}}\Bigg{]}$

In conclusion we have obtained

${\Pi^{*}_{\gamma_{0}}(\bm{\rho})\;=}$
${=\;\sum\limits_{t\in\mathcal{T}^{0}}\prod_{v\succeq 0}\Bigg{\\{}{1\over
s_{v}!}\sum_{(\gamma_{v^{1}},\dots,\gamma_{v^{s_{v}}})\in{\cal
P}^{s_{v}}}\prod^{s_{v}}_{i=1}\hbox to0.0pt{\mbox{\small\rm 1}\hss}\kern
1.49994pt1_{\\{\gamma_{v^{i}}\nsim\gamma_{v}\\}}\prod_{1\leq i<j\leq
s_{v}}\hbox to0.0pt{\mbox{\small\rm 1}\hss}\kern
1.49994pt1_{\\{\gamma_{v^{i}}\sim\gamma_{v^{j}}\\}}\rho_{\gamma_{v^{1}}}\dots\rho_{\gamma_{v^{s_{v}}}}\Bigg{\\}}}$
$None$

In a completely analogous way we also can obtain

${\Pi^{\rm Dob}_{\gamma_{0}}(\bm{\rho})\;=}$
${=\;\sum\limits_{t\in\mathcal{T}^{0}}\prod_{v\succeq 0}\Bigg{\\{}{1\over
s_{v}!}\sum_{(\gamma_{v^{1}},\dots,\gamma_{v^{s_{v}}})\in{\cal
P}^{s_{v}}}\prod^{s_{v}}_{i=1}\hbox to0.0pt{\mbox{\small\rm 1}\hss}\kern
1.49994pt1_{\\{\gamma_{v^{i}}\nsim\gamma_{v}\\}}\prod_{1\leq i<j\leq
s_{v}}\hbox to0.0pt{\mbox{\small\rm 1}\hss}\kern
1.49994pt1_{\\{\gamma_{v^{i}}\neq\gamma_{v^{j}}\\}}\rho_{\gamma_{v^{1}}}\dots\rho_{\gamma_{v^{s_{v}}}}\Bigg{\\}}}$
$None$

and

${\Pi^{\rm
KP}_{\gamma_{0}}(\bm{\rho})\;=\;\sum\limits_{t\in\mathcal{T}^{0}}\prod_{v\succeq
0}\Bigg{\\{}{1\over
s_{v}!}\sum_{(\gamma_{v^{1}},\dots,\gamma_{v^{s_{v}}})\in{\cal
P}^{s_{v}}}\prod^{s_{v}}_{i=1}\hbox to0.0pt{\mbox{\small\rm 1}\hss}\kern
1.49994pt1_{\\{\gamma_{v^{i}}\nsim\gamma_{v}\\}}\rho_{\gamma_{v^{1}}}\dots\rho_{\gamma_{v^{s_{v}}}}\Bigg{\\}}}$
$None$

#####  4.2.2 Trees and convergence

We start by defining a proper domain in the space of function
$(0,\infty)^{{\cal P}}$, i.e. the space of the functions $\bm{\mu}:{\cal
P}\to(0,\infty):\gamma\mapsto\mu_{\gamma}$. We are agree that given two
functions $\bm{\mu}$ and $\bm{\nu}$ in $(0,\infty)^{{\cal P}}$, we say that
$\bm{\mu}<\bm{\nu}$ if and only if $\mu_{\gamma}<\nu_{\gamma}$ for all
$\gamma\in{\cal P}$.

Given $n\in\mathbb{N}$ and $(\gamma_{0},\gamma_{1},\ldots,\gamma_{n})\in{\cal
P}^{n+1}$, let $b_{n}(\gamma_{0};\gamma_{1},\ldots,\gamma_{n})\geq 0$. Once
numbers $b_{n}(\gamma_{0};\gamma_{1},\ldots,\gamma_{n})$ are given for all
$n\in\mathbb{N}$ and $(\gamma_{0},\gamma_{1},\ldots,\gamma_{n})\in{\cal
P}^{n+1}$, we can define a function

$\varphi^{b}:(0,\infty)^{{\cal P}}\to(0,\infty]^{{\cal
P}}:\bm{u}\mapsto\varphi^{b}(\bm{u})$

with entries

$[\varphi^{b}(\bm{u})]_{\gamma}\doteq\varphi^{b}_{{\gamma}}(\bm{u})\;=\;1+\sum_{n\geq
1}\sum_{(\gamma_{1},\dots,\gamma_{n})\in{\cal
P}^{n}}b_{n}(\gamma;\gamma_{1},\ldots,\gamma_{n})\,u_{\gamma_{1}}\dots
u_{\gamma_{n}}$ $None$

We define the set

$\mathcal{D}^{b}=\Big{\\{}\bm{u}\in(0,\infty)^{{\cal
P}}:\;\;\varphi^{b}_{{\gamma}}(\bm{u})<+\infty,\,\,\forall\gamma\in{\cal
P}\Big{\\}}$

Then the restriction of $\varphi^{b}$ to $\mathcal{D}^{b}$ is a function in
$(0,\infty)^{\cal P}$, i.e.

$\varphi^{b}_{{\gamma}}(\bm{u})\;<\;\infty,\,\,\forall\gamma\in{\cal
P},\,\,\,\,\,\,\,\,\,\,\mbox{whenever $\bm{u}\in\mathcal{D}^{b}$}$

Note also that, if $\bm{u}\in\mathcal{D}^{b}$ and $\bm{u}^{\prime}<\bm{u}$
then also $\bm{u}^{\prime}\in\mathcal{D}^{b}$.

Let now $\bm{\mu}:{\cal P}\to(0,\infty)$ be a function in the set
$\mathcal{D}^{b}\subset(0,\infty)^{{\cal P}}$ and let
$\bm{r}\in(0,\infty)^{{\cal P}}$ be defined such that its entries
$r_{\gamma}$, as $\gamma$ varies in ${\cal P}$, are given by

$r_{\gamma}={\mu_{\gamma}\over\varphi^{b}_{{\gamma}}(\bm{\mu})}$ $None$

Note that $\bm{r}\in\mathcal{D}^{b}$ because $\bm{r}\leq\bm{\mu}$ by
construction, since $\varphi^{b}_{\gamma}(\bm{\mu})\geq 1$ for all
$\gamma\in{\cal P}$. Morevover the assumption $\bm{\mu}\in(0,\infty)^{{\cal
P}}$ is equivalent to say

$\mu_{\gamma}>0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{for all $\gamma_{0}\in{\cal
P}$}$ $None$

while assumption $\bm{\mu}\in\mathcal{D}^{b}$ means that

$\varphi^{b}_{\gamma}(\bm{\mu})<+\infty\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{for
all $\gamma\in{\cal P}$}$ $None$

Assumptions (4.31) and (4.32) guarantee that $\bm{r}\in(0,\infty)^{{\cal P}}$,
i.e.

$r_{\gamma}>0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{for all $\gamma\in{\cal P}$}$

Let us now consider, for any $\bm{\rho}\in\mathcal{D}^{p}$, the map
$T^{\bm{\rho}}=\bm{\rho}\varphi^{b}$. $T^{\bm{\rho}}$ is the map

$T^{\bm{\rho}}:(0,\infty)^{\cal P}\to(0,\infty]^{\cal P}:\bm{u}\mapsto
T^{\bm{\rho}}(\bm{u})$

with entries

$[T^{\bm{\rho}}(\bm{u})]_{\gamma}\doteq
T^{\bm{\rho}}_{\gamma}(\bm{u})=\rho_{\gamma}\varphi^{b}_{\gamma}(\bm{u})\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\gamma\in{\cal
P}$

From (4.30) we get

$\bm{\mu}=T^{\bm{r}}(\bm{\mu})$ $None$

I.e. $\bm{\mu}$ is fixed point for the map $T^{\bm{r}}$. So, by (4.33) we have
that, for all $\gamma_{0}\in{\cal P}$

$\mu_{\gamma_{0}}=T^{\bm{r}}_{\gamma_{0}}(\bm{\mu})=$
$=r_{\gamma_{0}}\;+\;r_{\gamma_{0}}\sum_{\gamma_{1}\in{\cal
P}}b_{1}(\gamma_{0};\gamma_{1})\,\mu_{\gamma_{1}}+r_{\gamma_{0}}\sum_{(\gamma_{1},\gamma_{2})\in{\cal
P}^{2}}b_{2}(\gamma_{0};\gamma_{1},\gamma_{2})\,\mu_{\gamma_{1}}\mu_{\gamma_{2}}+\;\dots$
$\dots\;+\,r_{\gamma_{0}}\sum_{(\gamma_{1},\dots,\gamma_{n})\in{\cal
P}^{n}}b_{n}(\gamma_{0};\gamma_{1},\ldots,\gamma_{n})\,\mu_{\gamma_{1}}\dots\mu_{\gamma_{n}}~{}+\dots$
$None$

Equation (4.34), recalling the definition (4.29) of
$\varphi_{\gamma}^{b}(\bm{\mu})$ can be visualized in the diagrammatic form

where

$\circ_{\gamma_{0}}=r_{\gamma_{0}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\bullet_{\gamma_{i}}=\mu_{\gamma_{i}}$

and, for any $n\geq 1$

The iteration $[{T^{r}}]^{2}(\mu)=T^{r}(T^{r}(\mu))$ corresponds to replacing
each of the bullets by each one of the diagrams of the expansion for $T^{r}$.

This leads to plane rooted trees of up to two generations, with open circles
at first-generation vertices and bullets at second-generation ones. The $k$-th
iteration of $T$ involves all possible _plane rooted trees_ previously seen
with generation up to $k$. In each tree of the expansion, vertices of the last
generation are occupied by bullets and all the others by open circles. Let us
recall that $\mathcal{T}^{0,k}$ denotes the set of trees with maximal depth
(or maximal generation number) equal to $k$. A straightforward inductive
argument shows that

$[T^{\bm{r}}]^{k}_{\gamma_{0}}(\bm{\mu})\;=\;r_{\gamma_{0}}\Bigl{[}\sum_{\ell=0}^{k-1}\Phi^{(\ell)}_{\gamma_{0}}(\bm{r})+R^{(k)}_{\gamma_{0}}(\bm{r},\bm{\mu})\Bigr{]}$
$None$

with

$\Phi^{(\ell)}_{\gamma_{0}}(\bm{r})\;=\,\sum_{t\in\mathcal{T}^{0,\ell}}\,\prod_{v\succeq
0}\Bigg{\\{}\sum_{(\gamma_{v^{1}},\dots,\gamma_{v^{s_{v}}})\in{\cal
P}^{s_{v}}}b_{s_{v}}(\gamma_{v};\gamma_{v^{1}},\ldots,\gamma_{v^{s_{v}}})\,r_{\gamma_{v^{1}}}\dots
r_{\gamma_{v^{s_{v}}}}\Bigg{\\}}$ $None$

while

$R^{(k)}_{\gamma_{0}}(\bm{r},\bm{\mu})=\,\sum_{t\in\mathcal{T}^{0,k}}\,\prod_{v\succeq
0}\Bigg{\\{}\sum_{(\gamma_{v^{1}},\dots,\gamma_{v^{s_{v}}})\in{\cal
P}^{s_{v}}}b_{s_{v}}(\gamma_{v};\gamma_{v^{1}},\ldots,\gamma_{v^{s_{v}}})\,\chi^{t}_{\gamma_{v^{1}}}\dots\chi^{t}_{\gamma_{v^{s_{v}}}}\Bigg{\\}}$
$None$

where

$\chi^{t}_{\gamma_{v}}=\begin{cases}r_{\gamma_{v}}&{\text{i}f}\,\,\,d_{t}(v)<k\\\
\mu_{\gamma_{v}}&{\text{i}f}\,\,\,d_{t}(v)=k\end{cases}$ $None$

with, we recall, $d_{t}(v)$ indicating the depth (distance from the root) of
$v$ in $t$. In other words $R^{(k)}_{\gamma_{0}}(\bm{r},\bm{\mu})$ has an
expression similar to $\Phi^{(k)}_{\gamma_{0}}(\bm{r})$ but with the
activities of the vertex of the $k$-th generation weighted by $\bm{\mu}$. Here
we agree that if $v$ is such that $s_{v}=0$ then $b_{0}(\gamma_{v})\equiv 1$.
Now, by (4.34) we have

$[T^{\bm{r}}]^{k}_{\gamma_{0}}(\bm{\mu})\;=\;\mu_{\gamma_{0}}$

which implies immediately, via (4.35),

$r_{\gamma_{0}}\sum_{\ell=0}^{k-1}\Phi^{(\ell)}_{\gamma_{0}}(\bm{r})\;\leq\;\mu_{\gamma_{0}}\;\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{for
all $k\in\mathbb{N}$ }$ $None$

Equation (4.39) immediately implies the following proposition.

###### Proposition 4.1

Let $\bm{\mu}$ be a function $\bm{\mu}\in(0,\infty)^{\cal P}$ and, for any
$\gamma\in{\cal P}$ let $\varphi^{b}_{{\gamma}}(\bm{\mu})$ be function defined
in (4.29) supposed to satisfy (4.32). Let $\bm{r}\in(0,\infty)^{\cal P}$ be
defined by (4.30). Then, for all $\bm{\rho}\leq\bm{r}$

  1. i)

The series

$\Phi^{b}_{\gamma_{0}}(\bm{\rho})\;:=\;\,\sum_{t\in\mathcal{T}^{0}}\,\prod_{v\succeq
0}\Bigg{\\{}\sum_{(\gamma_{v^{1}},\dots,\gamma_{v^{s_{v}}})\in{\cal
P}^{s_{v}}}b_{s_{v}}(\gamma_{v};\gamma_{v^{1}},\ldots,\gamma_{v^{s_{v}}})\,\rho_{\gamma_{v^{1}}}\dots\rho_{\gamma_{v^{s_{v}}}}\Bigg{\\}}$
$None$

converges for each $\gamma_{0}\in{\cal P}$ and admits, for each
$\gamma_{0}\in{\cal P}$, the bound

$\Phi^{b}_{\gamma_{0}}(\bm{\rho})\leq\Phi^{b}_{\gamma_{0}}(\bm{r})\leq\varphi^{b}_{{\gamma_{0}}}(\bm{\mu})$
$None$

  2. ii)

$\rho_{\gamma_{0}}\Phi^{b}_{\gamma_{0}}(\bm{\rho})=\lim_{n\to\infty}[T^{\bm{\rho}}]^{n}_{\gamma_{0}}(\bm{\rho})$
$None$

and $\rho_{\gamma_{0}}\Phi^{b}_{\gamma_{0}}(\bm{\rho})$ is solution of the
equation (4.34), i.e. is fixed point of the map $T^{\bm{\rho}}$, i.e.

$\rho_{\gamma_{0}}\Phi^{b}_{\gamma_{0}}(\bm{\rho})=T^{\bm{\rho}}_{\gamma_{0}}\Big{(}\rho_{\gamma}\Phi^{b}_{\gamma}(\bm{\rho})\Big{)}$
$None$

This proposition can be viewed as a generalization of the Lagrange inversion
formula for series depending on infinite (countable) variables. Proof.

i). By (4.35) we get

$r_{\gamma_{0}}\sum_{\ell=0}^{n}\Phi^{(\ell)}_{\gamma_{0}}(\bm{r})\,\,\leq\,\,[T^{\bm{r}}]_{\gamma_{0}}^{n+1}(\bm{\mu})\;\;\;\;\;\;\;\forall
n\in\mathbb{N}$

but, by definition (4.34) we have, for any $k\in\mathbb{N}$ that
$[T^{\bm{r}}]^{k}_{\gamma_{0}}(\bm{\mu})=\mu_{\gamma_{0}}$. So we obtain

$r_{\gamma_{0}}\sum_{\ell=0}^{n}\Phi^{(\ell)}_{\gamma_{0}}(\bm{r})\;\leq\;\mu_{\gamma_{0}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{for
all $n$}$

i.e., by (4.30),

$\sum_{\ell=0}^{n}\Phi^{(\ell)}_{\gamma_{0}}(\bm{r})\;\leq\;\varphi^{b}_{{\gamma_{0}}}(\bm{\mu})\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{for
all $n$}$

which implies

$\Phi^{b}_{\gamma_{0}}(\bm{r})\leq\varphi^{b}_{{\gamma_{0}}}(\bm{\mu})$

Therefore, by monotonicity, for any $\bm{\rho}\leq\bm{r}$

$\Phi^{b}_{\gamma_{0}}(\bm{\rho})\leq\Phi^{b}_{\gamma_{0}}(\bm{r})\leq\varphi^{b}_{{\gamma_{0}}}(\bm{\mu})$

ii) By (4.35) we have that

$[T^{\bm{\rho}}]_{\gamma_{0}}^{k}(\bm{\rho})\;=\;\rho_{\gamma_{0}}\Bigl{[}\sum_{\ell=0}^{k-1}\Phi^{(\ell)}_{\gamma_{0}}(\bm{\rho})+R^{(k)}_{\gamma_{0}}(\bm{\rho},\bm{\mu})|_{\bm{\mu}=\bm{\rho}}\Bigr{]}$

But, recalling definition (4.38)

$R^{(k)}_{\gamma_{0}}(\bm{\rho},\bm{\mu})|_{\bm{\mu}=\bm{\rho}}\;=$
$=\,\sum_{t\in\mathcal{T}^{0,k}}\,\prod_{v\succeq
0}\Bigg{\\{}\sum_{(\gamma_{v^{1}},\dots,\gamma_{v^{s_{v}}})\in{\cal
P}^{s_{v}}}b_{s_{v}}(\gamma_{v};\gamma_{v^{1}},\ldots,\gamma_{v^{s_{v}}})\,\rho_{\gamma_{v^{1}}}\dots\rho_{\gamma_{v^{s_{v}}}}\Bigg{\\}}\,\,=\,\,\Phi^{(k)}_{\gamma_{0}}(\bm{\rho})$

So

$[T^{\bm{\rho}}]^{k}_{\gamma_{0}}(\bm{\rho})\;=\;\rho_{\gamma_{0}}\sum_{\ell=0}^{k}\Phi^{(\ell)}_{\gamma_{0}}(\bm{\rho})$

Hence, for any $\bm{\rho}\leq\bm{r}$

$\lim_{k\to\infty}[T^{\bm{\rho}}]^{k}_{\gamma_{0}}(\bm{\rho})\,=\,\lim_{k\to\infty}\rho_{\gamma_{0}}\sum_{\ell=0}^{k}\Phi^{(\ell)}_{\gamma_{0}}(\bm{\rho})\,=\,\rho_{\gamma_{0}}\Phi^{b}_{\gamma_{0}}(\bm{\rho})$

Finally, for any $\bm{\rho}\leq\bm{r}$

$\displaystyle\rho_{\gamma_{0}}\Phi^{b}_{\gamma_{0}}(\bm{\rho})$
$\displaystyle=\lim_{n\to\infty}[T^{\bm{\rho}}]^{n+1}_{\gamma_{0}}(\bm{\rho})$
$\displaystyle=\,\lim_{n\to\infty}T^{\bm{\rho}}_{\gamma_{0}}\Bigl{(}[T^{\bm{\rho}}]^{n}_{\gamma_{0}}(\bm{\rho})\Bigr{)}$
$\displaystyle=\,\rho_{\gamma_{0}}\lim_{n\to\infty}\varphi^{b}_{{\gamma_{0}}}\Bigl{(}[T^{\bm{\rho}}]^{n}_{\gamma_{0}}(\bm{\rho})\Bigr{)}$
$\displaystyle=\,\rho_{\gamma_{0}}\varphi^{b}_{\gamma}\Bigl{(}\lim_{n\to\infty}[T^{\bm{\rho}}]^{n}_{\gamma_{0}}(\bm{\rho})\Bigr{)}$
$\displaystyle=\,\rho_{\gamma_{0}}\varphi^{b}_{{\gamma_{0}}}\Bigl{(}\rho_{\gamma_{0}}\Phi^{b}_{\gamma_{0}}(\bm{\rho})\Bigr{)}$
$\displaystyle=\,T^{\bm{\rho}}_{\gamma_{0}}\Bigl{(}\rho_{\gamma_{0}}\Phi^{b}_{\gamma_{0}}(\bm{\rho})\Bigr{)}$

$\Box$

#####  4.2.3 Convergence criteria

Fernández-Procacci criterion.

Let us now choose

$b_{n}(\gamma_{0};\gamma_{1},\ldots,\gamma_{n})=b^{*}_{n}(\gamma_{0};\gamma_{1},\ldots,\gamma_{n})\doteq{1\over
n!}\prod_{i=1}^{n}\hbox to0.0pt{\mbox{\small\rm 1}\hss}\kern
1.49994pt1_{\\{\gamma_{i}\not\sim\gamma_{0}\\}}\;\;\prod_{1\leq i<j\leq
n}\hbox to0.0pt{\mbox{\small\rm 1}\hss}\kern
1.49994pt1_{\\{\gamma_{i}\sim\gamma_{j}\\}}$ $None$

and thus

$\varphi^{b^{*}}_{\gamma_{0}}(\bm{\mu})\;=\;1+\sum_{n\geq
1}\frac{1}{n!}\,\sum_{(\gamma_{1},\dots,\gamma_{n})\in{\cal
P}^{n}\atop\gamma_{i}\not\sim\gamma_{0},\,\,\gamma_{i}\sim\gamma_{j}}{\mu_{\gamma_{1}}}\dots\mu_{\gamma_{n}}\;=\;\Xi_{{\cal
P}_{\gamma_{0}}}(\bm{\mu})$ $None$

Then, proposition 4.1 tells us that the series

$\Phi^{b^{*}}_{\gamma_{0}}(\bm{\rho})=$
$=\,\sum_{t\in\mathcal{T}^{0}}\,\prod_{v\succeq
0}\Bigg{\\{}\sum_{(\gamma_{v^{1}},\dots,\gamma_{v^{s_{v}}})\in{\cal
P}^{s_{v}}}{1\over s_{v}!}\prod_{i=1}^{s_{v}}\hbox to0.0pt{\mbox{\small\rm
1}\hss}\kern 1.49994pt1_{\\{\gamma_{v^{i}}\not\sim\gamma_{v}\\}}\prod_{1\leq
i<j\leq s_{v}}\hbox to0.0pt{\mbox{\small\rm 1}\hss}\kern
1.49994pt1_{\\{\gamma_{v^{i}}\sim\gamma_{v^{j}}\\}}\,\rho_{\gamma_{v^{1}}}\dots\rho_{\gamma_{v^{s_{v}}}}\Bigg{\\}}$
$None$

converges as soon as $\bm{\rho}\leq\bm{r}^{*}$ with

$r^{*}_{\gamma}={\mu_{\gamma}\over\Xi_{{\cal P}_{\gamma}}(\bm{\mu})}$ $None$

comparing (4.26) with (4.46) we immediately see that

$\Pi^{*}_{\gamma_{0}}(\bm{\rho})=\Phi^{b^{*}}_{\gamma_{0}}(\bm{\rho})$

So we immediately get, by Proposition 4.1, the following criterion for the
convergence of cluster expansions.

###### Theorem 4.1

Choose $\bm{\mu}\in\mathcal{D}^{b^{*}}\subset(0,\infty)^{\cal P}$ and let
$\bm{r}^{*}\in(0,\infty)^{\cal P}$ s.t.

$r^{*}_{\gamma_{0}}={\mu_{\gamma_{0}}\over\Xi_{{\cal
P}_{\gamma_{0}}}(\bm{\mu})}$

Let $\bm{\rho}$ such that

$\rho_{\gamma}\;\leq\;r^{*}_{\gamma}\,\,\,\,\,\,\,\,\,\,\,\,\forall\gamma\in{\cal
P}$ $None$

Then the series $|\Pi|_{\gamma_{0}}(\bm{\rho})$ defined in (4.9) is finite for
each $\gamma_{0}\in{\cal P}$ and

$|\Pi|_{\gamma_{0}}(\bm{\rho})\leq\Xi_{{\cal P}_{\gamma_{0}}}(\bm{\mu})$
$None$

and hence

$\rho_{\gamma_{0}}|\Pi|_{\gamma_{0}}(\bm{\rho})\leq\mu_{\gamma_{0}}$ $None$

for each $\gamma_{0}\in{\cal P}$.

Proof. By proposition 4.1 we have immediately that the series
$\Pi^{*}_{\gamma_{0}}(\bm{\rho})$ defined in (4.17) is finite for each
$\gamma_{0}\in{\cal P}$ and for all $\bm{\rho}$ such that $\rho_{\gamma}\leq
r^{*}_{\gamma}$ where $r^{*}_{\gamma}$ is defined in (4.48). Moreover

$\Pi^{*}_{\gamma_{0}}(\bm{\rho})\leq\Xi_{{\cal P}_{\gamma_{0}}}(\bm{\mu})$

for each $\gamma_{0}\in{\cal P}$. Now recalling (4.16) we obtain that the same
is true also for the series $|\Pi|_{\gamma_{0}}(\bm{\rho})$. $\Box$

Dobrushin criterion.

If we now choose

$b_{n}(\gamma_{0};\gamma_{1},\ldots,\gamma_{n})=b^{\rm
Dob}_{n}(\gamma_{0};\gamma_{1},\ldots,\gamma_{n})\doteq{1\over
n!}\prod_{i=1}^{n}\hbox to0.0pt{\mbox{\small\rm 1}\hss}\kern
1.49994pt1_{\\{\gamma_{i}\not\sim\gamma_{0}\\}}\;\;\prod_{1\leq i<j\leq
n}\hbox to0.0pt{\mbox{\small\rm 1}\hss}\kern
1.49994pt1_{\\{\gamma_{i}\neq\gamma_{j}\\}}$

and thus

$\varphi^{b^{\rm Dob}}_{\gamma_{0}}(\bm{\mu})\;=\;1+\sum_{n\geq
1}\frac{1}{n!}\,\sum_{(\gamma_{1},\dots,\gamma_{n})\in{\cal
P}^{n}\atop\gamma_{0}\nsim\gamma_{i}\,,\,\gamma_{i}\neq\gamma_{j}}{\mu_{\gamma_{1}}}\dots\mu_{\gamma_{n}}\;=\;\prod_{\gamma\nsim\gamma_{0}}[1+\mu_{\gamma}]\;,$
$None$

Then again proposition 4.1 tells us that the series

$\Phi^{b^{\rm Dob}}_{\gamma_{0}}(\bm{\rho})\;=$
$=\,\sum_{t\in\mathcal{T}^{0}}\,\prod_{v\succeq
0}\Bigg{\\{}\sum_{(\gamma_{v^{1}},\dots,\gamma_{v^{s_{v}}})\in{\cal
P}^{s_{v}}}{1\over s_{v}!}\prod_{i=1}^{s_{v}}\hbox to0.0pt{\mbox{\small\rm
1}\hss}\kern 1.49994pt1_{\\{\gamma_{v^{i}}\not\sim\gamma_{v}\\}}\prod_{1\leq
i<j\leq s_{v}}\hbox to0.0pt{\mbox{\small\rm 1}\hss}\kern
1.49994pt1_{\\{\gamma_{v^{i}}\neq\gamma_{v^{j}}\\}}\,\rho_{\gamma_{v^{1}}}\dots\rho_{\gamma_{v^{s_{v}}}}\Bigg{\\}}$
$None$

converges as soon as $\bm{\rho}\leq\bm{r}^{\rm Dob}$ with

$r^{\rm
Dob}_{\gamma}={\mu_{\gamma}\over\prod_{\gamma\nsim\gamma_{0}}[1+\mu_{\gamma}]}$
$None$

Comparing (4.27) with (4.52) we get

$\Pi^{\rm Dob}_{\gamma_{0}}(\bm{\rho})=\Phi^{b^{\rm
Dob}}_{\gamma_{0}}(\bm{\rho})$

So Proposition 4.1 also yields the following (weaker) criterion for the
convergence of cluster expansions.

######  Corollary 4.1 (Dobrushin)

Choose $\bm{\mu}\in{\mathcal{D}}^{b^{\rm Dob}}$ and let $\bm{r}^{\rm
Dob}\in(0,\infty)^{\cal P}$ s.t.

$r^{\rm
Dob}_{\gamma}={\mu_{\gamma}\over\prod_{{\tilde{\gamma}}\nsim\gamma}[1+\mu_{\tilde{\gamma}}]}$
$None$

Let $\bm{\rho}\in[0,\infty)^{\cal P}$ such that

$\rho_{\gamma}\;\leq\;r^{\rm
Dob}_{\gamma}\,={\mu_{\gamma}\over\prod_{{\tilde{\gamma}}\nsim\gamma}[1+\mu_{\tilde{\gamma}}]}\,\,\,\,\,\,\,\,\,\,\,\forall\gamma\in{\cal
P}$ $None$

Then the series $|\Pi|_{\gamma_{0}}(\bm{\rho})$ defined in (4.9) is finite for
each $\gamma\in{\cal P}$ and

$|\Pi|_{\gamma}(\bm{\rho})\leq\prod_{{\tilde{\gamma}}\nsim\gamma}[1+\mu_{\tilde{\gamma}}]$
$None$

or

$\rho_{\gamma}|\Pi|_{\gamma}(\bm{\rho})\leq\mu_{\gamma}$ $None$

for each $\gamma\in{\cal P}$.

For the benefit of the readers we stress that in the literature the Dobrushin
condition is generally written in a different (but equivalent) form. In
particular, in the Dobrushin paper [11] and also in [31] the condition (4.55)
is written as follows.

$\rho_{\gamma}\;\leq\;r^{\rm
Dob}_{\gamma}\;=(e^{{\tilde{\mu}}_{\gamma}}-1)e^{-\sum_{{\tilde{\gamma}}\in{\cal
P}:\,{\tilde{\gamma}}\not\sim\gamma}\tilde{\mu}_{\tilde{\gamma}}}\,\,\,\,\,\,\,\,\,\,\,\,\forall\gamma\in{\cal
P}$

This is clearly the same condition (4.55) by defining

$\tilde{\mu}_{\gamma}=\log[1+\mu_{\gamma}]$ $None$

Koptecký-Preiss criterion.

Finally, if we now choose

$b_{n}(\gamma_{0};\gamma_{1},\ldots,\gamma_{n})=b^{\rm
KP}_{n}(\gamma_{0};\gamma_{1},\ldots,\gamma_{n})\doteq{1\over
n!}\prod_{i=1}^{n}\hbox to0.0pt{\mbox{\small\rm 1}\hss}\kern
1.49994pt1_{\\{\gamma_{i}\not\sim\gamma_{0}\\}}$

and

$\varphi^{\rm KP}_{\gamma_{0}}(\bm{\mu})\;=\;1+\sum_{n\geq
1}\,\frac{1}{n!}\sum_{(\gamma_{1},\dots,\gamma_{n})\in{\cal
P}^{n}\atop\gamma_{0}\nsim\gamma_{i}\,,\,1\leq i\leq
n}{\mu_{\gamma_{1}}}\dots\mu_{\gamma_{n}}\;=\;\exp\Bigl{[}\sum_{\gamma\nsim\gamma_{0}}\mu_{\gamma}\Bigr{]}$
$None$

Then once again proposition 4.1 tells us that the series

$\Phi^{b^{\rm
KP}}_{\gamma_{0}}(\bm{\rho})\;=\;\,\sum_{t\in\mathcal{T}^{0}}\,\prod_{v\succeq
0}\Bigg{\\{}\sum_{(\gamma_{v^{1}},\dots,\gamma_{v^{s_{v}}})\in{\cal
P}^{s_{v}}}{1\over s_{v}!}\prod_{i=1}^{s_{v}}\hbox to0.0pt{\mbox{\small\rm
1}\hss}\kern
1.49994pt1_{\gamma_{v^{i}}\not\sim\gamma_{v}}\;\;\rho_{\gamma_{v^{1}}}\dots\rho_{\gamma_{v^{s_{v}}}}\Bigg{\\}}$
$None$

converges as soon as $\bm{\rho}\leq\bm{r}^{\rm KP}$ with

$r^{\rm
KP}_{\gamma}={\mu_{\gamma}\over\exp\Bigl{[}\sum_{{\tilde{\gamma}}\nsim\gamma}\mu_{\tilde{\gamma}}\Bigr{]}}$
$None$

Comparing (4.28) with (4.60) we get

$\Pi^{\rm KP}_{\gamma_{0}}(\bm{\rho})=\Phi^{b^{\rm
KP}}_{\gamma_{0}}(\bm{\rho})$

So we get the criterion of Kotecký and Preiss. Namely,

######  Corollary 4.2 (Kotecký-Preiss)

Choose $\bm{\mu}\in{\mathcal{D}}^{b^{\rm KP}}$ and let $\bm{r}^{\rm
KP}\in(0,\infty)^{\cal P}$ s.t.

$r^{\rm
KP}_{\gamma_{0}}={\mu_{\gamma_{0}}\over\exp\Bigl{[}\sum_{\gamma\nsim\gamma_{0}}\mu_{\gamma}\Bigr{]}}$
$None$

Let $\bm{\rho}\in[0,\infty)^{\cal P}$ such that

$\rho_{\gamma}\;\leq\;r^{\rm
KP}_{\gamma}\;=\;{\mu_{\gamma}\over\exp\Bigl{[}\sum_{{\tilde{\gamma}}\nsim\gamma}\mu_{\tilde{\gamma}}\Bigr{]}},\,\,\,\,\,\,\,\,\,\,\,\,\forall\gamma\in{\cal
P}$ $None$

Then the series $|\Pi|_{\gamma_{0}}(\bm{\rho})$ defined in (4.9) is finite for
each $\gamma_{0}\in{\cal P}$ and

$|\Pi|_{\gamma_{0}}(\bm{\rho})\leq\exp\Bigl{[}\sum_{\gamma\nsim\gamma_{0}}\mu_{\gamma}\Bigr{]}$
$None$

or

$\rho_{\gamma_{0}}|\Pi|_{\gamma_{0}}(\bm{\rho})\leq\mu_{\gamma_{0}}$

for each $\gamma_{0}\in{\cal P}$.

Remark. Again for the benefit of the readers we stress that in the literature
the Kotecky-Preiss condition is generally written in a different (but
equivalent) form. Namely in the original Kotecky-Preiss paper convergence is
guaranteed by choosing $\bm{\rho}$ such that there is a function $a_{\gamma}$
such that

$\sum_{{\tilde{\gamma}}\nsim\gamma}\rho_{\tilde{\gamma}}e^{a_{\tilde{\gamma}}}\leq
a_{\gamma}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall\gamma\in{\cal
P}$

This is clearly the same condition (4.62) by setting

$\mu_{\gamma}=\rho_{\gamma}e^{a_{\gamma}}$ $None$

Summing up, available convergence conditions are of the form

$\rho_{\gamma}\,\;\leq\;r_{\gamma}=\,{\mu_{\gamma}\over\varphi_{\gamma}(\mu)}$
$None$

with

$\varphi_{\gamma}(\bm{\mu})\;=\;\left\\{\begin{array}[]{ll}\exp\Bigl{[}\sum_{{\tilde{\gamma}}\nsim\gamma}\mu_{\tilde{\gamma}}\Bigr{]}&\mbox{
(Koteck\'{y}-Preiss)}\\\\[10.0pt]
\prod_{{\tilde{\gamma}}\nsim\gamma}\bigl{(}1+\mu_{\tilde{\gamma}}\bigr{)}&\mbox{
(Dobrushin)}\\\\[10.0pt] \Xi_{{\cal P}_{\gamma}}(\bm{\mu})&\mbox{
(Fern\'{a}ndez-Procacci)}\end{array}\right.$ $None$

Each condition is strictly weaker than the preceding one. Namely, since, for
fixed $\bm{\mu}\in(0,\infty)^{\cal P}$,

$\Xi_{{\cal
P}_{\gamma}}(\bm{\mu})\leq\prod_{{\tilde{\gamma}}\nsim\gamma}\bigl{[}1+\mu_{\tilde{\gamma}}\bigr{]}\leq\exp\Bigl{[}\sum_{\gamma\nsim\gamma_{0}}\mu_{\tilde{\gamma}}\Bigr{]}$

we get

$\bm{r}^{*}_{\gamma}\geq\bm{r}^{\rm Dob}_{\gamma}\geq\bm{r}_{\gamma}^{\rm KP}$

So the criterion given by the Corollary 4.2 (i.e. Kotecky-Preiss condition)
yields the worst estimate for convergence radius for the cluster expansion;
the Dobrishin Criterion of Corollary 4.1 gives an estimate which is better
(i.e. larger) than that given by the Kotecký-Preiss criterion for the same
radius and finally the criterion 4.1 give the best estimate for convergence
radius for the cluster expansion among the three proposed.

#####  4.2.4 Elementary Examples

In this section we give some elementary in order to illustrate how the
criterion (4.48) represents a sensible improvement on previous criteria in
applications.

Example 1. The Domino model on $\mathbb{Z}^{2}$. This model has also been
considered by Dobrushin in [11]. The elements of the polymer space ${\cal P}$
are in this case nearest neighbor bonds of the bidimensional cubic lattice.
For any $\gamma\in{\cal P}$ we put $\rho_{\gamma}=\rho$, where $\rho>0$ (all
polymers have the same activity). Two polymers are incompatible if and only if
they have non empty intersection. We can choose by symmetry that the function
$\mu_{\gamma}$ appearing in the Kotecky-Preiss, Dobrushin and Fernández-
Procacci criteria are constant at the value $\mu$.

The Kotecky-Preiss criterion (4.63) for the domino model then reads as

$\rho_{\gamma}\leq\mu_{\gamma}e^{-\sum_{\tilde{\gamma}\not\sim\gamma}\mu_{\tilde{\gamma}}}~{}~{}\Longleftrightarrow~{}~{}\rho\leq\mu
e^{-7\mu}$

which yelds at best

$\rho\leq{1\over 7e}\approx 0.0525$

On the other hand the Dobrushin condition (4.55) reads

$\rho_{\gamma}\leq{\mu_{\gamma}\over\prod_{\tilde{\gamma}\not\sim\gamma}[1+\mu_{\tilde{\gamma}}]}~{}~{}\Longleftrightarrow~{}~{}\rho\leq{\mu\over(1+\mu)^{7}}$

which yields at best

$\rho\leq{{1\over 6}\over(1+{1\over 6})^{7}}\approx 0.0566$

Finally, the condition (4.48) gives

$\rho_{\gamma}\leq{\mu_{\gamma}\over\Xi_{{\cal
P}_{\gamma}}(\bm{\mu})}~{}~{}\Longleftrightarrow~{}~{}\rho\leq{\mu\over
1+7\mu+9\mu^{2}}$

which yields at best

$\rho\leq{1\over 13}\approx 0.0769$

Example 2. The lattice gas on a bounded degree graph
$\mathbb{G}=(\mathbb{V},\mathbb{E})$ with hard core self repulsion and hard
core pair interaction and the triangular lattice on then plane Let
$\mathbb{G}=(\mathbb{V},\mathbb{E})$ be a bounded degree infinite graph with
vertex set $\mathbb{V}$ and edge set $\mathbb{E}$, and maximum degree
$\Delta$. A polymer system is obtained by choosing ${\cal P}=\mathbb{V}$ and
by defininig the incompatibility relation $\nsim$ by saying that two polymers
$\gamma$ and $\gamma^{\prime}$ (i.e. two vertices of $\mathbb{G}$) are
incompatible if and only if either $\gamma=\gamma^{\prime}$ or
$\\{\gamma,\gamma^{\prime}\\}\in\mathbb{E}$. This polymer gas realization is
called the self repulsive hard core lattice gas on $\mathbb{G}$. In this case
the polymers are the vertices of $\mathbb{G}$ and two polymers
$\\{x,y\\}\subset\mathbb{V}$ are incompatible if either $y=x$ (self repulsion)
or $\\{x,y\\}\in\mathbb{E}$ (hard core pair interaction). In general, since
polymers have no structure (they are just vertices in a graph) one can suppose
that the activity of a polymer $x\in{\cal P}$ is a constant, i.e.
$\rho_{x}=\rho$ for all $x\in\mathbb{V}$. Of course expect that the
convergence radius depends strongly on the topological structure of
$\mathbb{G}$. We first consider the worst case i.e. when the graph
$\mathbb{G}$ is such that the nearest neighbors of any vertex are pairwise
compatible. This happens e.g. if $\mathbb{G}$ is a tree or if it is the cubic
lattice $\mathbb{Z}^{d}$. The Kotecky-Preiss condition for this model then
reads as

$\rho_{x}\leq\mu_{x}e^{-\sum_{y\not\sim
x}\mu_{y}}~{}~{}\Longleftrightarrow~{}~{}\rho\leq\mu e^{-(\Delta+1)\mu}$

which yields at best

$\rho\leq{1\over(\Delta+1)e}$ $None$

On the other hand the Dobrushin condition reads

$\rho_{x}\leq{\mu_{x}\over\prod_{y\not\sim
x}[1+\mu_{y}]}~{}~{}\Longleftrightarrow~{}~{}\rho\leq{\mu\over(1+\mu)^{\Delta+1}}$

which yields at best

$\rho\leq{{1\over\Delta}\over(1+{1\over\Delta})^{\Delta+1}}={\Delta^{\Delta}\over(\Delta+1)^{\Delta+1}}={1\over\Delta+1}{1\over(1+{1\over\Delta})^{\Delta}}$
$None$

Finally, the condition (4.48) gives

$\rho_{x}\leq{\mu_{x}\over\Xi_{{\cal
P}_{x}}(\bm{\mu})}~{}~{}\Longleftrightarrow~{}~{}\rho\leq{\mu\over
1+(\Delta+1)\mu+\sum_{k=2}^{\Delta}{\Delta\choose
k}\mu^{k}}={\mu\over\mu+(1+\mu)^{\Delta}}$

which yields at best

$\rho\leq{{1\over\Delta-1}\over{1\over\Delta-1}+\left(1+{1\over\Delta-1}\right)^{\Delta}}={1\over
1+{\Delta^{\Delta}\over(\Delta-1)^{\Delta-1}}}={1\over\Delta(1+{1\over\Delta-1})^{\Delta-1}+1}$
$None$

To illustrate that Theorem 4.1 permits to improve this last bound (4.70) if we
know more about the topological structure of $\mathbb{G}$, we now consider a
case of the triangular lattice in $d=2$ (a regular graph with degree
$\Delta=6$, where our bound turns to be more efficient than the Dobrushin
bound and the Shearer-Sokal bound (4.68). For the triangular lattice the tree
bound (4.70) gives

$\varepsilon<{5^{5}\over 6^{6}}\approx 0,067$

while our bound gives

$\varepsilon\leq{c\over\Xi_{\gamma}(c)}={c\over 1+7c+9c^{2}+2c^{3}}$

The maximum occurs when $4c^{3}+9c^{2}-1=0$, which is somewhere between 1/3
and 3/10. For example choosing $c=1/3$ (which is not the best choice) we
obtain

$\varepsilon\leq{c\over\Xi_{\gamma}(c)}={{1\over 3}\over 1+{7\over
3}+{1}+{2\over 27}}\approx 0,075$

####  4.3 Gas of non overlapping finite subsets

In this section we will study a particular realization of the polymer gas
which appears in the most part of the examples in statistical mechanics.

We will suppose that it is given an infinite countable set $\mathbb{V}$, and
we define the space of polymers as

${\cal P}_{\mathbb{V}}=\\{R\subset\mathbb{V}:|R|<\infty\\}$

and the incompatibility relation in ${\cal P}_{\mathbb{V}}$ is defined as

$\gamma\not\sim{\tilde{\gamma}}\,\,\,\,\Longleftrightarrow\,\,\,\,\gamma\cap\tilde{\gamma}\neq\emptyset$

Note that now polymers have a cardinality, so that we can speak about big
polymers and small polymers. Of course, as before, to a polymer $\gamma$ is
associated an activity $\zeta(\gamma)$. We assume in general that
$\zeta(\gamma)\in\mathbb{C}$ as far as $\gamma\in{\cal P}_{\mathbb{V}}$ and we
set

$|\zeta(\gamma)|=\rho(\gamma)$ $None$

Note that here we allow the value $\zeta(\gamma)=0$ for some $\gamma$ in order
to stay more general. For example in the polymer expansion of high temperature
spin systems, the polymer space is always ${\cal P}_{\mathbb{V}}$ for some
suitable $\mathbb{V}$ but it happens that $\zeta(\gamma)=0$ whenever
$|\gamma|=1$.

In most of the physics realizations $\mathbb{V}$ is the vertex set of an
infinite graph $\mathbb{G}=(\mathbb{V},\mathbb{E})$ with edge set
$\mathbb{E}$. For example $\mathbb{V}=\mathbb{Z}^{d}$ and $\mathbb{E}$ is the
set of nearest neighbor in $\mathbb{Z}^{d}$. When $\mathbb{V}$ is the vertex
set of a graph $\mathbb{G}$ then $\mathbb{V}$ has a natural metric structure
induced by the graph distance in $\mathbb{G}$. This metric structure on
$\mathbb{V}$ allow us to talk about how spread is a polymers (a polymer is
spread if its points are far apart) and we can say now if two polymers
$\gamma$ and $\gamma^{\prime}$ are close or far apart. So, from the abstract
context we can pass to more concrete realizations which have richer
structures. Namely, if one suppose that polymers are finite subsets of an
underlying countable set $\mathbb{V}$ with $\not\sim=\cap$, the any polymer
has an activity and a cardinality, so we an distinguish between big and small
polymers. If we further suppose that the underlying set $\mathbb{V}$ is the
vertex set of some graph $\mathbb{G}$ we also can talk about distance between
polymers and spread polymers. In any case, in the whole section below we will
not suppose any graph structure for the set $\mathbb{V}$. Our abstract polymer
space is just the set of all finite subsets of a countable set with the
incompatibility relation being the the non void intersection.

Let now $\Lambda$ be a finite set of $\mathbb{V}$. A configuration of polymer
gas in $\Lambda$ is given once we specify the set of polymers which are
present in $\Lambda$. Of course this polymers must be pairwise compatible,
i.e. a configuration in $\Lambda$ is an unordered $n$-ple
$\\{\gamma_{1},\dots,\gamma_{n}\\}$ such that
$\gamma_{i}\cap\gamma_{j}=\emptyset$ for all $i,j=1,\dots,n$. The
“probability”111(4.79) is a real probability only if
$\zeta(\gamma)\in[0,+\infty)$ to see the configuration
$\\{\gamma_{1},\dots,\gamma_{n}\\}$ in the box $\Lambda$ is defined as

${\rm
Prob}_{\bm{\zeta}}(\gamma_{1},\dots,\gamma_{n})=\Xi_{\Lambda}^{-1}\prod_{i=1}^{n}\zeta(\gamma_{i})$
$None$

where $\Xi_{\Lambda}$ is the partition function defined as

$\Xi_{\Lambda}(\bm{\zeta})=1+\sum_{n\geq
1}\sum_{\\{\gamma_{1},\dots,\gamma_{n}\\}:~{}\gamma_{i}\subset\Lambda\atop\gamma_{i}\cap\gamma_{j}=\emptyset}\zeta(\gamma_{1})\dots\zeta(\gamma_{n})$
$None$

#####  4.3.1 Convergence via the abstract polymer criteria

We compare the three conditions for this model. Starting with the Kotecky-
Preiss condition, choosing $\mu(\gamma)=\rho(\gamma)e^{a|\gamma|}$ (recall:
$|\zeta(\gamma)|=\rho(\gamma)$), the condition (4.63) becomes the well known
inequality

$\sum_{{\tilde{\gamma}}\in{\cal
P}_{\mathbb{V}}\atop\tilde{\gamma}\not\sim\gamma}\rho({\tilde{\gamma}})\;e^{a|{\tilde{\gamma}}|}\>\leq\>a|\gamma|,\,\,\,\,\,\,\,\,\,\forall\gamma\in{\cal
P}$ $None$

Now using that $\tilde{\gamma}\not\sim\gamma$ means for the present model
${\tilde{\gamma}}\cap\gamma\neq\emptyset$ we have that

$\sum_{\tilde{\gamma}\not\sim\gamma}\rho({\tilde{\gamma}})e^{a|{\tilde{\gamma}}|}\leq|\gamma|\sup_{x\in\mathbb{V}}\sum_{\tilde{\gamma}\ni
x}\rho({\tilde{\gamma}})e^{a|{\tilde{\gamma}}|}$

Hence (4.74) becomes the well known condition

$\sup_{x\in\mathbb{V}}\,\sum_{\gamma\in{\cal P}_{\mathbb{V}}\atop\gamma\ni
x}\,\rho(\gamma)\,e^{a|\gamma|}\,\,\leq\,\,a$ $None$

On the other hand the Dobrushin condition (4.55) can be written as

$\rho(\gamma)\leq{c(\gamma)\over\prod_{{\tilde{\gamma}}\in{\cal
P}_{\mathbb{V}}:\,\tilde{\gamma}\not\sim\gamma}[1+c(\tilde{\gamma})]}$

choosing again $c(\gamma)=|\rho(\gamma)|e^{a|\gamma|}$ the condition above
becomes

$\prod_{{\tilde{\gamma}}\in{\cal
P}_{\mathbb{V}}\atop\tilde{\gamma}\not\sim\gamma}(1+\rho({\tilde{\gamma}})e^{a|{\tilde{\gamma}}|})\leq
e^{a|\gamma|},\,\,\,\,\,\,\,\,\,\forall\gamma\in{\cal P}$

i.e.

$\sum_{{\tilde{\gamma}}\in{\cal
P}_{\mathbb{V}}\atop\tilde{\gamma}\not\sim\gamma}\log(1+\rho({\tilde{\gamma}})e^{a|{\tilde{\gamma}}|})\leq
a|\gamma|\,\,\,\,\,\,\,\,\,\forall\gamma\in{\cal P}$ $None$

i.e.

$\sup_{x\in\mathbb{V}}\sum_{\gamma\in{\cal P}_{\mathbb{V}}\atop\gamma\ni
x}\log(1+\rho(\gamma)e^{a|\gamma|})\leq a$ $None$

which is slightly better than (4.75).

Finally, the condition (4.48), putting again
$c(\gamma)=\rho(\gamma)e^{a|\gamma|}$, becomes

$\Xi_{\,{\cal P}_{\gamma}}(c)\leq e^{a|\gamma|}$ $None$

where ${\cal P}_{\gamma}=\\{\gamma^{\prime}\in{\cal
P}_{\mathbb{V}}:\gamma^{\prime}\cap\gamma\neq\emptyset\\}$ and

$\Xi_{\,{\cal P}_{\gamma}}(c)=1+\sum_{n=1}^{|\gamma|}{1\over
n!}\sum_{(\gamma_{1},\dots,\gamma_{n})\in{\cal
P}_{\mathbb{V}}^{n}\atop\gamma_{i}\not\sim\gamma,\gamma_{i}\sim\gamma_{j}}\prod_{i=1}^{n}\rho(\gamma_{i})e^{a|\gamma_{i}|}$

We again use the fact that
$\gamma_{i}\not\sim\gamma_{j}\Longleftrightarrow\gamma_{i}\cap\gamma_{j}\neq\emptyset$
and
$\gamma_{i}\sim\gamma_{j}\Longleftrightarrow\gamma_{i}\cap\gamma_{j}=\emptyset$
to estimate the factor

$\sum_{(\gamma_{1},\dots,\gamma_{n})\in{\cal
P}_{\mathbb{V}}^{n}\atop\gamma_{i}\not\sim\gamma,\gamma_{i}\sim\gamma_{j}}\prod_{i=1}^{n}\rho(\gamma_{i})e^{a|\gamma_{i}|}$

Note that this factor is zero whenever $n>|\gamma|$, since there is no way to
choose $n$ subsets $\gamma_{i}$ of $\mathbb{V}$ such that they are all
pairwise compatible (i.e. non intersecting) and all incompatible (i.e.
intersecting) with a fixed subset $\gamma$ of $\mathbb{V}$ with a number of
elements equal to $|\gamma|$. On the other hand, when the sum above is not
zero, i.e. for $n\leq|\gamma|$, it can be bounded at least by (a very rough
bound)

$\displaystyle\sum_{(\gamma_{1},\dots,\gamma_{n})\in{\cal
P}^{n}\atop\gamma_{i}\not\sim\gamma,\gamma_{i}\sim\gamma_{j}}\prod_{i=1}^{n}\rho(\gamma_{i})e^{a|\gamma_{i}|}$
$\displaystyle\leq|\gamma|(|\gamma|-1)\cdots(|\gamma|-n+1)\Bigg{[}\sup_{x\in\mathbb{V}}\,\sum_{\gamma\in{\cal
P}_{\mathbb{V}}\atop x\in\gamma}{\rho(\gamma)}e^{a|\gamma|}\Bigg{]}^{n}$
$\displaystyle=~{}{|\gamma|\choose
n}n!\Bigg{[}\sup_{x\in\mathbb{V}}\,\sum_{\gamma\in{\cal P}_{\mathbb{V}}\atop
x\in\gamma}{\rho(\gamma)}e^{a|\gamma|}\Bigg{]}^{n}$

Thus

$\Xi^{\,\gamma}_{\,{\cal P}_{\mathbb{V}}}(c)\leq
1+\sum_{n=1}^{|\gamma|}{|\gamma|\choose
n}\Bigg{[}\sup_{x\in\mathbb{V}}\,\sum_{\gamma\in{\cal P}_{\mathbb{V}}\atop
x\in\gamma}{\rho(\gamma)}e^{a|\gamma|}\Bigg{]}^{n}=\Bigg{[}1+\sup_{x\in\mathbb{V}}\,\sum_{\gamma\in{\cal
P}_{\mathbb{V}}\atop
x\in\gamma}{\rho(\gamma)}e^{a|\gamma|}\Bigg{]}^{|\gamma|}$ $None$

Thus (4.78) can be written as

$\Bigg{[}1+\sup_{x\in\mathbb{V}}\,\sum_{\gamma\in{\cal P}_{\mathbb{V}}\atop
x\in\gamma}{\rho(\gamma)}e^{a|\gamma|}\Bigg{]}^{|\gamma|}\leq e^{a|\gamma|}$

i.e.

$\sup_{x\in\mathbb{V}}\,\sum_{\gamma\in{\cal P}_{\mathbb{V}}\atop
x\in\gamma}{\rho(\gamma)}e^{a|\gamma|}\leq e^{a}-1$ $None$

Note that by (4.49) we also get the upper bound

$\Pi_{\gamma}(\rho)\leq e^{a|\gamma|}$

We have therefore proved the following theorem

###### Theorem 4.2

Let $\mathbb{V}$ be a countable set and let ${\cal
P}_{\mathbb{V}}=\\{\gamma\subset\mathbb{V}:|\gamma|<\infty\\}$ be the polymer
set with incompatibility relation:
$\gamma\not\sim\gamma^{\prime}\,\,\Leftrightarrow\,\,\gamma\cap\gamma^{\prime}\neq\emptyset$
and (complex) activity $\zeta(\gamma)$. Assume that there is a positive number
$a>0$, such that, for all $x\in\mathbb{V}$

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

then, for all finite $\Lambda\subset\mathbb{V}$, we have that
${1\over|\Lambda|}\ln\Xi_{\Lambda}(\bm{\zeta})$, where
$\Xi_{\Lambda}(\bm{\zeta})$ is the partition function define in (4.73), can be
written as an absolutely convergent series uniformly in $\Lambda$.

#####  4.3.2 The subset gas: proof by Induction

We now illustrate the reasoning by induction to prove convergence of the
cluster expansion for the subset gas in the space ${\cal P}_{\mathbb{V}}$ in
the special case in which the activity is non negative (i.e. $\rho(\gamma)\geq
0$ for all $\gamma\in{\cal P}_{\mathbb{V}}$. We use the Miracle-sole approach.

###### Theorem 4.3

Let $\mathbb{V}$ be a countable set and let ${\cal
P}_{\mathbb{V}}=\\{\gamma\subset\mathbb{V}:|\gamma|<\infty\\}$ be the polymer
set with incompatibility relation:
$\gamma\not\sim\gamma^{\prime}\,\,\Leftrightarrow\,\,\gamma\cap\gamma^{\prime}\neq\emptyset$
and activity $\rho(\gamma)\geq 0$. Assume that there is a positive number
$a>0$, such that, for all $x\in\mathbb{V}$

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

(here of course $\rho\geq 0$) then, for all $\Lambda\in\mathbb{V}$, we have

$|\Sigma^{\Lambda}_{x}|(\rho)\leq a$ $None$

where $x\in\Lambda$ and

$|\Sigma^{\Lambda}_{x}|(\rho)=\sum_{n=1}^{\infty}{1\over
n!}\sum_{(\gamma_{1},\dots,\gamma_{n})\in\Lambda^{n}\atop\exists
i:~{}x\in\gamma_{i}}|\phi^{T}(\gamma_{1},\dots,\gamma_{n})|\;{\rho(\gamma_{1})}\dots\rho(\gamma_{n})$

Proof. First observe that, by the alternate sign property of
$\phi^{T}(\gamma_{1},\dots,\gamma_{n})$ we have that

$|\Sigma^{\Lambda}_{x}|(\rho)=-\log\Xi_{\Lambda}(-\rho)+\log\Xi_{\Lambda\backslash\\{x\\}}(-\rho)$
$None$

We will perform the proof by induction on $\Lambda$. We assume that, under the
condition (4.82), the inequality (4.83) is satisfied for a given $\Lambda$ and
any of its subsets. Take now $x\in\mathbb{Z}^{d}\backslash\Lambda$. We want to
bound

$|\Sigma^{\Lambda\cup\\{x\\}}_{x}|(\rho)=-\log\Xi_{\Lambda\cup\\{x\\}}(-\rho)+\log\Xi_{\Lambda}(-\rho)=-\log{\Xi_{\Lambda\cup\\{x\\}}(-\rho)\over\Xi_{\Lambda}(-\rho)}$

But now we know that

$\Xi_{\Lambda\cup\\{x\\}}(-\rho)=\Xi_{\Lambda}(-\rho)-\sum_{S\subset\Lambda}\rho(\\{x\\}\cup
S)\Xi_{\Lambda\backslash S}(-\rho)$

So

${\Xi_{\Lambda\cup\\{x\\}}(-\rho)\over\Xi_{\Lambda}(-\rho)}=1-\sum_{S\subset\Lambda}\rho(\\{x\\}\cup
S){\Xi_{\Lambda\backslash S}(-\rho)\over\Xi_{\Lambda}(-\rho)}$

Now, let $S=\\{y_{1},\dots,y_{k}\\}$ with of course $k=|S|$, then

${\Xi_{\Lambda\backslash
S}(-\rho)\over\Xi_{\Lambda}(-\rho)}={\Xi_{\Lambda\backslash\\{y_{1}\\}}(-\rho)\over\Xi_{\Lambda}(-\rho)}{\Xi_{\Lambda\backslash\\{y_{1},y_{2}\\}}(-\rho)\over\Xi_{\Lambda\backslash\\{y_{1}\\}}(-\rho)}\cdots{\Xi_{\Lambda\backslash\\{y_{1},\dots,y_{k}\\}}(-\rho)\over\Xi_{\Lambda\backslash\\{y_{1},\dots,y_{k-1}\\}}(-\rho)}$

The r.h.s. of equation above is a product of $k=|S|$ terms of the form

${\Xi_{\tilde{\Lambda}\backslash\\{z\\}}(-\rho)\over\Xi_{\tilde{\Lambda}}(-\rho)}$

with $\tilde{\Lambda}\subset\Lambda$ and $z\in\tilde{\Lambda}$. So by the
induction hypothesis for each of these terms we have

$\log{\Xi_{\tilde{\Lambda}\backslash\\{z\\}}(-\rho)\over\Xi_{\tilde{\Lambda}}(-\rho)}=\log\Xi_{\tilde{\Lambda}\backslash\\{z\\}}(-\rho)-\log\Xi_{\tilde{\Lambda}}(-\rho)=$
$=-\log\Xi_{\tilde{\Lambda}}(-\rho)+\log\Xi_{\tilde{\Lambda}\backslash\\{z\\}}(-\rho)=|\Sigma^{\tilde{\Lambda}}_{z}|(\rho)\leq
a$

So, adopting the convention
$\Lambda\backslash\\{y_{1},\dots,y_{i-1}\\}=\Lambda$ for $i=1$,

$\log{\Xi_{\Lambda\backslash
S}(-\rho)\over\Xi_{\Lambda}(-\rho)}=\sum_{i=1}^{k}\log{\Xi_{\Lambda\backslash\\{y_{1},\dots,y_{i}\\}}(-\rho)\over\Xi_{\Lambda\backslash\\{y_{1},\dots,y_{i-1}\\}}(-\rho)}\leq
k\,a=|S|\,a$

and hence

${\Xi_{\Lambda\backslash S}(-\rho)\over\Xi_{\Lambda}(-\rho)}\leq e^{a|S|}$

Thus

$|\Sigma^{\Lambda\cup\\{x\\}}_{x}|(\rho)=-\log\Xi_{\Lambda\cup\\{x\\}}(-\rho)+\log\Xi_{\Lambda}(-\rho)=-\log{\Xi_{\Lambda\cup\\{x\\}}(-\rho)\over\Xi_{\Lambda}(-\rho)}=$
$-\log\Bigg{[}1-\sum_{S\subset\Lambda}\rho(\\{x\\}\cup
S){\Xi_{\Lambda\backslash
S}(-\rho)\over\Xi_{\Lambda}(-\rho)}\Bigg{]}\leq-\log\Bigg{[}1-\sum_{S\subset\Lambda}\rho(\\{x\\}\cup
S)e^{a|S|}\Bigg{]}=$ $=-\log\Bigg{[}1-\sum_{S\subset\Lambda}\rho(\\{x\\}\cup
S)e^{a|S|}\Bigg{]}=-\log\Bigg{[}1-{1\over
e^{a}}\sum_{S\subset\Lambda}\rho(\\{x\\}\cup S)e^{a(|S|+1)}\Bigg{]}$
$\leq-\log\Bigg{[}1-{1\over e^{a}}(e^{a}-1)\Bigg{]}=a$

To complete the induction we have to show that the claim is true when
$\Lambda$ contains just one point, i.e. $\Lambda=\\{x\\}$. If
$\Lambda=\\{x\\}$ then we have only to check the function

$\displaystyle|\Sigma^{\\{x\\}}_{x}|(\rho)$
$\displaystyle=\sum_{n=1}^{\infty}{1\over
n!}\sum_{(\gamma_{1},\dots,\gamma_{n})\in(\\{x\\})^{n}\atop\exists
i:~{}x\in\gamma_{i}}|\phi^{T}(\gamma_{1},\dots,\gamma_{n})|\;{\rho(\gamma_{1})}\dots\rho(\gamma_{n})$
$\displaystyle=\sum_{n=1}^{\infty}{1\over
n!}|\phi^{T}(\\{x\\},\dots,\\{x\\})|\;[{\rho(\\{x\\})}]^{n}$
$\displaystyle=\sum_{n=1}^{\infty}{1\over n}\;[{\rho(\\{x\\})}]^{n}$
$\displaystyle\leq\sum_{n=1}^{\infty}{1\over n}\;\Big{[}1-{1\over
e^{a}}\Big{]}^{n}$ $\displaystyle=-\log\Big{[}1-\Big{(}1-{1\over
e^{a}}\Big{)}\Big{]}$ $\displaystyle=a$

where in the third line we have used that
$|\phi^{T}(\\{x\\},\dots,\\{x\\})|=\,(n-1)!$ and in the fourth line we used
that, by (4.82), $\rho(\\{x\\})\,\,\,\leq 1-{1\over e^{a}}$. So the induction
is completed. $\Box$

###  Chapter 5 Two systems in the cubic lattice

To show the utility of the polymer expansion in the study os discrete systems,
we will consider in this chapter two systems on the $d$-dimensional cubic
lattice $\mathbb{Z}^{d}$. We agree that $\mathbb{Z}^{d}$ is equipped with the
usual “graph distance” metric $|\cdot|$. Namely, given two points $x$ and $y$
in $\mathbb{Z}^{d}$, then $|x-y|$ denotes the graph distance between $x$ and
$y$ (i.e. the number of nearest neighbor edges of the shortest path between
$x$ and $y$).

####  5.1 The Self repuslive Lattice gas

The self repulsive lattice gas in $\mathbb{Z}^{d}$ is a system formed by
“particles” which may or may not occupy the vertices $x$ of the unit cubic
lattice $\mathbb{Z}^{d}$. In other words we associate to any site $x$ of
$\mathbb{Z}^{d}$ a variable $n_{x}$ taking values in the set $\\{0,1\\}$. When
$n_{x}=1$ a particle is occupying the site $x$, and when $n_{x}=0$, the site
$x$ is empty. The name self-repulsive is because each site can be occupied by
at most one particle. Let us then suppose that each particle occupying the
site $x$ has an activity $\lambda_{x}$ assumed to be constant, i.e.
$\lambda_{x}=\lambda$ for all $x\in\mathbb{Z}^{d}$. Finally, let us suppose
that particles in the self repulsive lattice gas interact via a pair potential
$V(x,y)$ which we assume to be symmetric $V(x,y)=V(y,x)$ (and, of course, self
repulsive, i.e. only one particle can occupy a site $x\in\mathbb{Z}^{d}$).
Namely:

$V(x,x)=+\infty\,\,\,\,\,\,\,\,\,\ \mbox{for all $x\in\mathbb{Z}^{d}$}$ $None$

We will also assume that the pair potential is summable (regular) in the
following sense

$\sup_{x\in\mathbb{Z}^{d}}\sum_{y\in\mathbb{Z}^{d}\atop y\neq
x}|V(x,y)|=J<\infty$ $None$

Note that the assumptions (5.2) and (5.2) automatically guarantee that the
pair potential is stable with stability constant not greater that $J/2$.
Indeed for any $n\in{}\mathbb{N}$ and
$(x_{1},\dots,x_{n})\in(\mathbb{Z}^{d})^{n}$,

$\sum_{1\leq i<j\leq n}V(x_{i},x_{j})\geq-{1\over 2}\sum_{(i,j)\in[n]^{2}\atop
i\neq j}|V(x_{i},x_{j})|=-{1\over 2}\sum_{i=1}^{n}\sum_{i\in[n]\atop j\neq
1}|V(x_{i},x_{j})|\geq-{J\over 2}n,$ $None$

The Grand Canonical partition function of such a lattice gas is

$Z_{\Lambda}(\beta,\lambda)=~{}\sum_{n=0}^{\infty}{\lambda^{n}\over
n!}\sum_{(x_{1},\dots,x_{n})\in\Lambda^{n}}~{}e^{-\beta\sum_{1\leq i<j\leq
n}V(x_{i},x_{j})}$

The specific case in which

$V_{\rm h.c.}(x,y)=\begin{cases}+\infty&{\rm if}~{}|x-y|\leq 1\\\ 0&{\rm
otherwise}\end{cases}$

belongs to the class of pair potentials satisfying assumptions (5.2) and
(5.3). so by the discussion done in Section 4.2.4, if $V(x_{i},x_{j})=V_{\rm
h.c.}(x_{i},x_{j})$ then the finite volume pressure
$p_{\Lambda}(\lambda)={1\over|\Lambda|}\ln Z_{\Lambda}(\lambda)$ (now
independent from $\beta$) converge absolutely and it is bounded by
$(2d-1)^{-1}$ uniformly in $\Lambda$ as soon as

$\lambda\leq\lambda_{c}\equiv{1\over 1+{(2d)^{2d}\over(2d-1)^{2d-1}}}$ $None$

In other words, $p_{\Lambda}(\beta,\lambda)$ is analytic in the disk
$|\lambda|\leq\lambda_{c}$.

#####  5.1.1 Covergence by direct Mayer expansion

As far as the general case is concerned (i.e. $V(x,y)$ regular and stable pair
potential $\beta$), we can obtain the convergent condition for the high
tempretaure/low activity phase of such lattice gas by directly perform a Mayer
expansion of the grand canonical partition function similarly to what we did
in Secrion3.1.2. Let us sketch how.  The Grand Canonical partition function is

$Z_{\Lambda}(\beta,\lambda)=~{}\sum_{n=0}^{\infty}{\lambda^{n}\over
n!}\sum_{(x_{1},\dots,x_{n})\in\Lambda^{n}}~{}e^{-\beta\sum_{1\leq i<j\leq
n}V(x_{i},x_{j})}$

where of course we are assuming in that $V(x,x)=+\infty$. By Mayer expansion
on the factor $e^{-\beta\sum_{i<j}V(x_{i},x_{j})}$ we find, as usual

$\ln Z_{\Lambda}(\beta,\lambda)~{}=~{}\sum_{n=1}^{\infty}{\lambda^{n}\over
n!}\sum_{(x_{1},\dots,x_{n})\in\Lambda^{n}}~{}\sum_{g\in{G}_{n}}\prod_{\\{i,j\\}\in
E_{g}}(e^{-\beta V(x_{i},x_{j})}-1)$

Then, by Theorem 3.8 and tree graph inequality (3.70), we have, using
stability also condition (5.3),

$|\sum_{g\in{G}_{n}}\prod_{\\{i,j\\}\in E_{g}}(e^{-\beta
V(x_{i},x_{j})}-1)|\leq e^{{\beta J\over
2}n}\sum_{\tau\in{T}_{n}}\prod_{\\{i,j\\}\in
E_{\tau}}(1-e^{-\beta|V(x_{i},x_{j})|})$

hence

$|\ln
Z_{\Lambda}(\beta,\lambda)|\leq\sum_{n~{}=~{}1}^{\infty}{|\lambda|^{n}\over
n!}\sum_{(x_{1},\dots,x_{n})\in\Lambda^{n}}~{}e^{{\beta J\over
2}n}\sum_{\tau\in{T}_{n}}\prod_{\\{i,j\\}\in
E_{\tau}}(1-e^{-\beta|V(x_{i},x_{j})|})\leq$
$\leq\sum_{n~{}=~{}1}^{\infty}{|\lambda|^{n}\over n!}~{}e^{{\beta J\over
2}n}\sum_{\tau\in{T}_{n}}\sum_{(x_{1},\dots,x_{n})\in\Lambda^{n}}\prod_{\\{i,j\\}\in
E_{\tau}}(1-e^{-\beta|V(x_{i},x_{j})|})$

but similarly to what we saw in Proposition 3.1, formula (3.17) it is not
difficult to show that

$\sum_{(x_{1},\dots,x_{n})\in\Lambda^{n}}\prod_{\\{i,j\\}\in
E_{\tau}}(1-e^{-\beta|V(x_{i},x_{j})|})\leq|\Lambda|\left[\sup_{x\in\mathbb{Z}^{d}}\sum_{y\in\mathbb{Z}^{d}}(1-e^{-\beta|V(x,y)|})\right]^{n-1}$

note that the sum over $y$ after the sup includes also $y=x$ where
$V(x,x)~{}=~{}\infty$. Observing that

$\sup_{x\in\mathbb{Z}^{d}}\sum_{y\in\mathbb{Z}^{d}\atop y\neq
x}(1-e^{-\beta|V(x,y)|})\leq\sup_{x\in\mathbb{Z}^{d}}\sum_{y\in\mathbb{Z}^{d}\atop
y\neq x}\beta|V(x,y)|\leq\beta J$

we get

$\sum_{(x_{1},\dots,x_{n})\in\Lambda^{n}}\prod_{\\{i,j\\}\in
E_{\tau}}(1-e^{-\beta|V(x_{i},x_{j})|})\leq|\Lambda|\left[1+\beta
J\right]^{n-1}$

and therefore

$\displaystyle|\ln Z_{\Lambda}(\beta,\lambda)|$
$\displaystyle\leq\sum_{n=1}^{\infty}{|\lambda|^{n}\over n!}~{}e^{{\beta
J\over 2}n}|\Lambda|[1+\beta J]^{n-1}n^{n-2}$
$\displaystyle\leq\sum_{n=1}^{\infty}{|\lambda|^{n}\over n^{2}}~{}e^{{\beta
J\over 2}n}|\Lambda|[1+\beta J]^{n-1}{n^{n}\over n!}$
$\displaystyle\leq\sum_{n=1}^{\infty}{|\lambda|^{n}\over n^{2}}~{}e^{{\beta
J\over 2}n}|\Lambda|[1+\beta J]^{n-1}e^{n}$

thus in this case the condition for the convergence is

$|\lambda|e^{{\beta J\over 2}+1}[1+\beta J]<1$ $None$

This condition is quite unsatisfactory since it says that, for any
temperature, even very high (i.e even for $\beta$ very small), one needs to
set the activity smaller than $1/e$ to ensure convergence.

#####  5.1.2 High temperature polymer expansion of the lattice gas

We now obtain the convergence condition for the same $\ln
Z_{\Lambda}(\beta,\lambda)$ performing a first example of high temperature
polymer expansion. We will get a much satisfactory bound and this will
illustrate quite well how convenient can be to perform polymer expansion in
discrete systems when possible!

The Grand canonical partition function of the same lattice gas enclosed in
$\Lambda\subset\mathbb{Z}^{d}$, with activity $\lambda$, inverse temperature
$\beta$, interacting via a pair potential $V(x,y)$ ($x$ and $y$ sites in
$\Lambda$) such that $V(x,x)~{}=~{}\infty$ (a site $x$ can be occupied at most
by one particle) can be written as follows

$\displaystyle Z_{\Lambda}(\beta,\lambda){}$
$\displaystyle=~{}\sum_{n=0}^{\infty}{\lambda^{n}\over
n!}\sum_{(x_{1},\dots,x_{n})\in\Lambda^{n}}~{}e^{-\beta\sum_{i<j}V(x_{i},x_{j})}$
$\displaystyle=~{}\sum_{n=0}^{|\Lambda|}{\lambda^{n}\over
n!}\sum_{(x_{1},\dots,x_{n})\in\Lambda^{n}\atop x_{i}\neq
x_{j}}~{}e^{-\beta\sum_{i<j}V(x_{i},x_{j})}$
$\displaystyle=~{}\sum_{n=0}^{|\Lambda|}\lambda^{n}\sum_{\\{x_{1},\dots,x_{n}\\}\subset\Lambda}~{}e^{-\beta\sum_{i<j}V(x_{i},x_{j})}$
$\displaystyle=~{}\sum_{S\subset\Lambda}\lambda^{|S|}~{}e^{-\beta\sum_{\\{x,y\\}\subset
S}V(x,y)}$

Define then for $x\in\Lambda$ the variable $n_{x}$ taking values in the set
$\\{0,1\\}$ ($n_{x}$ can be interpreted as the occupation number of the site
$x$: $n_{x}=0$ means that the site is empty and $n_{x}=1$ means that the site
is occupied).

We can rewrite the partition function above by considering that to each
$S\subset\Lambda$ we can associate a unique function
$\bm{n}_{\Lambda}:\Lambda\to\\{0,1\\}:x\mapsto n_{x}$ such that
$\bm{n}^{-1}(1)=S$ (we remind that $\bm{n}_{\Lambda}$ is usually called a
configuration of the lattice gas in $\Lambda$ and it specifies which sites are
occupied by a particle and which sites are empty). We let $N_{\Lambda}$ be the
set of all such functions. Clearly, if $|N_{\Lambda}|$ denotes the cardinality
of $N_{\Lambda}$ and $|\Lambda|$ is the number of sites in $\Lambda$, then
$|N_{\Lambda}|=2^{|\Lambda|}$. Moreover, in general, if
$R\subset\mathbb{Z}^{d}$, we denote by $\bm{n}_{R}$ a possible configuration
in $R$ and by $N_{R}$ the set of all possible configurations $n_{R}$.

As remarked earlier, there is a one to one correspondence between
$\bm{n}_{\Lambda}\in N_{\Lambda}$ and $S\subset\Lambda$ by defining
$S(n_{\Lambda})=\\{x\in\Lambda:n_{x}=1\\}$. Therefore we can write

$\displaystyle Z_{\Lambda}(\beta,\lambda)$
$\displaystyle=~{}\sum_{S\subset\Lambda}\,\lambda^{|S|}~{}e^{-\beta\sum_{\\{x,y\\}\subset
S}V(x,y)}$ $None$ $\displaystyle=~{}\sum_{\bm{n}_{\Lambda}\in
N_{\Lambda}}\lambda^{\sum_{x\in\Lambda}n_{x}}e^{-\beta\sum_{\\{x,y\\}\subset\Lambda}n_{x}n_{y}V(x,y)}$

Expand now the exponential in (5.6)

$e^{-\beta\sum_{\\{x,y\\}\subset\Lambda}n_{x}n_{y}V(x,y)}~{}=~{}\prod_{\\{x,y\\}\subset\Lambda}[e^{-\beta
n_{x}n_{y}V(x,y)}-1+1]~{}=~{}$
$~{}=~{}\sum_{s=1}^{|\Lambda|}\sum_{\\{R_{1},\dots,R_{s}\\}\in\pi(\Lambda)}\rho(R_{1})\cdots\rho(R_{r})$

where $\pi(\Lambda)~{}=~{}$ set of all partitions of $\Lambda$ (so that
$\\{R_{1},\dots,R_{s}\\}$ is a partition of $\Lambda$), and

$\rho(R)~{}=~{}\begin{cases}1&{\text{i}f}|R|~{}=~{}1\\\
\displaystyle{\sum\limits_{g\in G_{R}}\prod\limits_{\\{x,y\\}\in
E_{g}}[e^{-\beta n_{x}n_{y}V(x,y)}-1]}&{\text{i}f}~{}|R|\geq 2\end{cases}$

where $G_{R}$ is the set of connected graph with vertex set $R$.

Thus (5.6) can be written as,

$\displaystyle Z_{\Lambda}(\beta,\lambda)$
$\displaystyle=~{}\sum_{\bm{n}_{\Lambda}\in
N_{\Lambda}}\lambda^{\sum_{x\in\Lambda}n_{x}}\sum_{s=1}^{|\Lambda|}\sum_{R_{1},\dots,R_{s}\in\pi(\Lambda)}\rho(R_{1})\cdots\rho(R_{s})$
$\displaystyle=~{}\sum_{s=1}^{|\Lambda|}\sum_{R_{1},\dots,R_{s}\in\pi(\Lambda)}\sum_{\bm{n}_{R_{1}}\in
N_{R_{1}}}\rho(R_{1})\lambda^{\sum\limits_{x\in
R_{1}}n_{x}}\cdots\\!\\!\\!\sum_{\bm{n}_{R_{s}}\in
N_{R_{s}}}\\!\\!\\!\rho(R_{s})\lambda^{\sum\limits_{x\in R_{s}}n_{x}}$

Define now

${\tilde{\rho}}(R)\,~{}=~{}\,\sum_{n_{R}\in
N_{R}}\;\rho(R)\;\lambda^{\sum_{x\in R}\;n_{x}}$

Now observe that, for any $g\in G_{R}$, the factor

$\prod_{\\{x,y\\}\in E_{g}}[e^{-\beta n_{x}n_{y}V(x,y)}-1]\neq 0$

is different from zero only for the configuration $\bm{n}_{R}$ such that
$n_{x}=1$ for all $x\in R$. Therefore

${\tilde{\rho}}(R)~{}=~{}\begin{cases}1+\lambda&{\rm if}~{}|R|=1\\\
\lambda^{|R|}\sum\limits_{g\in G_{R}}\prod\limits_{\\{x,y\\}\in
E_{g}}[e^{-\beta V(x,y)}-1]&{\rm if}~{}|R|\geq 2\end{cases}$

Defining now

$\zeta(R)~{}=~{}\begin{cases}1&{\text{i}f}~{}|R|~{}=~{}1\\\
{\lambda^{|R|}\over(1+\lambda)^{|R|}}\sum\limits_{g\in
G_{R}}\prod\limits_{\\{x,y\\}\in E_{g}}[e^{-\beta
V(x,y)}-1]&{\text{i}f}~{}|R|\geq 2\end{cases}$ $None$

we obtain

$\displaystyle Z_{\Lambda}(\beta,\lambda)$ $\displaystyle=$
$\displaystyle(1+\lambda)^{|\Lambda|}\sum_{s=1}^{|\Lambda|}\sum_{R_{1},\dots,R_{s}\in\pi(\Lambda)}\zeta(R_{1})\cdots\zeta(R_{s})$
$\displaystyle=$ $\displaystyle(1+\lambda)^{|\Lambda|}\sum_{s\geq
0}\sum_{\\{R_{1},\dots,R_{s}\\}\subset\Lambda\atop|R_{i}|\geq 2,~{}R_{i}\cap
R_{j}=\emptyset}\zeta(R_{1})\cdots\zeta(R_{s})$

where the term $s=0$ in the last sum is equal to 1 and corresponds to the
partition of $\Lambda$ in $|\Lambda|$ subsets each of cardinality 1. Clearly

$\displaystyle\Xi_{\Lambda}(\beta,\lambda)$ $\displaystyle\doteq\sum_{s\geq
0}\sum_{\\{R_{1},\dots,R_{s}\\}\subset\Lambda\atop|R_{i}|\geq 2,~{}R_{i}\cap
R_{j}=\emptyset}\zeta(R_{1})\cdots\zeta(R_{s})$
$\displaystyle=~{}1+\sum_{n\geq 1}{1\over
n!}\sum_{R_{1},\dots,R_{n}\atop|R_{i}|\geq 2,\,R_{i}\cap
R_{j}~{}=~{}\emptyset}\zeta(R_{1})\cdots\zeta(R_{n})$

I.e. $\Xi_{\Lambda}(\beta,\lambda)$ is the grand canonical partition function
of a polymer gas in which polymers are finite subsets of
$R\subset\mathbb{Z}^{d}$ such that $|R|\geq 2$ and with activity $\zeta(R)$.

We finally have

$Z_{\Lambda}(\beta,\lambda)=(1+\lambda)^{|\Lambda|}\Xi_{\Lambda}(\beta,\lambda)$

and hence

$\ln
Z_{\Lambda}(\beta,\lambda)=|\Lambda|\ln(1+\lambda)+\ln\Xi_{\Lambda}(\beta,\lambda)$

Hence the pressure of the lattice gas ${1\over|\Lambda|}\log
Z_{\Lambda}(\beta,\lambda)$ converges absolutely uniformly in $\Lambda$ if
${1\over|\Lambda|}\log\Xi_{\Lambda}(\beta,\lambda)$ does.

I.e., by (4.80) conditions if,

$\sum_{n\geq
2}e^{an}\sup_{x\in\mathbb{Z}^{d}}\sum_{R\subset\mathbb{Z}^{d}:\atop x\in
R,~{}|R|=n}|\zeta(R)|\leq e^{a}-1$ $None$

or a little bit “roughly” (i.e choosing $a=\ln 2$ which is not an optimal
choice)

$\sum_{n\geq
2}2^{n}\sup_{x\in\mathbb{Z}^{d}}\sum_{R\subset\mathbb{Z}^{d}:\atop x\in
R,~{}|R|=n}|\zeta(R)|~{}\leq~{}1$ $None$

Recalling definition (5.7) of the activity $\zeta(R)$ and setting
${\tilde{\lambda}}~{}=~{}{\lambda\over(1+\lambda)}$ we have

$\sup_{x\in Z^{d}}\sum_{R\subset\mathbb{Z}^{d}:\atop x\in
R,~{}|R|=n}|\zeta(R)|~{}=~{}{|{\tilde{\lambda}}|}^{n}\sup_{x\in
Z^{d}}\sum_{R\subset\mathbb{Z}^{d}:\atop x\in R,~{}|R|=n}\bigg{|}\sum_{g\in
G_{R}}\prod_{\\{x,y\\}\in E_{g}}[e^{-\beta V(x,y)}-1]\bigg{|}$

Now note that

$\sum_{R\subset\mathbb{Z}^{d}:\atop x\in
R,~{}|R|=n}~{}=~{}{1\over(n-1)!}\sum_{(x_{1},\dots,x_{n})\in\mathbb{Z}^{dn}\atop
x_{1}=x,\;x_{i}\neq x_{j}\ {\rm for}\ i\neq j}$

Hence

$\sup_{x\in\mathbb{Z}^{d}}\sum_{R\subset\mathbb{Z}^{d}:\atop x\in
R,~{}|R|=n}|\zeta(R)|~{}=$
$={{|{\tilde{\lambda}}|}^{n}\over(n-1)!}\sup_{x\in\mathbb{Z}^{d}}\sum_{(x_{1},\dots,x_{n})\in\mathbb{Z}^{dn}\atop
x_{1}=x,\;x_{i}\neq x_{j}\ {\rm for}\ i\neq j}\bigg{|}\sum_{g\in
G_{n}}\prod_{\\{i,j\\}\in E_{g}}[e^{-\beta V(x_{i},x_{j})}-1]\bigg{|}$ $None$

Once again, by Theorem 3.8, tree graph inequality (3.69) and using stability
condition (5.3) we have the estimate

$\bigg{|}\sum_{g\in G_{n}}\prod_{\\{i,j\\}\in E_{g}}[e^{-\beta
V(x_{i},x_{j})}-1]\bigg{|}\leq e^{{\beta J\over 2}n}\sum_{\tau\in
T_{n}}\prod_{\\{i,j\\}\in E_{\tau}}(1-e^{-\beta|V(x_{i},x_{j})|})$

Now observe that, for any $\tau\in T_{n}$

$\displaystyle\sum_{(x_{1},\dots,x_{n})\in\mathbb{Z}^{dn}\atop
x_{1}=x,\;x_{i}\neq x_{j}\ {\rm for}\ i\neq j}\prod_{\\{i,j\\}\in
E_{\tau}}(1-e^{-\beta|V(x_{i},x_{j})|})$
$\displaystyle\leq~{}\Big{[}\sup_{x\in\mathbb{Z}^{d}}\sum_{y\in\mathbb{Z}^{d}\atop
y\neq x}\beta|V(x,y)|\Big{]}^{n-1}$ $\displaystyle=~{}[\beta J]^{n-1}$

Therefore, recalling also Cayley formula (3.6), we get

$\displaystyle\sup_{x\in Z^{d}}\sum_{R\subset\mathbb{Z}^{d}:\atop x\in
R,~{}|R|=n}|\zeta(R)|$ $\displaystyle\leq~{}{n^{n-2}\over(n-1)!}[\beta
J]^{n-1}(|{\tilde{\lambda}}|e^{\beta J\over 2})^{n}$
$\displaystyle\leq~{}|{\tilde{\lambda}}|e^{\beta J\over
2}\Big{[}|{\tilde{\lambda}}|e^{{\beta J\over 2}+1}\,\beta J\Big{]}^{n-1}$

Hence the condition for convergence (5.8) is satisfied if

$2|{\tilde{\lambda}}|e^{\beta J\over 2}\sum_{n\geq
2}\Big{[}2|{\tilde{\lambda}}|e^{{\beta J\over 2}+1}\,\beta
J\Big{]}^{n-1}~{}\leq~{}1$ $None$

which, after some calculus, is satisfied if

$|{\tilde{\lambda}}|e^{{\beta J\over 2}+1}\leq{1\over\beta J}{1\over
1+\sqrt{1+{4\over e\beta J}}}$ $None$

This condition is much better than (5.5). Indeed, denoting by $\beta_{c}$ the
unique solution of the equation

$e^{{\beta J\over 2}+1}~{}=~{}{1\over\beta J}{1\over 1+\sqrt{1+{4\over e\beta
J}}}$

and recalling that ${\tilde{\lambda}}=\lambda/(1+\lambda)$, we get that, as
soon as $\beta\leq\beta_{c}$, the pressure of the lattice gas is an absolute
convergent expansion for all $\lambda$ real. The temperature $\beta_{c}$ is a
first example of critical temperature: below $\beta_{c}$ the lattice gas is in
a pure phase for any activity $\lambda>0$.

####  5.2 The BEG model at low temperature

As an example, we consider the Blume-Emery-Griffiths (BEG) model in the low
temperature disordered phase. The model is defined on the cubic unit lattice
in $d$-dimensions $\mathbb{Z}^{d}$ by supposing that in each vertex
$x\in\mathbb{Z}^{d}$ there is a spin variable $\sigma_{x}$ taking values in
the set $\\{0,-1,+1\\}$. Given a volume $\Lambda\subset\mathbb{Z}^{d}$
(typically, a cube centered at the origin), a spin configuration in $\Lambda$
is a function $\bm{\sigma}:\Lambda\to\\{0,-1,+1\\}:x\mapsto\sigma_{x}$, We
denote by $\Sigma_{\Lambda}$ the set of all spin configurations in $\Lambda$
(note that $|\Sigma_{\Lambda}|=3^{|\Lambda|}$). Given
$\bm{\sigma}\in\Sigma_{\Lambda}$, its energy $H_{\Lambda}(\bm{\sigma})$ is
given by the Hamiltonian

$H_{\Lambda}(\bm{\sigma})=-\sum_{\\{x,y\\}\subset\Lambda\atop|x-y|=1}[\sigma_{x}\sigma_{y}+{\rm
Y}\sigma^{2}_{x}\sigma^{2}_{y}]+{\rm X}\sum_{x\in\Lambda}\sigma_{x}^{2}$
$None$

where in general ${\rm X}$ and ${\rm Y}$ are real parameters. In other words
spins interacts via a nearest neighbor pair potential. We will further suppose
that

${\rm X}>d(1+|{\rm Y}|)$ $None$

With this hypothesis it is not difficult to check that the minimum of
$H_{\Lambda}(\bm{\sigma})$ is reached at the configuration $\bm{\sigma}$ such
that $\sigma_{x}=0$ for all $\xi\in\Lambda$. This region is called the
disordered phase. Via the Hamiltonian (5.13) we can assign a probability
$Prob(\bm{\sigma})$ to each configuration $\bm{\sigma}$ in $\Lambda$. Namely,

$Prob(\bm{\sigma})={e^{-\beta H_{\Lambda}(\bm{\sigma})}\over
Z_{\Lambda}(\beta)}$

where $\beta>0$ is the inverse temperature and

$Z_{\Lambda}(\beta)=\sum_{\bm{\sigma}\in\Sigma_{\Lambda}}e^{-\beta
H(\bm{\sigma})}$ $None$

is the partition function of the system. The thermodynamics of the BEG model
is recovered from the the partition function $Z_{\Lambda}(\beta)$. namely, the
(finite volume) free energy of the model is given by

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

We’ll show in this section that the free energy (5.16) of the BEG model is
analytic as function of $\beta$ for $\beta$ sufficiently large (i.e low
temperature). This is achieved by rewriting $Z_{\Lambda}(\beta)$ defined in
(5.15) as the partition function of a polymer system of the type considered in
the previous sections. Hence the free energy of the BEG model coincides with
the pressure of this polymer system. Then, using theorem 4.2, we will prove
that, in the disordered phase (5.14) and with the assumptions (5.14), the
pressure of such polymer system converges absolutely for $\beta$ large enough
(i.e. for sufficiently low temperatures).

In order to do that, let us define, for a fixed spin configuration
$\bm{\sigma}\in\Sigma_{\Lambda}$, the subset of $\Lambda$ given by
$P=\\{x\in\Lambda:\sigma_{x}\neq 0\\}$. We view this set as the union of its
connected components, i.e. $P=\cup_{i=1}^{n}p_{i}$ with each set
$p_{i}\subset\Lambda$ being connected in the sense that for each partition
$A,B$ of $p_{i}$ (i.e. $A\cup B=p_{i}$ and $A\cap B=\emptyset$) there exist
$x\in A$ and $y\in B$ such that $|x-y|=1$. The configuration $\bm{\sigma}$
induces a (non zero) spin configuration $\bm{s}_{p_{i}}$ on each connected
component $p_{i}$ of $P$ which is a function
$\bm{s}_{p_{i}}:p_{i}\to\\{-1,+1\\}:x\mapsto s_{x}$. The pairs
${\bm{p}}_{i}=(p_{i},\bm{s}_{p_{i}})$ are the polymers associated to the
configuration $\bm{\sigma}$.

We denote by ${\mathcal{Q}}$ the set of all subsets of $\mathbb{Z}^{d}$ which
are connected and finite. Given $p\in{\mathcal{Q}}$, we denote by $S_{p}$ the
set of all (non zero) spin configurations $\bm{s}_{p}$ in $p$ (note that
$|S_{p}|=2^{|p|})$. By construction the correspondence
$\bm{\sigma}\leftrightarrow\\{\bm{p}_{1},\dots,\bm{p}_{n}\\}$ is one to one.
The distance between two polymers $\bm{p}=(p,\bm{s}_{p})$ and
$\tilde{\bm{p}}=(\tilde{p},\bm{s}_{\tilde{p}})$ is the number
$d(p,\tilde{p})=\min_{x\in p,\,\,y\in\tilde{p}}|x-y|$. Note that if
$\\{\bm{p}_{1},\dots,\bm{p}_{n}\\}$ are the polymers associated to the
configuration $\bm{\sigma}$, then necessarily $d(p_{i},p_{j})\geq 2$ for all
$\\{i,j\\}\subset\\{1,\dots,n\\}$.

With these definitions, given $\bm{\sigma}\in\Sigma_{\Lambda}$ such that
$\bm{\sigma}\leftrightarrow\\{\bm{p}_{1},\dots,\bm{p}_{n}\\}$, we can rewrite
the Hamiltonian (5.13) as

$H_{\Lambda}(\bm{s})=\sum_{i=1}^{n}\Big{[}{\rm X}|p_{i}|-A(\bm{p}_{i})\Big{]}$

where

$A(\bm{p}_{i})=\sum_{\\{x,y\\}\subset p_{i}\atop|x-y|=1}[s_{x}s_{y}+{\rm Y}]$
$None$

Observe now that to sum over configuration $\bm{\sigma}\in\Sigma_{\Lambda}$ is
equivalent to sum over polymers configurations
$\\{\bm{p}_{1},\dots,\bm{p}_{n}\\}$ in $\Lambda$ such that $n\geq 0$ ($n=0$,
i.e. no polymers, is the lowest energy configuration $\sigma_{x}=0$ for all
$x\in\Lambda$) and $d(p_{i},p_{j})\geq 2$ for all pairs
$\\{i,j\\}\subset\\{1,\dots,n\\}$. Hence the partition function of the system,
at inverse temperature $\beta$ and with free boundary conditions, is rewritten
as

$\displaystyle Z_{\Lambda}(\beta)$
$\displaystyle=~{}\sum_{\sigma_{\Lambda}}e^{-\beta H(\sigma_{\Lambda})}$
$None$ $\displaystyle=~{}1~{}+~{}\sum_{n\geq 1}{1\over
n!}\sum_{(\bm{p}_{1},\dots,\bm{p}_{n})\in{\cal P}^{n}\atop
p_{i}\subset\Lambda,\;d(p_{i},p_{j})\geq
2}\rho_{\bm{p}_{1}}\dots\rho_{\bm{p}_{n}}$

where

$\rho_{\bm{p}}=e^{-\beta\big{[}{\rm X}|p|-A(\bm{p})\big{]}}$ $None$

and ${\cal P}$ is the set

${\cal P}=\Big{\\{}\bm{p}=(p,\bm{s}_{p}):\mbox{$p\in{\mathcal{Q}}$ and
$\bm{s}_{p}\in S_{p}$}\Big{\\}}$ $None$

Thus we have rewritten the partition function $Z_{\Lambda}(\beta)$ of the BEG
model as the partition function of an hard-core polymer gas in which the space
of polymers is the set given in (5.20). Each polymer $\bm{p}\in{\cal P}$ has
activity $\rho_{\bm{p}}$ given by (5.19) and the hard core interaction
$V(\bm{p}_{i},\bm{p}_{j})$ between pair of polymers $p_{i},p_{j}$ is

$V(\bm{p}_{i},\bm{p}_{j})=\begin{cases}0&{\rm if}~{}d(p_{i},p_{j})\geq 2\\\
+\infty&{\rm otherwise}\end{cases}$ $None$

With these definitions it is immediate to see that r.h.s. of (5.18) can be
written as

$\displaystyle Z_{\Lambda}(\beta)$ $\displaystyle=1+\sum_{n\geq 1}{1\over
n!}\sum_{(\bm{p}_{1},\dots,\bm{p}_{n})\in{\cal
P}^{n}_{\Lambda}}\rho_{\bm{p}_{1}}\dots\rho_{\bm{p}_{n}}e^{-\sum_{1\leq
i<j\leq n}V(\bm{p}_{i},\bm{p}_{j})}$ $None$ $\displaystyle=1+\sum_{n\geq
1}{1\over n!}\sum_{(\bm{p}_{1},\dots,\bm{p}_{n})\in{\cal
P}^{n}_{\Lambda}\atop\bm{p}_{i}\sim\bm{p}_{j}}\rho_{\bm{p}_{1}}\dots\rho_{\bm{p}_{n}}$

where we have denoted shortly ${\cal P}_{\Lambda}=\\{(p,\bm{s}_{p})\in{\cal
P}:\;p\subset\Lambda\\}$ and $\bm{p}_{i}\sim\bm{p}_{j}$ means that
$d(p_{i},p_{j})\geq 2$ (and similarly ahead we use the symbol
$\bm{p}_{i}\nsim\bm{p}_{j}$ when $d(p_{i},p_{j})<2$).

So, by Theorem 4.1, the pressure of this polymer gas (i.e. the free energy of
our BEG model) is absolutely convergent if there exist
$\\{\mu_{\bm{p}}\\}_{\bm{p}\in{\cal P}}$ such that such that

$\rho_{\bm{p}}\leq{\mu_{\bm{p}}\over\Xi_{\bm{p}}(\bm{\mu})}\;\;\;\,\,\,\,\,\,\,\forall\gamma\in{\cal
P}$ $None$

with

$\Xi_{\bm{p}}(\bm{\mu})=1+\sum_{n\geq
1}\frac{1}{n!}\,\sum_{(\bm{p}_{1},\dots,\bm{p}_{n})\in{\cal
P}^{n}\atop\bm{p}_{i}\not\sim\bm{p},\,\,\bm{p}_{i}\sim\bm{p}_{j}}{\mu_{\bm{p}_{1}}}\dots\mu_{\bm{p}_{n}}$

Given $p\in{\mathcal{Q}}$ let $\partial p=\\{x\in\mathbb{Z}^{d}\setminus
p:\exists y\in p~{}s.t.~{}|x-y|=1\\}$ be the the boundary of $p$ and let
$\bar{p}=p\cup\partial p$. Given $\alpha>0$, we set

$\mu_{\bm{p}}=\rho_{\bm{p}}e^{\alpha|\bar{p}|}$ $None$

Hence, inserting (5.24) in (5.23), we obtain that the pressure of such contour
gas can be written in terms of an absolutely convergent series if, for some
$\alpha>0$

$\Xi_{\bm{p}}(\bm{\mu})\leq e^{\alpha|\bar{p}|}$ $None$

Now, similarly to what we did in Section 4.3.1 we can bound

$\Xi_{\bm{p}}(\bm{\mu})\leq\Bigg{[}1+\sup_{x\in\mathbb{Z}^{d}}\,\sum_{\bm{p}\in{\cal
P}\atop x\in p}\rho_{\bm{p}}e^{\alpha|\bar{p}|}\Bigg{]}^{|\bar{p}|}$

so that condition (5.25) is satisfied if

$\sup_{x\in\mathbb{Z}^{d}}\sum_{\bm{p}\in{\cal P}\atop x\in
p}\rho_{\bm{p}}e^{\alpha|\bar{p}|}\leq e^{\alpha}-1$ $None$

Now, recalling (5.19), observe that

$A(\bm{p}_{i})=\sum_{\\{x,y\\}\subset p_{i}\atop|x-y|=1}[s_{x}s_{y}+{\rm
Y}]\leq d(1+|{\rm Y}|)|p_{i}|$

and so

$\rho_{\bm{p}}\leq e^{-\beta D|p|}$

where

$D={\rm X}-d(1+|{\rm Y}|)$

and observe that $D>0$ by hypotheis (5.14). Therefore, choosing for simpliciy
$\alpha=\ln 2$, condition (5.26) is satisfied if

$\sup_{x\in\mathbb{Z}^{d}}\sum_{\bm{p}\in{\cal P}\atop x\in p}e^{-\beta
D|p|}2^{|\bar{p}|}\leq 1$ $None$

Now observe that surely $|\bar{p}|\leq 2d|p|+2\leq 3d|p|$. Therefore we get
that condition (5.27) is satisfied if

$\sum_{n=1}^{\infty}(2^{3d+1}e^{-\beta
D})^{n}\sup_{x\in\mathbb{Z}^{d}}\sum_{\bm{p}\in{\cal P}\atop x\in
p,\;|p|=n}1\leq 1$ $None$

Now we have that

$\sup_{x\in\mathbb{Z}^{d}}\sum_{\bm{p}\in{\cal P}\atop x\in
p,\;|p|=n}1=2^{n}\sup_{x\in\mathbb{Z}^{d}}\sum_{p\in{\mathcal{Q}}\atop x\in
p,\;|p|=n}1=2^{n}C_{n}$

where $C_{n}$ is the number of connected sets of vertices of $\mathbb{Z}^{d}$
with cardinality $n$ containing a fixed point $x$ of $\mathbb{Z}^{d}$ (this
number does not depend on $x$ and can be chosen to be the orgin) and the extra
factor $2^{n}$ counts the number of functions $\sigma_{p}$ from $p$ to
$\\{-1,+1\\}$ when $|p|=n$. $C_{n}$ can be easily bounded by $C^{n}$ for some
$C$, e.g. one can take $C_{n}\leq(4d)^{n}$. So condition (5.28) becomes

$\sum_{n=1}^{\infty}(d2^{3d+3}e^{-\beta D})^{n}\leq 1$

which is satisfied if

$\beta\geq\beta^{*}\doteq{\ln(d2^{3d+4})\over D}$ $None$

Therefore for all $\beta\geq\beta^{*}$ the free energy (5.16) of the BEG mode
under the condition (5.14) is analytic in $\beta$ an no phase transitions
occur.

###  Chapter 6 Spin systems in a lattice

We denote by $\mathbb{Z}^{d}$ the unit lattice in $d$ dimensions, i.e. an
element $x\in\mathbb{Z}^{d}$ is an ordered $d$-ple
$x~{}=~{}(n_{1},\dots,n_{d}):n_{i}\in\mathbb{Z}$. We will consider
$\mathbb{Z}^{d}$ embedded in the standard way in $\mathbb{R}^{d}$. Let
$\Lambda$ be a finite subset of ${\mathbb{Z}}^{d}$, e.g.usually one takes al
set $\Lambda$ the set
$\Lambda~{}=~{}\\{(n_{1},\dots,n_{d}):n_{i}\in\mathbb{Z};-{L\over 2}\leq
n_{i}\leq{L\over 2},\forall i~{}=~{}1,\dots,d\\}$, i.e. all sites of the unit
lattice inside a cube with the center in the origin and with size $L$.  The
definition of a spin system on ${\mathbb{Z}}^{d}$ begins by associating to
each site $x\in\Lambda$ a variable $\phi_{x}$, (the ”spin” at the site $x$).
The spin $\phi_{x}$ takes values in some space $\Omega$ (which is the same for
all $x\in\mathbb{Z}^{d}$) equipped with a probability measure $d\mu(\phi_{x})$
and with a norm $\|\phi_{x}\|$. The single spin probability space $\Omega$
depends of the type of system we are treating and, in general, its structure
can be very different from case to case. The probability measure
$d\mu(\phi_{x})$ is called sometimes the single spin a priori distribution.

The simplest choice for which $\Omega$ is assuming that $\Omega$ is a finite
set, with $d\mu$ assigning equal probability to any value of the spin in this
set $\Omega$. E.g. the choice $\Omega~{}=~{}\\{1,-1\\}$ (i.e. the spin
$\phi_{x}$ at a site $x$ can take only the two values $\phi_{x}~{}=~{}\pm 1$)
and $d\mu(\phi_{x})~{}=~{}{1\over 2}\delta_{\phi_{x},+1}+{1\over
2}\delta_{\phi_{x},-1}$, (i.e. the probability that the spin $\phi_{x}$ at the
site $x$ has the value $+1$ is ${1\over 2}$ and so for $\phi_{x}~{}=~{}-1$)
corresponds to Ising systems. For such systems the random variable $\phi_{x}$
is usually denote in the literature by $\sigma_{x}$ and when
$\sigma_{x}~{}=~{}1$ people say that the spin is “up” while
$\sigma_{x}~{}=~{}-1$ is referred as spin “down”.

The choice $\Omega~{}=~{}\\{0,1\\}$ and $d\mu(\phi_{x})~{}=~{}{1\over
2}\delta_{\phi_{x},+1}+{1\over 2}\delta_{\phi_{x},0}$ corresponds to the
Lattice gas considered in the previous sections. In this case the variable
$\phi_{x}$ is usually denoted by $n_{x}$ and interpreted as an occupation
number, namely if $n_{x}=1$ then we say that a particle is present at the site
$x$ and the site $x$ is occupied, while if $n_{x}=0$ then no particle is
present at $x$ and the site $x$ is empty.

The choice $\Omega~{}=~{}\\{1,2,\dots,q\\}$ with $q$ positive integer and
$d\mu(\phi_{x})~{}=~{}{1\over q}[\delta_{\phi_{x},1}+{1\over
q}\delta_{\phi_{x},2}+\dots+{1\over q}\delta_{\phi_{x},q}]$ correspond to the
so called Potts model.

A less simple choice for $\Omega$ is to assume $\Omega$ infinite but compact.
A typical example is the (classical) Heisenberg model for which
$\Omega~{}=~{}\\{\phi_{x}\in\mathbb{R}^{d}:\|\phi_{x}\|~{}=~{}1\\}$, i.e.
$\Omega$ is the unit sphere in $d$ dimensions, and $d\mu(\phi_{x})~{}=~{}[{\rm
volume}~{}\Omega]^{-1}d\Omega(\phi_{x})$, where $d\Omega(\phi_{x})$ is the
usual Rienmann measure on the sphere.

Spin systems for which $\Omega$ is a compact space (in the sense of the norm
$\|\phi_{x}\|$), are usually called bounded spin systems. But we could also
consider the case in which $\Omega$ is not a compact space. Spin systems for
which the single spin space $\Omega$ is a non compact space are usually known
as unbounded spin systems. In general bounded spin systems are treated easier
than unbounded spin systems.

As an example of a unbounded spin system we will consider ahead the so called
$\lambda\phi^{4}$ field theory, for which $\Omega~{}=~{}\mathbb{R}$, (i.e. the
spin take values in $\mathbb{R}$) and

$d\mu(\phi_{x})~{}=~{}{Z}^{-1}e^{-U(\phi_{x})}d\phi_{x}$

where $U(\phi_{x})~{}=~{}\lambda\phi^{4}$ with $\lambda>0$ and
$Z~{}=~{}\int_{\mathbb{R}}e^{-U(\phi_{x})}d\phi_{x}$. In this case $\phi_{x}$
is interpreted as the value of the field at the site $x$.

A configuration $\phi$ of a general spin system in $\mathbb{Z}^{d}$ is given
once it has been specified the value of $\phi_{x}\in\Omega$ for all
$x\in\mathbb{Z}^{d}$. More precisely, a configuration $\phi$ is a function
$\phi:\mathbb{Z}^{d}\to\Omega$. The set of all configuration $\phi$ of the
system is the set $\Phi=\Omega^{\mathbb{Z}^{d}}$. So a configuration $\phi$ is
an element of a very big space $\Phi$ and in general it is mathematically
problematic to define a probability measure on this space. Even the product
measure $\prod_{x\in\mathbb{Z}^{d}}\mu(\phi_{x})$ can be difficult to define
rigorously depending on the structure of the space $\Omega$ (e.g. in case that
$\Omega$ in non compact).

Hence, next step towards the definition of a spin system in the lattice it to
fix some finite subset $\Lambda\subset\mathbb{Z}^{d}$ and then consider our
system of spins restricted to this set $\Lambda$. So a configuration
$\phi_{\Lambda}$ of the system restricted to $\Lambda$ is given once it has
been specified the value of $\phi_{x}\in\Omega$ for all $x\in\Lambda$. We
denote by $\Phi_{\Lambda}=\prod_{x\in\Lambda}\Omega$ the set of all spin
configurations restricted to $\Lambda$. Now the product measure
$d\mu(\phi_{\Lambda})=\prod_{x\in\Lambda}d\mu(\phi_{x})$ in $\Phi_{\Lambda}$
is rigorously defined without no particular difficulties. If $X$ is any subset
of $\Lambda$, then symbols as $\Phi_{X}$ and $d\mu(\phi_{X})$ are defined in
the obvious way.

The boundary conditions are in general chosen by fixing a spin configuration
$\phi_{\Lambda^{c}}$ for the spins outside $\Lambda$ (i.e. in the set
$\Lambda^{c}~{}=~{}\mathbb{V}-\Lambda$). Such choice can be sometimes limited
due to the structure of the single spin space $\Omega_{x}$. But other choices
are also possible. E.g. two very popular choices of the boundary conditions
are the so-called free boundary conditions and periodic boundary conditions.
In particular, the free boundary condition choice is to say that “there is no
outside”, i.e. simply to study the spin system in the finite volume $\Lambda$
with the identification $\mathbb{V}=\Lambda$. Also the periodic boundary
conditions follows the same philosophy. However to be stated $\mathbb{V}$ has
to possesses some additional structure. For example, if
$\mathbb{V}=\mathbb{Z}^{d}$, one has imposed boundary conditions by saying
that $\Lambda$ is a torus in $\mathbb{Z}^{d}$.

Finally, the Hamiltonian or interaction between spins, denoted here by
$H(\phi_{\Lambda},\phi_{\Lambda^{c}})$, is in general expressed in terms of a
potential $\Psi$, i.e. an assignment to each nonempty subset
$X\subset\mathbb{V}$ of real valued measurable function $\Psi(X,\phi_{X})$
definied in $\Omega_{X}$. Once $\Psi$ has been given, and boundary conditions
are give through a fixed configuration $\phi_{\Lambda^{c}}$ in
$\mathbb{V}\backslash\Lambda$, the Hamiltonian
$H(\phi_{\Lambda},\phi_{\Lambda^{c}})$ of the discrete spin system confined in
the volume $\Lambda$ is given (when boundary conditions are give through a
fixed configuration $\phi_{\Lambda^{c}}$ in $\mathbb{V}\backslash\Lambda$) by

$H(\phi_{\Lambda},\phi_{\Lambda^{c}})=\sum_{X:X\cap\Lambda\neq\emptyset\atop|X|\geq
2}\Psi(X,\phi_{X})$ $None$

In the majority of the applications presented in this book the potential
$\Psi$ will be defined only for pairs of spins, namely $\Psi(X,\phi)\neq 0$
only for $|X|=2$. In this case $\Psi$ is called a pair potential.

Note that the Hamiltonian (6.1) can be also written as

$V(\phi_{\Lambda},\phi_{\Lambda^{c}})~{}=~{}H_{\rm
bulk}(\phi_{\Lambda})+W(\phi_{\Lambda},\phi_{\Lambda^{c}})$ $None$

where

$H_{\rm bulk}(\phi_{\Lambda})~{}=~{}\sum_{X\subset\Lambda}\Psi(X,\phi_{X})$

and

$W(\phi_{\Lambda},\phi_{\Lambda^{c}})~{}=~{}\sum_{X:X\cap\Lambda\neq\emptyset\atop
X\cap\Lambda^{c}\neq\emptyset}\Psi(X,\phi_{X})$

where evidently the term $H_{\rm bulk}$ represent the “internal” energy inside
$\Lambda$ of the system, which does not depend on boundary conditions, while
$W(\phi_{\Lambda},\phi_{\Lambda^{c}})$, which is the boundary condition term,
represents the interaction of the system in $\Lambda$ with the “outside”. Note
that in the free boundary conditions one sets
$W(\phi_{\Lambda},\phi_{\Lambda^{c}})=0$, which indeed corresponds to suppose
that there is no outside.

Once we give explicitly $(\Omega,\mu)$, the boundary conditions and the
Hamiltonian, then the measure $\Pi$ of the lattice system is defined by

$d\Pi(\phi_{\Lambda})={d\mu(\phi_{\Lambda})e^{-\beta
H(\phi_{\Lambda},\phi_{\Lambda^{c}})}\over\Xi_{\Lambda}(\beta)}$ $None$

where $\beta>0$ is a parameterusually interpreted in statistical mechanics as
the inverse temperature of the system and the normalization constant
$\Xi_{\Lambda}(\beta)$, given by

$Z_{\Lambda}(\beta)=\int d\mu(\phi_{\Lambda})e^{-\beta
H(\phi_{\Lambda},\phi_{\Lambda^{c}})}$ $None$

is usually called the Partition functions of the system.

####  6.1 Some examples

1\. The celebrated Ising model falls easily in the structure defined above .
Here the set $\mathbb{V}$ is typically the lattice $\mathbb{Z}^{d}$ in $d$
dimensions, and the model is defined on a box $\Lambda\subset\mathbb{Z}^{d}$.
The role of the spin $\phi_{x}$, $x\in\Lambda$, is played by a random variable
usually denoted by $\sigma_{x}$ which can take one of the two possible value
$\sigma_{x}~{}=~{}\pm 1$. A configuration
${\bf\sigma}_{\Lambda}\in\\{-1,1\\}^{|\Lambda|}$ of the system is given once
the value of the spin $\sigma_{x}$ for each site $x\in\Lambda$ is specified.
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
$\sigma_{\Lambda}:\Lambda\to\\{+1,-1\\}$. Observe that the total number of
configurations of the system in $\Lambda$ is $2^{|\Lambda|}$.

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

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

where $h$ is an “external magnetic field”, $J$ is a real constant representing
the intensity of the nearest neighbors interaction, and the model is said
ferromagnetic when $J>0$ and antiferromagnetic otherwise. ${\cal
B}(\sigma_{\Lambda})$ represents the interaction of the spins inside the box
$\Lambda$ with a fixed configuration outside $\Lambda$ (i.e. in
$\mathbb{Z}^{d}\backslash\Lambda$).

The statistical mechanics of the Ising model 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

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

is the partition function. The term $e^{-\beta
H_{\Lambda}({\bf\sigma}_{\Lambda})}$ is called Boltzmann weight of the
configuration ${\bf\sigma}_{\Lambda}$.

The structure of the probability distribution above shows that if the system
is ferromagnetic the spins tend to align, i.e. the configurations have higher
probability if they have a large number of nearest neighbors assuming the same
direction of the spin, while in the antiferromagnetic case the configuration
with nearest neighbors of the opposite sign are privileged.

In order to point out the ingredients of the general structure of the discrete
systems described above, we will consider for simplicity free boundary
conditions, namely we put ${\cal B}(\sigma_{\Lambda})=0$ in (6.5). Then
observe that

$Z_{\Lambda}(\beta,h)~{}=~{}\sum_{{\bf\sigma}_{\Lambda}}\Big{[}\prod_{x\in\Lambda}e^{h\sigma_{x}}\Big{]}e^{-\beta
J\sum_{\\{x,y\\}\subset\Lambda\atop x,y~{}{\rm
near.~{}neigh.}}\sigma_{x}\sigma_{y}}=$

$~{}=~{}[2\cosh(\beta h)]^{|\Lambda|}\Xi_{\Lambda}(\beta,h)$

where

$\Xi_{\Lambda}(\beta,h)=\int d\mu(\sigma_{\Lambda})e^{-\beta
J\sum_{\\{x,y\\}\subset\Lambda\atop x,y~{}{\rm
near.~{}neigh.}}\sigma_{x}\sigma_{y}}$ $None$

It is clear now that giving to the term $J\sum_{\\{x,y\\}\subset\Lambda\atop
x,y~{}{\rm near.~{}neigh.}}\sigma_{x}\sigma_{y}$ the role of the potential $V$
defined in (6.1), we have that the Ising model is a lattice system in which
$\Omega_{x}=\\{1,-1\\}$, the single spin distribution is the discrete measure
given by

$\mu(\sigma_{x}=+1)={e^{\beta Jh}\over\cosh\beta
h},~{}~{}~{}~{}~{}\mu(\sigma_{x}=-1)={e^{-\beta Jh}\over\cos h\beta Jh}$

and the interactions $\Psi(X,\phi_{X})$ are defined only for $X$ of the form
$X=\\{x,y\\}$ with $x$ and $y$ nearest neighbors.

The Ising model, despite its simplicity, has a very rich structure which make
it one of the most interesting model in statistical mechanics. We shall
investigate some of its properties in chapter ??.

2\. The Potts model is a generalization of the Ising model. The system is
again defined on a box $\Lambda\subset\mathbb{Z}^{d}$ which is a square subset
of the lattice $\mathbb{Z}^{d}$ in $d$ dimensions, but here the spin
$\sigma_{x}$ varies in the set $\\{1,2,\dots,q\\}$ where $q$ is an integer.
The Hamiltonian is given by

$H_{\Lambda}(\sigma_{\Lambda})=-J\sum_{\\{x,y\\}\subset\Lambda\atop x,y~{}{\rm
nearest~{}neighbors}}\delta_{\sigma_{x}\sigma_{y}}$ $None$

where $\delta_{\sigma_{x}\sigma_{y}}$ is the Kronecker symbol which is equal
to one when $\sigma_{x}=\sigma_{y}$ and zero otherwise. Again, the coupling
$J$ is called ferromagnetic if $J>0$ and anti-ferromagnetic if $J<0$.

As in the case of the Ising model the statistical mechanics of the system can
be done by introducing the Boltzmann weight of a configuration
$\sigma_{\Lambda}$, defined a s $\exp\\{-\beta
H_{\Lambda}(\sigma_{\Lambda})\\}$. Then the probability to find the system in
the configuration $\sigma_{\Lambda}$ is given b y

${\rm Prob}(\sigma_{\Lambda})={e^{-\beta H_{\Lambda}(\sigma_{\Lambda})}\over
Z_{\Lambda}(q,\beta)}$ $None$

Again the normalization constant in the denominator is called the partition
function and is given by

$Z_{\Lambda}(q,\beta)=\sum_{\sigma_{\Lambda}\in\\{1,2,...,q\\}^{|\Lambda|}}e^{-\beta
H_{\Lambda}(\sigma_{\Lambda})}$ $None$

The case $\beta J=-\infty$ is the anti-ferromagnetic and zero temperature
Potts model with $q$-states. In this case configurations with non zero
probability are only those in which adjacent spins have different values (or
colors) and $Z_{\Lambda}(q,\beta)$ becomes simply the number of all allowed
configurations (proper colorings).

The Potts model is obviously in the class above. The spin $\sigma_{x}$ plays
the role of $\phi_{x}$, $\Omega_{x}=\\{1,\dots,q\\}$, the single spin
distribution is the uniform distribution on $\\{1,\dots,q\\}$ and as above the
interaction is defined only for $X$ of the form $X=\\{x,y\\}$ wit h $x$ and
$y$ nearest neighbors.

Namely, we can write

$Z_{\Lambda}(q,\beta)=q^{|\Lambda|}\int d\mu(\sigma_{\Lambda})\in e^{-\beta
H_{\Lambda}(\sigma_{\Lambda})}$

where $\Omega_{x}=\\{1,2,\dots,q\\}$, and the single spin distribution is the
discrete uniform measure on $\Omega_{x}=\\{1,2,\dots,q\\}$ given by

$\mu(\sigma_{x}=i)={1\over q}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall
i\in\\{1,2,\dots,q\\}$

and the interactions $\Psi(X,\phi_{X})$ are defined only for $X$ of the form
$X=\\{x,y\\}$ with $x$ and $y$ nearest neighbors.

3\. Let us now consider the example of lattice gases interacting through pair
potential. This will be a crucial example in the following sections.

In statistical mechanics (see also below, section ??), the gran canonical
partition function of a lattice gas of identical particles enclosed in a cubic
volume $\Lambda\subset\mathbb{Z}^{d}$ with activity $\lambda$, interacting via
a pair potential $V(x,y)$ ($x$ and $y$ sites in $\Lambda$) is given by

$Z_{\Lambda}(\beta,\lambda)=\sum_{n=0}^{\infty}{\lambda^{n}\over
n!}\sum_{(x_{1},...,x_{n})\in\Lambda^{n}}e^{-\beta\sum_{i,j}V(x_{i},x_{j})}$
$None$

As far as lattice gas is concerned, it is natural to assume that at most one
particle can sit in each site $x$, hence we suppose that $V(x,x)=+\infty$. In
this case (6.11) can be written in the following way

$Z_{\Lambda}(\beta,\lambda)=\sum_{\sigma_{\Lambda}\in\\{0,1\\}^{|\Lambda|}}\lambda^{\sum_{x\in\Lambda}\sigma_{x}}e^{-\beta\sum_{\\{x,y\\}\subset\Lambda}\sigma_{x}\sigma_{y}V(x,y)}$
$None$

where $\sum_{\sigma_{\Lambda}\in\\{0,1\\}^{|\Lambda|}}$ means sum over all
configurations $\\{\sigma_{x}\\}$ with $x\in I$ and the variables $\sigma_{x}$
are such that $\sigma_{x}=1$ if the site is occupied by a particle and
$\sigma_{x}=0$ if it is empty.

It is easy to see that this model falls in the class of discrete systems
defined in (6.3) and (6.4). Namely we can write

$Z_{\Lambda}(\beta,\lambda)=(1+\lambda)^{|\Lambda|}\int
d\mu(\sigma_{\Lambda})e^{-\beta\sum_{\\{x,y\\}\subset\Lambda}\sigma_{x}\sigma_{y}V(x,y)}$

where $\Omega_{x}=\\{0,1\\}$, the single spin distribution is

$\mu(\sigma_{x}=0)={1\over
1+\lambda},~{}~{}~{}~{}~{}\mu(\sigma_{x}=1)={\lambda\over 1+\lambda}$

and the interactions $\Psi(X,\phi_{X})$ are defined only for $X=\\{x,y\\}$ and
have the form $\Psi(X,\phi_{X})=\sigma_{x}\sigma_{y}V(x,y)$.

The model can be easily generalized to a lattice gas with a many body
potential writing

$\Xi_{\Lambda}(\beta,\lambda)=\sum_{\sigma_{\Lambda}\in\\{0,1\\}^{|\Lambda|}}\lambda^{\sum_{x\in\Lambda}\sigma_{x}}e^{-\beta\sum_{X\subset\Lambda}\sigma_{X}V(X)}$
$None$

where $\sigma_{X}=\prod_{x\in X}\sigma_{x}$

4\. An example of discrete system in which the single spin state space is
continuous but compact is the so-called Heisemberg model. The system has the
same structure of the Ising model, but the spins are unit vector in
$\mathbb{R}^{d}$, i.e. $\sigma_{x}\in S_{1}^{d}$, where $S_{1}^{d}$ is the
unit sphere in $\mathbb{R}^{d}$. We will denote with $a\cdot b$ the scalar
product between the two vectors $a$ and $b$. Hence the Hamiltonian of the
system is given by

$H_{\Lambda}({\bf\sigma}_{\Lambda})~{}=~{}-J\sum_{\\{x,y\\}\subset\Lambda\atop
x,y~{}{\rm
nearest~{}neighbors}}\sigma_{x}\cdot\sigma_{y}-\sum_{x\in\Lambda}h\cdot\sigma_{x}$

where $h\in\mathbb{R}^{d}$ is an “external magnetic field”, $J$ represents
again the intensity of the nearest neighbors interaction, and the model is
said ferromagnetic when $J>0$ and antiferromagnetic otherwise. We are using
once again free boundary condition for sake of simplicity.

The partition function of the Heisemberg model is

$Z_{\Lambda}(\beta,h)~{}=~{}\int_{(S_{1}^{d})^{|\Lambda|}}\prod_{x\in\Lambda}d\sigma_{x}e^{-\beta
H_{\Lambda}({\bf\sigma}_{\Lambda})}$

The Heisemberg model falls in the class of the discrete systems defined above.
We can indeed write

$Z_{\Lambda}(\beta,h)~{}=~{}M^{|\Lambda|}\int
d\mu(\sigma_{x})\prod_{x\in\Lambda}d\sigma_{x}e^{-J\sum_{\\{x,y\\}\subset\Lambda\atop
x,y~{}{\rm nearest~{}neighbors}}\sigma_{x}\cdot\sigma_{y}}$

where

$d\mu(\sigma_{x})={d\sigma_{x}\ e^{\beta h\cdot\sigma_{x}}\over M}\qquad
M=\int_{S_{1}^{d}}d\sigma_{x}\ e^{\beta h\cdot\sigma_{x}}$

and $d\sigma_{x}$ denotes the Lebesgue measure on the surface of the sphere
$S_{1}^{d}$.

5\. As a final example we introduce here a discrete model in which the single
spin state space in non-compact. This model is very important, both in
statistical mechanics and in field theory.

We take as $\mathbb{V}$ the lattice $\mathbb{Z}^{d}$ equipped with the usual
Euclidean distance. Let $\Lambda\subset\mathbb{Z}^{d}$ be a cubic subset of
${\mathbb{Z}}^{d}$.

The spin variable $\phi_{x}$, called also field, takes values in
${\mathbb{R}}$. A configuration $\phi_{\Lambda}$ is a function
$\phi:\Lambda\to{\mathbb{R}}$.

We define the boundary conditions of the system fixing a value of the field
$\phi_{x}$ for all $x\in\mathbb{Z}^{d}\setminus\Lambda$ We consider the
discrete model described by the (Gibbs) measure

$\mu_{\Lambda}^{\omega}(\cdot)={1\over
Z(\Lambda,\omega)}\int\prod_{x\in\Lambda}d\phi_{x}e^{-H(\phi_{\Lambda},\omega)}(\cdot)$
$None$

where $d\phi_{x}$ is the Lebesgue measure in $\mathbb{R}$ and the partition
function $Z(\Lambda,\omega)$ is defined by

$Z(\Lambda,\omega)=\int\prod_{x\in\Lambda}d\phi_{x}e^{-H(\phi_{\Lambda},\omega)}$
$None$

The Hamiltonian of the system is

$H(\phi_{\Lambda},\omega)=\sum_{x\in\Lambda}U(\phi_{x})-\sum_{\\{x,y\\}\cap\Lambda\neq\emptyset}J_{xy}\phi_{x}\phi_{y}+\sum_{x\in\Lambda}h_{x}\phi_{x}=$
$=\sum_{x\in\Lambda}\left[U(\phi_{x})-\phi_{x}\omega_{x}\right]-\sum_{\\{x,y\\}\subset\Lambda}J_{xy}\phi_{x}\phi_{y}$
$None$

$U(x)$ is an even polynomial of degree $2k$, $k>1$, of the form

$U(x)=x^{2k}+\sum_{i=0}^{k-1}u_{2i}x^{2i}$ $None$

with $u_{2i}\in{\mathbb{R}}$; the pair potential $J_{xy}$ is such that

$\sup_{x\in{\bf Z}^{d}}\sum_{y\neq x}|J_{xy}|=J<+\infty$ $None$

$h_{x}\in\mathbb{R}$ represents the external field, and in the last line of
(6.16) we defined

$\omega_{x}=-h_{x}+\sum_{y\in\Lambda^{c}}J_{xy}\phi_{y}$ $None$

The boundary fields $\phi_{y}$ with $y\in\Lambda^{c}$ must be chosen in such
way that $\sum_{y\in\Lambda^{c}}J_{xy}\phi_{y}$ is finite for all $x$.

This model is again in the class defined above. We can write

$Z(\Lambda,\omega)=M^{|\Lambda|}\int\prod_{x\in\Lambda}d\mu(\phi_{x})e^{-\sum_{\\{x,y\\}\subset\Lambda}J_{xy}\phi_{x}\phi_{y}}$
$None$

The single spin distribution is in this case dependent form the site $x$:

$d\mu(\phi_{x})={d\phi_{x}e^{-U(\phi_{x})-\phi_{x}\omega_{x}}\over M}$

where

$M=\int d\phi_{x}e^{-U(\phi_{x})-\phi_{x}\omega_{x}}$

The system we just defined has a lot of very interesting features, and some of
them will be extensively discussed in details in section ??

####  6.2 Polymer gas representation of $Z_{\Lambda}(\beta)$

We need to recall here some definitions and notations about graphs and more
generally hypergraphs.

If $A$ is any finite set, we denote by $|A|$ the number of elements of $A$. An
hypergraph $g$ is a pair $g=(V,E)$ where $V$ is a finite or countable set and
$E$ is a collection of distinct subsets of $V$ with cardinality greater or
equal than 2. The elements $x\in V$ are called vertices of the hypergraph $g$
and $V$ is the set of vertices of $g$. The elements $e\in E$ are called edges
of the hypergraph and $E$ is the set of edges of $g$. A graph $g$ such that
$|e|=2$ for all $e\in E$ is called a graph. Let $g=(V,E)$ be a hypergraph, let
$U\subset V$ and let $E|_{U}=\\{e\in E:e\subset U\\}$. Then the graph
$g|_{U}=(U,E|_{U})$ is called the restriction of $G$ to $U$. A hypergraph
$g^{\prime}=(V^{\prime},E^{\prime})$ is a subgraph of $g=(V,E)$ if
$V^{\prime}\subset V$, and $E^{\prime}\subset E|_{V^{\prime}}$.

An hypergraph $g=(V_{g},E_{g})$ is said to be connected if for any pair $B,C$
of subsets of $V$ such that $B\cup C=V_{g}$ and $B\cap C=\emptyset$, there is
a $e\in E_{g}$ such that $e\cap B\neq\emptyset$ and $e\cap C\neq\emptyset$. A
connected component of an hypergraph $g=(V_{g},E_{g})$ is a a subset $U\subset
V$ such that $g|_{U}$ is connected and for any $U^{\prime}$ such that
$V\supseteq U^{\prime}\supset U$, $g|_{U^{\prime}}$ is not connected. Note
that a connected component of $g=(V_{g},E_{g})$ can be a singleton. The set of
all connected components of $g=(V_{g},E_{g})$ form a partition of $V_{g}$. We
denote by ${\cal G}_{V}$ the set of all hypergraphs with vertex set $V$ and by
${G}_{V}$ the set of all connected hypergraphs with vertex set $V$.

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

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

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

$Z_{\Lambda}(\beta)~{}=~{}\int
d\mu(\phi_{\Lambda})\prod_{X\subset\Lambda}\exp\left[-\beta\Psi(X,\phi_{X})\right]~{}=~{}$
$~{}=~{}\int
d\mu(\phi_{\Lambda})\prod_{X\subset\Lambda}[(e^{-\beta\Psi(X,\phi_{X})}-1)+1]~{}=~{}$

Now, posing $U_{X}=e^{-\beta\Psi(X,\phi_{X})}-1$ we see that

$\prod_{X\subset\Lambda}[U_{X}+1]=\sum_{g\in{\cal G}_{\Lambda}}\prod_{X\in
E_{g}}U_{X}$

(here we agree that the graph with no edges, i.e. $g$ such that with
$E_{g}=\emptyset$ yields the contribution 1 to the sum above) and hence we get

$Z_{\Lambda}(\beta)~{}=~{}\int d\mu(\phi_{\Lambda})\sum_{g\in{\cal
G}_{\Lambda}}\prod_{X\in E_{g}}(e^{-\beta\Psi(X,\phi_{X})}-1)$

Note again that the graph with empty edge set yields the contribution 1 in
equation above, since by definition $\int d\mu(\phi_{\Lambda})1~{}=~{}1$.

We now can group together connected components in $\sum_{g\in{\cal
G}_{\Lambda}}$ as follows. First note every graph $g\in{\cal G}_{\Lambda}$
determines a partition of $\Lambda$ into its connected components. Hence

$\sum_{g\in{\cal G}_{\Lambda}}\prod_{X\in
g}(e^{-\beta\Psi(X,\phi_{X})}-1)~{}=~{}\sum_{k=0}^{|\Lambda|}\sum_{\\{R_{1},\dots
R_{k}\\}\atop{\rm
partition~{}of~{}\Lambda}}\prod_{i~{}=~{}1}^{k}\sum_{g\in{{G}_{R_{i}}}}\prod_{X\in
E_{g}}(e^{-\beta\Psi(X,\phi_{X})}-1)$

where by convention the term corresponding to $k=0$ is set equal to 1 and
$\sum_{g\in{{G}_{R}}}\prod_{X\in g}(e^{-\beta\Psi(X,\phi_{X})}-1)=1$ if
$|R|=1$.

Hence we can also write

$Z_{\Lambda}(\beta)=\int
d\mu(\phi_{\Lambda})\sum_{k=0}^{|\Lambda|}\sum_{\\{R_{1},\dots
R_{k}\\}:~{}R_{i}\subset\Lambda\atop R_{i}\cap
R_{j}~{}=~{}\emptyset,~{}|R_{i}|\geq
2}\prod_{i~{}=~{}1}^{k}\sum_{g\in{{G}_{R_{i}}}}\prod_{X\in
g}(e^{-\beta\Psi(X,\phi_{X})}-1)$

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

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

with

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

and evidently

$\sum_{k=0}^{|\Lambda|}\sum_{\\{R_{1},\dots
R_{k}\\}:~{}R_{i}\subset\Lambda\atop R_{i}\cap
R_{j}~{}=~{}\emptyset,~{}|R_{i}|\geq
2}~{}~{}\rho_{\beta}(R_{1})\dots\rho_{\beta}(R_{k})=~{}~{}~{}~{}\sum_{k\geq
0}{1\over k!}\sum_{(R_{1},\dots,R_{k}):~{}R_{i}\subset\Lambda\atop R_{i}\cap
R_{j}=\emptyset,~{}|R_{i}|\geq
2}~{}~{}~{}~{}\rho_{\beta}(R_{1})\dots\rho_{\beta}(R_{k})$

where in the l.h.s we are summing over unordered $k$-uples $\\{R_{1},\dots
R_{k}\\}$ while in the r.h.s we are summing over ordered $k$-uples
$(R_{1},\dots R_{k})$.

So we have shown that the free boundary condition partition function
$Z_{\Lambda}(\beta)$ defined in (6.21) can always be written as

$Z_{\Lambda}(\beta)~{}=~{}\sum_{k\geq 0}{1\over
k!}\sum_{(R_{1},\dots,R_{k}):~{}R_{i}\subset\Lambda\atop R_{i}\cap
R_{j}~{}=~{}\emptyset,~{}|R_{i}|\geq
2}\rho_{\beta}(R_{1})\dots\rho_{\beta}(R_{k})$ $None$

It is now easy to convince one-self that expression above is actually the
grand canonical partition function of a gas of particle of different species
in which the particles (or “polymers”) $R$ are subsets of $\Lambda$ with
cardinality greater or equal 2, with activity $\rho(R)$. Note that the
activity is not the same for every polymer, i.e. different polymers have in
principle different activities. These polymers are interacting via a two-body
interaction which prevent the overlapping between different polymers. Indeed,
the only interaction between polymers is a repulsive hard core pair
interaction. Namely, if we define for any pair of polymers $R_{i},R_{j}$

$U(R_{i},R_{j})~{}=~{}\begin{cases}0&ifR_{i}\cap R_{j}~{}=~{}\emptyset\\\