High-Temperature Cluster Expansion for Classical and Quantum Spin Lattice Systems With Multi-Body Interactions

Abstract

We develop a novel cluster expansion for finite-spin lattice systems subject to multi-body quantum —and, in particular, classical— interactions. Our approach is based on the use of “decoupling parameters”, advocated by Park (J. Stat. Phys. 27, 553–576 (1982)), which relates partition functions with successive additional interaction terms. Our treatment, however, leads to an explicit expansion in a $\beta$-dependent effective fugacity that permits an explicit evaluation of free energy and correlation functions at small $\beta$. To determine its convergence region we adopt a relatively recent cluster summation scheme that replaces the traditional use of Kikwood-Salzburg-like integral equations by more precise sums in terms of particular tree-diagrams Bissacot et al. (J. Stat. Phys. 139, 598–617 (2010)). As an application we show that our lower bound of the radius of $\beta$-analyticity is larger than Park’s for quantum systems two-body interactions.

Appendices

Appendix A Gentle Introduction to KMS States and Araki-Ion’s Gibbs Condition

This topic is not easy to access to the uninitiated because available references (e.g., [1, 23, Section III.3], [3, Sections 5.3$-$5.4], [36, Sections IV.4–IV.5]) opt for general and elegant expositions that fail to reveal the naturalness of the underlying ideas.

1.1 KMS States

While classical Gibbs states can be defined through DLR equations (see e.g., [23, Section III.1], [14, Chapter 2], [12, Chapter 6]), quantum states are defined by their tracial properties. For inspiration, it is useful to start by the well known finite-volume states. The tracial characterization of the free —that is, no interacting— state is contained in the following equivalence.

Lemma 43

[36, Lemma IV.4.1] If $\Theta ^0_\Lambda$ is a normalized linear functional on ${\mathcal {A}}_\Lambda$ [that is, $\Theta ^0_\Lambda ({\textbf{1}})=1$], then

$$\begin{aligned} \Theta ^0_\Lambda (AB)=\Theta ^0_\Lambda (BA)\quad \Longleftrightarrow \quad \Theta ^0_\Lambda =\textrm{tr}_\Lambda \end{aligned}$$

(222)

where the right-hand side is the normalized trace defined in (18).

The presence of the operator $\textrm{e}^{-\beta H_\Lambda }$ changes the tracial properties of the Gibbs states $\pi ^\beta _\Lambda$ defined in (32). The right property emerges from the following calculation

$$\begin{aligned} \textrm{tr}_\Lambda \bigl (A B \,\textrm{e}^{-\beta H_\Lambda }\bigr )\;=\; \textrm{tr}_\Lambda \bigl (B \,\textrm{e}^{-\beta H_\Lambda }\,A\bigr )\;=\; \textrm{tr}_\Lambda \bigl (B \,\bigl [\textrm{e}^{-\beta H_\Lambda }\,A \,\textrm{e}^{\beta H_\Lambda }\,\bigr ] \,\textrm{e}^{-\beta H_\Lambda }\bigr ). \end{aligned}$$

(223)

Therefore,

$$\begin{aligned} \pi ^\beta _\Lambda (AB)\;=\; \pi ^\beta _\Lambda \bigl ( B\; \Delta ^{\beta \Phi }_\Lambda (A)\bigr ) \end{aligned}$$

(224)

with

$$\begin{aligned} \Delta ^{\beta \Phi }_\Lambda (A)\;=\; \textrm{e}^{-\beta H_\Lambda }\,A \,\textrm{e}^{\beta H_\Lambda }. \end{aligned}$$

(225)

The passage to infinite volume relies on the following theorem due, with increasing levels of generality, to Streater, Robinson and Ruelle (see references in the paragraph above the reference given for the theorem)

Theorem 44

[36, Theorem IV.3.3] If $\Phi \in {\mathbb {B}}^{(\alpha )}$ for some $\alpha >0$, then there exists a $\star$-automorphism $\Delta ^{\beta \Phi }$ on ${\mathcal {A}}$ such that

(226)

for each $A\in {\mathcal {A}}$.

Note that ${\mathbb {B}}^{(\alpha )}$ is the Banach space of interactions with finite $\Vert \Phi \Vert _\alpha$ defined in Subsection  2.2. The operator $\Delta ^{\beta \Phi }$ is defined by an expansion in iterated commutators. Here is the infinite-volume version of (224).

Definition 45

A state $\Theta$ on ${\mathcal {A}}$ is a KMS state for an interaction $\Phi \in {\mathbb {B}}^{(\alpha )}$, $\alpha >0$, if

$$\begin{aligned} \Theta (AB)\;=\; \Theta \bigl ( B\; \Delta ^{\beta \Phi } (A)\bigr ) \end{aligned}$$

(227)

for each $A,B\in {\mathcal {A}}$.

Given the continuity of states, (227) is equivalent to

$$\begin{aligned} \Theta (AB)\;=\; \lim _{\Lambda \rightarrow {\mathbb {L}}}\Theta \bigl ( B\; \Delta ^{\beta \Phi }_\Lambda (A)\bigr ). \end{aligned}$$

(228)

This expression may be more suitable for concrete arguments and computations, given the explicit character (225) of the finite-volume approximations. The proof that KMS becomes DLR for classical interactions can be found in [23, Section III.5].

1.2 Gibbs Condition

The previous notion of KMS states, even when it fully characterizes what a quantum statistical mechanical state should be, has two drawbacks. The first one is that, it does not lead to a direct proof that translation-invariant states coincide with the solutions of a variational approach. The second one is that, unlike DLR equations, it does not make reference to finite-volume objects and relations (other than that contained in the limit (228)). An alternative approach, due to Araki and Ion [1], solves both issues.

A natural way to mimic the classical case would be to consider all limits of states $\pi ^\Theta _\Lambda (\bullet )$ for all possible quantum boundary conditions $\Theta$. In the classical case, these limits attains all extremal Gibbs states, and hence they are sufficient to generate the full phase diagram. No similar result has been proved, however, for extremal KMS states, and the proposed approach is, instead, based on a factorization idea. The derivation of the DLR equations relies in the decomposition

$$\begin{aligned} \textrm{e}^{-\beta H_{\Lambda '}}\;=\; \textrm{e}^{-\beta H_{\Lambda }}\, \textrm{e}^{-\beta W_{\Lambda ,\Lambda '}} \,\textrm{e}^{-\beta H_{\Lambda '\setminus \Lambda }} \end{aligned}$$

(229)

for $\Lambda '\supset \Lambda$. Here

$$\begin{aligned} W_{\Lambda ,\Lambda '}\;=\; \sum _{X\in {\mathcal {P}}_{\partial \Lambda }\cap {\mathcal {P}}_{\Lambda '}} \Phi (X)\; \end{aligned}$$

(230)

The factorization (229) does not hold in the quantum case because the corresponding operators do not commute. The following rewriting, however, is valid also for quantum system

$$\begin{aligned} \textrm{e}^{-\beta (H_{\Lambda '}-W_{\Lambda ,\Lambda '})}\;=\; \textrm{e}^{-\beta H_{\Lambda }}\, \textrm{e}^{-\beta H_{\Lambda '\setminus \Lambda }} \end{aligned}$$

(231)

because operators with disjoint support commute. To make sense of this factorizationn when $\Lambda '\rightarrow {\mathbb {L}}$, each exponential must be associated to a homomorphism $\Delta ^\beta$ and the resulting decomposition should be interpreted in the following way: If for each finite $\Lambda$ we “perturb” a KMS state $\Theta$ so to “substract” $W_\Lambda$, the resulting state becomes the product of a $\Delta ^{\beta \Phi }_\Lambda$-KMS state —namely $\pi ^\beta _\Lambda$ — and a KMS state for $\Delta ^{\beta \Phi }_{{\mathbb {L}}\setminus \Lambda }$. Denoting $\Theta ^{-W_\Lambda }$ the proposed perturbation, (231) transcribes into the identity

$$\begin{aligned} \Theta ^{-W_\Lambda }\;=\; \pi _\Lambda ^\beta \otimes {\widetilde{\Theta }}_{{\mathbb {L}}\setminus \Lambda } \end{aligned}$$

(232)

with ${\widetilde{\Theta }}_{{\mathbb {L}}\setminus \Lambda }$ a state on ${\mathcal {A}}_{{\mathbb {L}}\setminus \Lambda }$. To define the perturbed state $\Theta ^{-W_\Lambda }$, let us again look first on finite volumes. In this case, to substract $W_{\Lambda ,\Lambda '}$ in the state $\pi ^\beta _{\Lambda '}$ means to pass to

$$\begin{aligned} \pi ^{-W_{\Lambda ,\Lambda '}}_{\Lambda '}(A)\;=\; \frac{\textrm{tr}_{\Lambda '} \bigl (A \,\textrm{e}^{-\beta H_\Lambda }\,\textrm{e}^{-\beta H_{\Lambda '\setminus \Lambda }}\bigr )}{\textrm{tr}_{\Lambda '} \bigl (\textrm{e}^{-\beta H_\Lambda }\,\textrm{e}^{-\beta H_{\Lambda '\setminus \Lambda }}\bigr )} \;=\; \bigl [\pi _\Lambda ^\beta \otimes \pi ^\beta _{\Lambda '\setminus \Lambda }\bigr ](A). \end{aligned}$$

(233)

Such a state can be constructed from $\pi _{\Lambda '}^\beta$ by changing the exponential weight according to the trivial identity

$$\begin{aligned} \textrm{e}^{-\beta (H_{\Lambda '}-W_{\Lambda ,\Lambda '})}\;=\; \bigl [\textrm{e}^{-\beta (H_{\Lambda '}-W_{\Lambda ,\Lambda '})}\; \textrm{e}^{\beta H_{\Lambda '}}\bigr ]\,\textrm{e}^{-\beta H_{\Lambda '}}\;=:\; \Gamma _\Lambda ^{\beta ,\Lambda '} \,\textrm{e}^{-\beta H_{\Lambda '}}. \end{aligned}$$

(234)

Therefore, the removal of the term $W_{\Lambda ,\Lambda '}$ yields a perturbed state

$$\begin{aligned} \pi ^{-W_{\Lambda ,\Lambda '}}_{\Lambda '}(A)\;=\; \frac{\pi ^\beta _{\Lambda '}\bigl (A\,\Gamma _\Lambda ^{\beta ,\Lambda '} \bigr )}{\pi ^\beta _{\Lambda '}\bigl (\Gamma _\Lambda ^{\beta ,\Lambda '}\bigr ) }. \end{aligned}$$

(235)

A time-ordered expansion (see, e.g., the proof of Theorem IV.5.5 in [36]) shows that the operators $\Gamma _\Lambda ^{\beta ,\Lambda '}$ converge, as $\Lambda ' \rightarrow {\mathbb {L}}$, to a bounded operator

$$\begin{aligned} \Gamma ^\beta _\Lambda \;=\; \lim _{\Lambda '\rightarrow {\mathbb {L}}} \textrm{e}^{-\beta (H_{\Lambda '}-W_{\Lambda ,\Lambda '})}\; \textrm{e}^{\beta H_{\Lambda '}} \end{aligned}$$

(236)

which is used to define the infinite-volume analogue of (235)

Definition 46

The $-W_\Lambda$-perturbation of a state $\Theta$ is the state such for each $A\in {\mathcal {A}}$

$$\begin{aligned} \Theta ^{-W_\Lambda }(A)\;=\; \frac{\Theta \bigl (A\,\Gamma ^\beta _\Lambda \bigr )}{\Theta \bigl (\Gamma ^\beta _\Lambda \bigr )} \;=\; \lim _{\Lambda '\rightarrow {\mathbb {L}}} \frac{\Theta \bigl (A\,\textrm{e}^{-\beta (H_{\Lambda '}-W_{\Lambda ,\Lambda '})}\; \textrm{e}^{\beta H_{\Lambda '}}\bigr )}{\Theta \bigl (\textrm{e}^{-\beta (H_{\Lambda '}-W_{\Lambda ,\Lambda '}\bigr )}\; \textrm{e}^{\beta H_{\Lambda '}}\bigr )} \end{aligned}$$

(237)

It is relatively simple to verify that if $\Theta$ is a $\Delta ^{\beta \Phi }$-KMS state, then $\Theta ^{-W_\Lambda }$ is a $\bigl (\Delta ^{\beta \Phi }_\Lambda \,\Delta ^{\beta \Phi }_{{\mathbb {L}}\setminus \Lambda }\bigr )$-KMS state.

Definition 47

A state $\Theta$ on ${\mathcal {A}}$ is called a Gibbs state for an interaction $\Phi \in {\mathbb {B}}^{(\alpha )}$, for some $\alpha >0$, if for each finite $\Lambda$ there exists a state ${\widetilde{\Theta }}_{{\mathbb {L}}\setminus \Lambda }$ on ${\mathcal {A}}_{{\mathbb {L}}\setminus \Lambda }$ such that (232) holds.

A more constructive version of this definition is obtained by “perturbing back” the factorized state into the original $\Theta$ through the inverse of the first identity in (237):

$$\begin{aligned} \Theta (A)\;=\; \frac{\Theta ^{-W_\Lambda }\bigl (A\,\bigl [\Gamma ^\beta _\Lambda \bigr ]^{-1}\bigr )}{\Theta ^{-W_\Lambda }\bigl (\bigl [\Gamma ^\beta _\Lambda \bigr ]^{-1}\bigr )} \end{aligned}$$

(238)

with

$$\begin{aligned} \bigl [\Gamma ^\beta _\Lambda \bigr ]^{-1}\;=\; \lim _{\Lambda '\rightarrow {\mathbb {L}}} \textrm{e}^{-\beta H_{\Lambda '}}\,\textrm{e}^{\beta (H_{\Lambda '}-W_{\Lambda ,\Lambda '})}\; \end{aligned}$$

(239)

In this way, we obtain the following, more cumbersome but constructive definition (which is actually the original definition given by Araki and Ion in their seminal work [1]).

Proposition 48

(Alternative definition of Gibbs states) A state $\Theta$ on ${\mathcal {A}}$ is a Gibbs state for an interaction $\Phi \in {\mathbb {B}}^{(\alpha )}$, for some $\alpha >0$, if and only if, for each finite $\Lambda$ there exists a state ${\widetilde{\Theta }}_{{\mathbb {L}}\setminus \Lambda }$ on ${\mathcal {A}}_{{\mathbb {L}}\setminus \Lambda }$ such that

$$\begin{aligned} \Theta (A)= & {} \frac{\pi _\Lambda ^\beta \otimes {\widetilde{\Theta }}_{{\mathbb {L}}\setminus \Lambda }\bigl (A\,\bigl [\Gamma ^\beta _\Lambda \bigr ]^{-1}\bigr )}{\pi _\Lambda ^\beta \otimes {\widetilde{\Theta }}_{{\mathbb {L}}\setminus \Lambda }\bigl (\bigl [\Gamma ^\beta _\Lambda \bigr ]^{-1}\bigr )} \end{aligned}$$

(240)

$$\begin{aligned}= & {} \lim _{\Lambda '\rightarrow {\mathbb {L}}} \frac{\pi _\Lambda ^\beta \otimes {\widetilde{\Theta }}_{{\mathbb {L}}\setminus \Lambda }\bigl (A\,\textrm{e}^{-\beta H_{\Lambda '}}\,\textrm{e}^{\beta (H_{\Lambda '}-W_{\Lambda ,\Lambda '})}\bigr )}{\pi _\Lambda ^\beta \otimes {\widetilde{\Theta }}_{{\mathbb {L}}\setminus \Lambda }\bigl (\textrm{e}^{-\beta H_{\Lambda '}}\,\textrm{e}^{\beta (H_{\Lambda '}-W_{\Lambda ,\Lambda '})}\bigr )} \end{aligned}$$

(241)

The second equality is the working definition adopted in Sect. 2.4 above. The proof that the sets of KMS and Gibbs states coincide can be found, for instance, in [1, 3, Sections 5.4] and [36, Sections IV.5].

Appendix B Disguises of the Möbius Transform

The Möbius transform is often surreptitiously introduced through very elegant alternative treatments. Let us spell out the two more popular ones.

1.1 Park’s $\delta$’s and $\varepsilon$’s

Park [29] rewrites the right-hand side of (42) in terms of operators $\delta ^x$ which amounts to a discrete derivative. To prove the inverse formula in the left-hand side of this expression, he introduces a sort of inverse operators $\varepsilon ^x$. Here is a transcription of his approach (our $\varepsilon$ is slightly different from Park’s).

Proposition 49

Consider the following operators on the vector space of functions $F: \{\text{ parts } \text{ of } {\mathcal {S}}\} \rightarrow {\mathbb {V}}$ where ${\mathcal {S}}$ is a finite set and ${\mathbb {V}}$ a vector space. Define the following operators for each $x\in \Lambda$:

$$\begin{aligned} \delta ^x F(A)= & {} F\bigl (A\cup \{x\}\bigr ) - F(A) \end{aligned}$$

(242)

$$\begin{aligned} \varepsilon ^x F(A)= & {} F\bigl (A\cup \{x\}\bigr ) + F(A) \end{aligned}$$

(243)

The following properties are easily verified:

• (a) These operators commute between themselves.

• (b) $\displaystyle \bigl (1+\delta ^x\bigr )F(A) = F\bigl (A\cup \{x\}\bigr ) = \bigl (\varepsilon ^x-1\bigr )F(A)$

[As a matter of fact, Park’s definition of $\delta ^x F(A)$ is $F\bigl (A\cup \{x\}\bigr ) - F\bigl (A\setminus \{x\}\bigr )$. Hence, it coincide with ours each time that $x\not \in A$, as in the application below.]

The commutativity properties justifies the definition of $\delta ^S:=\prod _{x\in S} \delta ^x$ and $\varepsilon ^S:=\prod _{x\in S} \varepsilon ^x$, for each $S\subset {\mathcal {S}}$. In particular, part (b) of the proposition implies

$$\begin{aligned} \bigl (1+\delta \bigr )^SF(\emptyset ) = F(S) = \bigl (\varepsilon -1\bigr )^SF(\emptyset ). \end{aligned}$$

(244)

The interest of these operators stems from the following lemma who follows from a two-line induction argument.

Lemma 50

For each function F and sets $B\subset {\mathcal {S}}$

$$\begin{aligned} \delta ^BF(\emptyset )= & {} \sum _{A\subset B} (-1)^{\left| B\setminus A \right| } F(A) \end{aligned}$$

(245)

$$\begin{aligned} \varepsilon ^BF(\emptyset )= & {} \sum _{A\subset B} F(A). \end{aligned}$$

(246)

Identities (244)–(246) readily imply the inclusion–exclusion relation.

Theorem 51

(Möbius transform) For each set $S\subset \Lambda$,

• (M1) $\displaystyle G(S)=\delta ^S F(\emptyset )\;\Longrightarrow \; \varepsilon ^S G(\emptyset )=F(S)$ .

• (M2) $\displaystyle F(S)=\varepsilon ^S G(\emptyset )\;\Longrightarrow \; \delta ^S F(\emptyset )=G(S)$ .

Proof

[(M1)]

$$\begin{aligned} \varepsilon ^S G(\emptyset )\;=\; \sum _{A\subset S} G(A)\;=\; \sum _{A\subset S} \delta ^A F(\emptyset ) \;=\; \bigl (1+\delta \bigr )^S F(\emptyset )\;=\;F(S). \end{aligned}$$

[(M2)]

$$\begin{aligned} \delta ^S F(\emptyset )\;=\; \sum _{A\subset S} (-1)^{\left| S\setminus A \right| } F(A)\;=\; \sum _{A\subset S} (-1)^{\left| S\setminus A \right| } \varepsilon ^A G(\emptyset ) \;=\; \bigl (\varepsilon -1\bigr )^S G(\emptyset )\;=\;G(S). \end{aligned}$$

$\square$

1.2 Use of Decoupling Parameters

This is a time-honored technique to represent the right-hand side of (42). In our general setup it involves the introduction of a parameter $s_x\in [0,1]$ associated to each point of ${\mathcal {S}}$. A point x is thought as decoupled if $s_x=0$ and fully coupled if $s_x=1$. The association is made in reference to a given function F on subsets and it is defined so to establish associated function ${{\widehat{F}}}:[0,1]^{{\mathcal {S}}} \longrightarrow {\mathbb {V}}$ such that

• (i) The function ${\underline{s}} \rightarrow {{\widehat{F}}}({\underline{s}})$ is smooth (at least with first-order partial derivatives).

• (ii) For each $A\subset {\mathcal {S}}$, $F(A) \;=\; {{\widehat{F}}}\bigl ({\textbf{1}}_A 0_{{{\mathcal {S}}}\setminus A}\bigr )$.

The example of interest in Park’s work is for ${{\mathcal {S}}}= {\mathcal {B}}_\Lambda$ and a function ${{\widehat{F}}}({\mathcal {B}})=Z_{{\mathcal {B}}}^\Lambda$. The “partially decoupled” associated function is ${{\widehat{F}}}({\underline{s}})=Z_{\Lambda }^{s\Phi }$ in which is term $\Phi (X)$ is multiplied by a factor $s_X$ that interpolates between decoupling and full coupling.

These parameters allow to express the relation $G(S)=\delta ^S F$ as an application of the fundamental theorem of calculus.

Proposition 52

$$\begin{aligned} \sum _{A\subset B} (-1)^{\left| B\setminus A \right| } {{\widehat{F}}}\bigl ({\textbf{1}}_{B} 0_{{{\mathcal {S}}}\setminus A}\bigr ) \;=\; \left\{ \prod _{x\in B}\int _0^1 ds_x \frac{\partial }{\partial s_x}\right\} {{\widehat{F}}}({\underline{0}}). \end{aligned}$$

(247)

Proof

By the Fundamental Theorem of Calculus,

$$\begin{aligned} \int _0^1 ds_x \frac{\partial }{\partial s_x} {{\widehat{F}}}({\underline{s}})\;=\; {{\widehat{F}}}\bigl (1_x \, s_{{{\mathcal {S}}}\setminus \{x\}}\bigr ) - {{\widehat{F}}}\bigl (0_x s_{{{\mathcal {S}}}\setminus \{x\}}\bigr ). \end{aligned}$$

(248)

This operation is, therefore, equivalent to the operator $\delta ^x$. The proof then follows easily by induction on $\left| B \right|$. $\square$

Appendix C Tree Expansions and Their Summability

In this section we summarize the key ingredients behind the summability criteria of Theorem 19 and Proposition 26.

1.1 Penrose Partition

Here are the steps, proposed by Penrose, to associate to each graph ${\mathbb {G}}$ a unique tree ${\mathcal {T}}({\mathbb {G}})$ . The starting step is to fix an order of the vertices $v_i$ of ${\mathbb {G}}$.

Generation 0:

Choose $v_0$ as root and consider the resulting graph distance (= minimal number of links needed to attain the root) of the remaining vertices $v_i$. Let ${\mathcal {K}}_k$ denote the set of vertices at graph distance k from the root. The tree ${\mathcal {T}}$ keeps the vertices at their respective distances, it only removes some of the links in ${\mathbb {G}}$.

Generation 1:

Label the different vertices in ${\mathcal {K}}_1$ in the form $v_{(1,i_1)}$ and keep the corresponding edges $(v_0,v_{(1,i)})$. The labels (1, i) are chosen in increasing vertex order. All edges in ${\mathbb {G}}$ linking siblings, that is bonds of the form $(v_{(1,i_1)},v_{(1,j)})$— are omitted.

Generation ${\varvec{k}}$ $\varvec{ (2\le k\le n)}$:

After determining the tree up to the $(k-1)$-th generation, the vertices in ${\mathcal {K}}_{k-1}$ are labelled in the form $v_{(1,i_1,\ldots ,i_{k-1})}$. We start by the smallest such vertex –which is labelled by the sequence ${\varvec{i}}_{k-1}:=(1,i_1,\ldots ,i_{k-1})$ which is smallest in lexicographic order— and keep all the links between it and vertices in ${\mathcal {K}}_k$. Label then in the form $({\varvec{i}}_{k-1},i_k)$ with $i_k$ respecting vertex order. Then continue with the second vertex in ${\mathcal {K}}_{k-1}$, including all its links with the remaining vertices in ${\mathcal {K}}_k$. Proceed in this fashion through the successive vertices in ${\mathcal {K}}_{k-1}$ following the order of increasing subscripts. Omit all the remaining edges in ${\mathbb {G}}$ linking vertices in ${\mathcal {K}}_k$ with vertices in ${\mathcal {K}}_{k-1}$ or between vertices in ${\mathcal {K}}_k$.

1.2 Tree Summability

The resulting Penrose trees are planar trees uniquely characterized by the absence of links between (i) vertices at the same graph distance of the root $X_0$, and (ii) a vertex labelled ${\varvec{i}}_{k-1}$ with a vertex labelled $({\varvec{i}}'_{k-1},i_k)$ with ${\varvec{i}}_{k-1}<{\varvec{i}}'_{k-1}$.

Both in the proof of Theorem 19 and in Proposition 26 we have a graph ${\mathbb {G}}$ with a countable number of vertices and a family of labels $X\in {\mathcal {P}}$. Let us denote ${{\mathbb {G}}}(X_0,X_1,\ldots , X_n)$ the restriction to an n-vertex labelled graph with $X_i$ being the label of the vertex $v_i$. The issue is to bound sums of the form

$$\begin{aligned} T(X_0)\;=\; 1+ \sum _{n\ge 1} \sum _{\begin{array}{c} \{X_1,\ldots , X_n\}\\ X_i\in {\mathcal {P}}, X_i\ne X_0 \end{array}} F\bigl [{\mathcal {T}}\bigl ({{\mathbb {G}}}(X_0,X_1,\ldots , X_n)\bigr )\bigr ] \prod _{i=1}^n W(X_i), \end{aligned}$$

(249)

with F such that the omission of restriction (ii) above leads to a vertex-factorized upper bound $T(X_0)\le {{\widehat{T}}}(X_0)$ with

$$\begin{aligned} {{\widehat{T}}}(X_0)\;=\; 1+ \sum _{n\ge 1} \sum _{\tau \in {\mathbb {T}}_n} \sum _{\begin{array}{c} \{X_1,\ldots , X_n\}\\ X_i\in {\mathcal {B}}_{\mathbb {L}}, X_i\ne X_0 \end{array}} \prod _{i=0}^n c_{s_i}\bigl (X_i,X_{(i,1)},\ldots , X_{(i,s_i)}\bigr )\prod _{j=1}^n W(X_j) \end{aligned}$$

(250)

with $c_n$ an appropriate function (see examples below). Here ${\mathbb {T}}_n$ is the family of (unlabelled) planar trees with n vertices, $s_i$ the number of children of the vertex i and $(i,1),\ldots , (i,s_i)$ the corresponding children.

The following is the canonical way of summing tree expansions of the form (250) with positive weights. As trees with $n+1$ generations are inductively generated by placing n-generation trees on the leaves of first-generation trees, it is necessary and sufficient to bound the latter to generate an upper bound that guarantees convergence of the positive series. This bound requires the existence of positive weights $\zeta$ such that for each X, the X-rooted first-generation trees corresponding to the right-hand side of (101),

$$\begin{aligned} \varphi _X\;=\; 1+ \sum _{s\ge 1}\sum _{\begin{array}{c} \{X_1,\ldots , X_n\}\\ X_i\in {\mathcal {B}}_{\mathbb {L}}, X_i\ne X \end{array}} c_{s}\bigl (X,X_1,\ldots , X_s\bigr )\prod _{j=1}^n \zeta (X_j) \end{aligned}$$

(251)

satisfy

$$\begin{aligned} W(X)\,\varphi _X\;\le \; \zeta (X). \end{aligned}$$

(252)

Working iteratively in different generations, this bound extends then to the inequality $W(X)\, T(X)\le \zeta (X)$. In this way, we obtain the following fundamental result.

Theorem 53

For weights $\zeta$ satisfying (252) the following is true

• (a) The expansions (250) —and hence (249)— converge absolutely and uniformly in $\Lambda$ in the polydisc $\left| W(Y) \right| \le \zeta _Y$, $Y\in {\mathcal {P}}$.

• (b) For each $\varvec{\lambda }\in [0,\infty )^{\mathcal {P}}$ let ${\varvec{T}}_{\varvec{\lambda }}(\varvec{\mu })=\varvec{\lambda }\varvec{\varphi }(\varvec{\mu })$ be the map from $[0,+\infty )^{{\mathcal {P}}}$ to $[0,+\infty ]^{{\mathcal {P}}}$ defined by

  $$\begin{aligned} \bigl [{\varvec{T}}_{\varvec{\lambda }}(\varvec{\mu })\bigr ]_Y\;=\;\lambda _Y\,\varvec{\varphi }_Y(\varvec{\mu }) \end{aligned}$$

  (253)

  for each $Y\in {\mathcal {P}}$. Then

  • (i) There exist $\varvec{\lambda }^*, \,\varvec{T}_{\varvec{\lambda }}^{\infty }(\varvec{\mu })\in [0,+\infty )^{{\mathcal {P}}}$ such that

    $$\begin{aligned} \varvec{T}^n_{\varvec{\lambda }}(\varvec{\lambda })\underset{n\rightarrow \infty }{\nearrow }\varvec{\lambda }^*,\,\varvec{T}^n_{\varvec{\lambda }}(\varvec{\lambda })\underset{n\rightarrow \infty }{\searrow }\varvec{T}_{\varvec{\lambda }}^{\infty }(\varvec{\mu }) \end{aligned}$$

    (254)

    and $\varvec{T}_{\varvec{\lambda }}(\varvec{\lambda }^*)=\varvec{\lambda }^*$.

  • (ii) $\left| \varvec{\Sigma } \right| =\lim _\Lambda \left| \varvec{\Sigma } \right| ^\Lambda$ exists and satisfies, for each $n\in {\mathbb {N}}$,

    $$\begin{aligned} \varvec{\lambda }\left| \varvec{\Sigma } \right| \!(\varvec{\lambda })\;\le \; \varvec{\lambda }^*\;\le \; \varvec{T}^{\infty }_{\varvec{\lambda }}(\varvec{\mu })\;\le \; \varvec{T}^{n+1}_{\varvec{\lambda }}(\varvec{\mu })\le \varvec{T}^{n}_{\varvec{\lambda }}(\varvec{\mu })\;\le \; \varvec{\mu }. \end{aligned}$$

    (255)

See the comments below Theorem 19 for proofs of this result. Most proofs —an exception is the original proof in [10]— are focused on the particular case (256), but the arguments are general.

1.3 Sketch of the Proof of Theorem 19

The proof has two steps

• (S1) Use Penrose decomposition to bound (67) by a sum of the form (250) with

  $$\begin{aligned} c_{s_i}\bigl (X_i,X_{(i,1)},\ldots , X_{(i,s_i)}\;=\; \prod _{j=1}^n\mathbb {1}_{\left\{ X_{(i,j)}\not \sim X_i \right\} } \prod _{1\le k<\ell \le n}\mathbb {1}_{\left\{ X_{(i,k)} \sim X_{(i,\ell )}\right\} }. \end{aligned}$$

  (256)

• (S2) Apply Theorem 53.

1.4 Proof of Proposition 27

Again, two steps:

• (S1) Apply Theorem 53 to the expression given in Proposition 26.

• (S2) Use that

  $$\begin{aligned} {{\widehat{T}}}_1(X)\;=\;1 + \sum _{n\ge 1} \sum _{\begin{array}{c} \{X_1,\ldots , X_n\}\\ X_i\in {\mathcal {B}}_{\mathbb {L}}, X_i\ne X\\ X_i\cap X\ne \emptyset \end{array} } \prod _{j=1}^n \zeta (X_j) \;=\; \prod _{\begin{array}{c} Y\in {\mathcal {B}}_{\mathbb {L}},\,Y\ne X\\ Y\cap X\ne \emptyset \end{array}}\bigl [1+\zeta (Y)\bigr ]. \end{aligned}$$

  (257)

Appendix D Comparison with Park’s Results

In this subsection, we compare our estimations with the results provided by Y. M. Park in [29]. To describe the latter, let us denote

$$\begin{aligned} b_{\textrm{p}}:= & {} \sup _{x\in {\mathbb {Z}}^d}\sum _{X\ni x}\Vert \Phi (X)\Vert \ge \Vert \Phi (X)\Vert _{\infty } \end{aligned}$$

(258)

$$\begin{aligned} c_{\textrm{p}}:= & {} \sup _{x\in {\mathbb {Z}}^d}\sum _{\begin{array}{c} x\in X\subset {\mathbb {Z}}^{d}\\ X\,\textrm{finite} \end{array}}\Vert \Phi (X)\Vert \textrm{e}^{(\alpha /2)\,|X|} \end{aligned}$$

(259)

$$\begin{aligned} \Vert \Phi (X)\Vert _{\infty }:= & {} \sup _{n}\sup _{\begin{array}{c} x\in X\subset {\mathbb {Z}}^d\\ |X|=n \end{array}}\Vert \Phi (X)\Vert \end{aligned}$$

(260)

$$\begin{aligned} \Vert \Phi (X)\Vert _{\alpha /2}:= & {} \sup _n\sup _{x\in {\mathbb {Z}}^d}\sum _{\begin{array}{c} x\in X\subset {\mathbb {Z}}^d\\ |X|=n \end{array}}\Vert \Phi (X)\Vert \textrm{e}^{\alpha /2|X|} \end{aligned}$$

(261)

Park uses Kirkwood-Salzburg equations, rather than full-fledge cluster expansions; as a consequence, he does not address analyticity of free energies but of correlation functions. Park’s domain of $\beta$-analyticity takes the form

$$\begin{aligned} {\mathcal {D}}_{\textrm{p}}:=\left\{ \beta \in {\mathbb {C}}:|\beta |< \frac{\alpha }{4},\;A(|\beta |)\textrm{e}^{-\frac{\alpha }{4}+2b|\beta |}<1\right\} \end{aligned}$$

where

$$\begin{aligned} A(|\beta |)=1+c_1|\beta |\frac{\textrm{e}^{\frac{-\alpha }{4}+b_{\textrm{p}}|\beta |}}{1-\textrm{e}^{\frac{-\alpha }{4}+b_{\textrm{p}}|\beta |}}. \end{aligned}$$

Let us denote

$$\begin{aligned} F(\beta ):= \,A(\beta )\textrm{e}^{-\alpha /4+2\,b_{\textrm{p}}|\beta |}-1 \end{aligned}$$

Elementary analysis show that $F(\beta )$ is strictly increasing function, so F has a unique root $\beta ^*\in (0,\alpha /(b_{\textrm{p}}8))$. As a consequence,

$$\begin{aligned} c_1|\beta ^*|\textrm{e}^{b_{\textrm{p}}|\beta ^*|}= & {} \left[ \textrm{e}^{\alpha /4-2\,b_{\textrm{p}}|\beta ^*|}-1\right] \left[ \textrm{e}^{\alpha /4}-\textrm{e}^{b_{\textrm{p}}|\beta ^*|}\right] . \end{aligned}$$

(262)

For the sake of simplicity, we focus on lattice systems having only nearest-neighbor pair-interactions. Without loss of generality we assume that $\Vert \Phi (X)\Vert _{\infty }=1$, so $b_{\textrm{p}}=2d$ and $c_1>c_{\textrm{p}}=2d\textrm{e}^{\alpha }$. Replacing $c_1$ by $c_{\textrm{p}}$ we obtain that the solution of the equation

$$\begin{aligned} 2d\textrm{e}^{\alpha }|\beta ^*|\textrm{e}^{2d|\beta ^*|}= & {} \left[ \textrm{e}^{\alpha /4-2d|\beta |^*}-1\right] \left[ \textrm{e}^{\alpha /4}-\textrm{e}^{2d|\beta ^*|}\right] \end{aligned}$$

(263)

yields a lower bound on the radius of convergence.

Figure 1 shows the dependence of $y=2d|\beta |$ on $x=\textrm{e}^{\alpha /4}$.

Fig. 1

The graphic of the solution $2d\beta$

Note that (263) implies that $\beta \rightarrow 0$ when $|\alpha |\rightarrow +\infty$. From Figure 1, we see that $2d\beta ^*<0.06$. Hence,

$$\begin{aligned} \Vert \Phi \Vert _{\infty }|\beta |\textrm{e}^{\Vert \Phi \Vert _{\infty }|\beta |}\approx \frac{0.06}{2d}\left( 1+\frac{0.06}{2d}\right) <\frac{0.0921428}{2d}. \end{aligned}$$

The last result is our bound obtained in previous subsection.