Polymer analyticity and dilute continuum correlations

Fix $n\ge 2$ and a tuple $({\gamma }_1,\dots ,{\gamma }_n)$, and let $H$ be the simple graph on $[n]$ whose edge $\{i,j\}$ is present exactly when ${\gamma }_i\sim {\gamma }_j$. By [F1], $U_n={\sum }_{G\subseteq H,G\text{ connected}}(-1{)}^{\mathrm{\mid }E(G)\mathrm{\mid }}$. Here the complete graph on $[n]$ has edge set $\left(\genfrac{}{}{0px}{}{[n]}{2}\right)$, and $G\subseteq H$ means that $G$ has the same vertex set and $E(G)\subseteq E(H)$. Represent each edge $\{i,j\}$ with $i<j$ by $(i,j)$ and order these pairs lexicographically, first by $i$ and then by $j$; list the edges as $e_1{<}_{\mathrm{l}\mathrm{e}\mathrm{x}}\cdots {<}_{\mathrm{l}\mathrm{e}\mathrm{x}}{e}_m$ and assign rank $\rho (e_k)=k\cdot 1_{\mathbb{R}}$. For a tree $T$, its rank sum is ${\sum }_{e\in E(T)}\rho (e)$. By [F8], each connected graph $G$ has a spanning tree; by [F9], its spanning trees form a finite nonempty set, so their finite real rank sums have a minimum. A tree has a unique path between any two vertices: connectedness gives a path, and two distinct paths would contain a cycle between their first divergence and next common vertex, contradicting acyclicity. Here is the cut and exchange argument used below. Let $f=\{u,v\}$ be an edge of a tree $T$. There is no $u$-$v$ path in $T-f$, since such a path together with $f$ would be a cycle in $T$. For any vertex $w$, the unique $T$-path from $u$ to $w$ either avoids $f$, or crosses $f$ from $u$ to $v$ and then has a suffix from $v$ to $w$ avoiding $f$. Thus every vertex is reachable from $u$ or from $v$ in $T-f$, but not both: reachability from both would concatenate to a $u$-$v$ walk avoiding $f$, and deleting repeated-vertex portions of a finite walk gives a path, contradicting the preceding observation. Vertices reachable from the same endpoint are joined by concatenating their endpoint paths and deleting repetitions. Hence $T-f$ has exactly two connected components, with vertex sets $C_u,C_v$, and no remaining tree edge crosses this partition; call it the cut of $f$. If $e=\{x,y\}\mathrm{\notin }E(T)$ and $f$ lies on its unique $T$-path, then $x,y$ lie in opposite components, for otherwise a path in $T-f$ would contradict uniqueness of the $T$-path. Adding $e$ to $T-f$ joins the two components, so $T-f+e$ is connected. It is acyclic: a cycle would have to use $e$ since $T-f$ is acyclic, and removing $e$ from that cycle would give a path in $T-f$ between opposite components. Thus $T-f+e$ is a spanning tree on the same vertex set. Every path from one side of any such cut to the other has a crossing edge, by considering the first vertex at which its finite vertex list changes sides. The minimum-rank-sum spanning tree is unique: if distinct minimizing trees $T,S$ existed, take the least-ranked edge $e$ in their symmetric difference and, after swapping their names if needed, take $e\in T\setminus S$. Deleting $e$ from $T$ gives a cut crossed by the $S$-path between the endpoints of $e$; a crossing edge $f$ on that path lies in $S\setminus T$, and replacing $f$ by the earlier edge $e$ lowers the rank sum of $S$ by [F10], a contradiction. For a tree $T$, let $A(T)\subseteq \left(\genfrac{}{}{0px}{}{[n]}{2}\right)\setminus E(T)$ consist of the edges later than every edge on their unique $T$-path. If $G$ contains $T$ and a non-tree edge $e$ is earlier than a path edge $f$, then $T-f+e$ is a spanning tree of lower rank sum by [F10], so $T$ is not minimum. Conversely, if every non-tree edge of $G$ is later than every edge on its $T$-path, then for each $f\in T$ every other edge of $G$ crossing the cut made by deleting $f$ is later than $f$. Choose a minimum-rank-sum spanning tree $S$ of $G$, which exists by [F8]–[F9]. If $S$ omitted $f$, the $S$-path between the endpoints of $f$ crosses this cut at an edge $e$; adding $f$ to $S$ and deleting that path edge $e$ gives a spanning tree with lower rank sum by [F10], contradicting minimality of $S$. Hence $S$ contains every edge $f$ of $T$ and equals $T$: any additional edge would form a cycle with its $T$-path. Thus $T$ is the unique minimum tree of $G$ exactly when $T\subseteq G\subseteq T\cup A(T)$. Consequently, connected subgraphs of $H$ split into disjoint classes indexed by trees $T\subseteq H$, each class consisting of $T\cup A$ for $A\subseteq A(T)\cap E(H)$. The signed sum in one class is $(-1{)}^{n-1}(1-1{)}^{\mathrm{\mid }A(T)\cap E(H)\mathrm{\mid }}$, whose real absolute value is at most $1$. Applying the finite-sum triangle inequality from [F5, F7] to the class contributions gives $\mathrm{\mid }{U}_n({\gamma }_1,\dots ,{\gamma }_n)\mathrm{\mid }\le {\sum }_{T\in {\mathcal{T}}_n}{\prod }_{\{i,j\}\in E(T)}{\mathbf{1}}_{\{{\gamma }_i\sim {\gamma }_j\}}$. For $n=1$ both sides equal $1$.

A rooted tree means a tree with one distinguished vertex; its height is the greatest number of edges in a path from that root. A size-$n$ labelled tree here has vertex set $[n]$, and in $S_{\gamma }^{(k)}$ its root is label $1$; removing the root separates it into child subtrees. For fixed root label $1$ of type $\gamma$, let $S_{\gamma }^{(k)}$ be the nonnegative sum over all such finite rooted labelled trees of height at most $k$, with all other vertex types summed over $\mathrm{\Gamma }$, each non-root vertex of type $\eta$ weighted by $\mathrm{\mid }{z}_{\eta }\mathrm{\mid }$, each edge weighted by ${\mathbf{1}}_{\{\eta \sim {\eta }^{\mathrm{\prime }}\}}$, and each size-$n$ sum divided by $(n-1)!$. The root has weight $1$, so $S_{\gamma }^{(0)}=1$. At height at most $k+1$, the root has $m$ child subtrees of height at most $k$. For ordered subtree sizes $n_1,\dots ,n_m$, assigning the $n-1$ labels to the ordered branch vertex sets contributes $(n-1)!\mathrm{/}(n_1!\cdots n_m!)$. In each branch there are $n_j$ choices for its child-root label; with the fixed-root subtree sum normalized by $(n_j-1)!$, the factor $n_j(n_j-1)!\mathrm{/}{n}_j!$ is $1$. The parent normalization $(n-1)!$ therefore leaves the product of the branch weights, and division by $m!$ removes the ordering of the branches. Since $\mathrm{\Gamma }$ is finite or in bijection with $\mathbb{N}$, increasing finite type and size cutoffs exhaust the choices; by the nonnegative-sum convention [F6], the branch decomposition gives the corresponding countable-sum identity. Now suppose $S_{\eta }^{(k)}\le e^{a_{\eta }}$ for every $\eta$. The sum $A_{\gamma }^{(k)}:={\sum }_{\eta \sim \gamma }\mathrm{\mid }{z}_{\eta }\mathrm{\mid }{S}_{\eta }^{(k)}$ then satisfies $0\le A_{\gamma }^{(k)}\le a_{\gamma }<\mathrm{\infty }$ by [A1]. The branch identity is therefore the convergent exponential series $S_{\gamma }^{(k+1)}={\sum }_{m=0}^{\mathrm{\infty }}(A_{\gamma }^{(k)}{)}^m\mathrm{/}m!=e^{A_{\gamma }^{(k)}}$, so monotonicity of the real exponential [F3] gives $S_{\gamma }^{(k+1)}\le e^{a_{\gamma }}$. Since $S_{\gamma }^{(0)}=1\le e^{a_{\gamma }}$ by [F2], induction gives this bound for every $k$. The nonnegative sums increase with $k$ and every finite tree has finite height, so their limit is the full rooted tree sum and is at most $e^{a_{\gamma }}$.

Apply the pointwise graph-to-tree bound of step 1.1 to each tuple in the definition of $R_{\gamma }$. All weights are nonnegative after taking absolute values, so summing over the countable polymer labels and then over $n$ preserves the inequality; the right side is the rooted tree sum bounded in step 1.2. Hence $R_{\gamma }\le e^{a_{\gamma }}$.

Write $f_{ij}=-{\mathbf{1}}_{\{{\gamma }_i\sim {\gamma }_j\}}$. Reflexivity makes ${\prod }_{i<j}(1+f_{ij})$ zero when a tuple repeats a polymer, and otherwise this product is one exactly for a compatible subset. Hence $Z_V(t)={\sum }_{n\ge 0}\frac{t^n}{n!}{\sum }_{\vec{\gamma }\in V^n}({\prod }_i{w}_{{\gamma }_i}){\prod }_{i<j}(1+f_{ij})$ as a formal series. Expanding the pair product gives all simple graphs. A set partition of $[n]$ is a finite family of nonempty pairwise-disjoint blocks whose union is $[n]$; write ${\mathrm{\Pi }}_n$ for the set of such partitions. This set is finite since it is contained in $\mathcal{P}(\mathcal{P}([n]))$. The connected components of each finite graph give its unique partition into connected graph blocks. Each graph's edge weight factors over these blocks. The coefficient of $t^n$ in $\exp (L_V(t))$ is $\frac{1}{n!}{\sum }_{\pi \in {\mathrm{\Pi }}_n}{\sum }_{\vec{\gamma }\in V^n}({\prod }_{B\in \pi }{U}_{\mathrm{\mid }B\mathrm{\mid }}({\vec{\gamma }}_B))({\prod }_{i=1}^n{w}_{{\gamma }_i})$: for ordered block sizes $m_1,\dots ,m_k$, assigning the $n$ labels to those blocks contributes $n!\mathrm{/}(m_1!\cdots m_k!)$, and the exponential's factor $1\mathrm{/}k!$ removes the ordering of the $k$ blocks. These factors leave exactly the connected-component expansion of the all-graph coefficient. Thus $Z_V$ and $\exp (L_V)$ have identical formal coefficients.

Let $d_n$ be the coefficient of $t^n$ in $L_V$, and apply [F2] to the finite polymer set $V$ with activities $r{w}_{\eta }$. Since $r>0$, [F5] gives $\mathrm{\mid }r{w}_{\eta }\mathrm{\mid }=r\mathrm{\mid }{w}_{\eta }\mathrm{\mid }$, so [A1] implies the rooted criterion. For each fixed root ${\gamma }_1$, the $n$th nonnegative term of that rooted sum is at most $e^{a_{{\gamma }_1}}$. The finite-sum triangle inequality, multiplicativity in [F5], and agreement of the complex modulus with real absolute value give $r^n\mathrm{\mid }{d}_n\mathrm{\mid }\le \frac{1}{n!}{\sum }_{{\gamma }_1\in V}r\mathrm{\mid }{w}_{{\gamma }_1}\mathrm{\mid }{\sum }_{{\gamma }_2,\dots ,{\gamma }_n\in V}\mathrm{\mid }{U}_n({\gamma }_1,\dots ,{\gamma }_n)\mathrm{\mid }{\prod }_{j=2}^n\mathrm{\mid }r{w}_{{\gamma }_j}\mathrm{\mid }\le \frac{1}{n}{\sum }_{\gamma \in V}r\mathrm{\mid }{w}_{\gamma }\mathrm{\mid }{e}^{a_{\gamma }}$. Thus on $\mathrm{\mid }t\mathrm{\mid }\le s<r$, the series is dominated by a constant times ${\sum }_{n\ge 1}(s\mathrm{/}r{)}^n\mathrm{/}n$, and its derivative series by a constant times ${\sum }_{n\ge 1}(s\mathrm{/}r{)}^{n-1}$. Both converge uniformly on every such closed subdisc.

By [F4], $L_V$ differentiates term by term throughout $\mathrm{\mid }t\mathrm{\mid }<r$. For each fixed $t$ with $\mathrm{\mid }t\mathrm{\mid }<r$, step 1.2 gives ${\sum }_{n\ge 1}\mathrm{\mid }{d}_n\mathrm{\mid }\mathrm{\mid }t{\mathrm{\mid }}^n<\mathrm{\infty }$, so the total absolute sum after expanding $\exp (L_V(t))$ is at most ${\sum }_{k\ge 0}({\sum }_{n\ge 1}\mathrm{\mid }{d}_n\mathrm{\mid }\mathrm{\mid }t{\mathrm{\mid }}^n{)}^k\mathrm{/}k!<\mathrm{\infty }$. The products may therefore be regrouped by total degree; their coefficients are the formal coefficients identified in step 1.1, so $Z_V(t)=\exp (L_V(t))$ on the disc. By [F3], $\exp (L_V(t))\exp (-L_V(t))=1$, so $Z_V(t)$ is nonzero and $L_V$ is its analytic logarithm with $L_V(0)=0$.

Fix $n\ge 2$ and positions $x_1,\dots ,x_n$, and put $f_{ij}=f_{\beta }(x_i-x_j)$. For each nonempty label set $I$, let $U_I={\sum }_{i<j,i,j\in I}v(x_i-x_j)$. If $U_I=+\mathrm{\infty }$, the adopted convention gives $e^{-2\beta U_I}=0$; otherwise all pair values in the sum are finite. In either case the exponential definition and addition law [F2], together with evenness of $v$, give ${\prod }_{i\in I}{\prod }_{j\in I\setminus \{i\}}\mathrm{\mid }1+f_{ij}\mathrm{\mid }=e^{-2\beta U_I}$. Stability and monotonicity [F1, F2] imply this product is at most $e^{2\beta B\mathrm{\mid }I\mathrm{\mid }}$. Writing $q_i={\prod }_{j\in I\setminus \{i\}}\mathrm{\mid }1+f_{ij}\mathrm{\mid }$, all $q_i$ are nonnegative and their product is at most $e^{2\beta B\mathrm{\mid }I\mathrm{\mid }}$; hence some $r\in I$ has $q_r\le e^{2\beta B}$.

The maps $(x_1,\dots ,x_n)\mapsto x_i-x_j$ are continuous and hence Borel under [A1, F4]. Since $v$ is Borel [F1], composition with $v$ is Borel [F4]. The adopted map $u\mapsto e^{-\beta u}$ on $\mathbb{R}\cup \{+\mathrm{\infty }\}$ is Borel: on $\mathbb{R}$ it is continuous by [F2], and its value at the Borel singleton $+\mathrm{\infty }$ is $0$ by [F4]. Thus $f_{\beta }$ is Borel, and $U_n$ is a finite sum of finite products of Borel factors [F4]. The same composition and arithmetic facts on ${\mathbb{R}}^{dn}$ make each tree weight Borel. Repeated application of the two-factor identity in [F5] identifies the finite product Borel sigma-algebra, and the Borel-to-Lebesgue clause in [F5] makes the tree weight measurable for product Lebesgue measure.

For disjoint label sets $I,J$, define $g(I,J)$ as the finite sum of ${\prod }_{\{i,j\}\in E(G)}{f}_{ij}$ over simple graphs on $I\cup J$ in which every connected component meets $I$; set $g(\mathrm{\varnothing },\mathrm{\varnothing })=1$ and $g(\mathrm{\varnothing },J)=0$ for nonempty $J$. Evenness in [F1] makes each unordered-edge weight independent of which endpoint is written first. Choose $r\in I$ as in step 1.1 and put $I^{\mathrm{\prime }}=I\setminus \{r\}$. In each graph let $K\subseteq J$ be the neighbors of $r$ in $J$. Edges from $r$ to $I^{\mathrm{\prime }}$ are independently present or absent, and summing their two weights gives ${\prod }_{j\in I^{\mathrm{\prime }}}(1+f_{rj})$. Deleting $r$ leaves a graph all of whose components meet $I^{\mathrm{\prime }}\cup K$, and conversely any such graph reconstructs exactly one original graph after adjoining the chosen $r$-edges. Thus $g(I,J)=({\prod }_{j\in I^{\mathrm{\prime }}}(1+f_{rj})){\sum }_{K\subseteq J}\left({\prod }_{k\in K}{f}_{rk}\right)g(I^{\mathrm{\prime }}\cup K,J\setminus K)$.

Under Countable Choice [A1], the product Lebesgue spaces are sigma-finite by [F3], so Tonelli [F6] applies. Root a labelled tree $T$ at $1$. A finite tree with at least two vertices has at least two leaves: a longest simple path exists in a finite graph, and each endpoint has exactly one incident edge, since another incident edge would either extend the path or create a cycle. At least one of the two leaves differs from the root. Deleting a leaf preserves connectedness and acyclicity on the remaining vertices, so the remaining graph is a tree unless only the root remains. Delete such a leaf and integrate its coordinate first. Translation invariance [F7] and evenness of $f_{\beta }$ give ${\int }_{{\mathbb{R}}^d}\mathrm{\mid }{f}_{\beta }(x_{\mathrm{\ell }}-x_j)\mathrm{\mid }d{x}_{\mathrm{\ell }}=C_{\beta }$. Repeating after each leaf deletion proves that the pinned tree integral is $C_{\beta }^{n-1}$.

Let ${\mathcal{F}}_I(J)$ be the forests on $I\cup J$ with exactly one vertex from $I$ in each tree, and define $h(I,J)=e^{2\beta B(\mathrm{\mid }I\mathrm{\mid }+\mathrm{\mid }J\mathrm{\mid })}{\sum }_{F\in {\mathcal{F}}_I(J)}{\prod }_{\{i,j\}\in E(F)}\mathrm{\mid }{f}_{ij}\mathrm{\mid }$, with the same empty-root convention as $g$. For nonempty $I$, delete the root $r$ selected in step 1.1. Its component cannot contain another vertex of $I$, and the set $K\subseteq J$ of its neighbors gives exactly the roots of the resulting branches; this is a bijection with forests in ${\mathcal{F}}_{I^{\mathrm{\prime }}\cup K}(J\setminus K)$. The exponential addition law [F2] factors the size weight, so $h(I,J)=e^{2\beta B}{\sum }_{K\subseteq J}({\prod }_{k\in K}\mathrm{\mid }{f}_{rk}\mathrm{\mid })h(I^{\mathrm{\prime }}\cup K,J\setminus K)$. If $I=\mathrm{\varnothing }$, both $g$ and $h$ are $1$ for $J=\mathrm{\varnothing }$ and $0$ otherwise. Induct on $\mathrm{\mid }I\mathrm{\mid }+\mathrm{\mid }J\mathrm{\mid }$ from these base cases. Step 1.1 supplies the root bound and step 2.1 supplies the graph recurrence; applying the finite-sum real triangle inequality from [F12, F13] gives $\mathrm{\mid }g(I,J)\mathrm{\mid }\le q_r{\sum }_K({\prod }_{k\in K}\mathrm{\mid }{f}_{rk}\mathrm{\mid })\mathrm{\mid }g(I^{\mathrm{\prime }}\cup K,J\setminus K)\mathrm{\mid }$. The induction hypothesis and $q_r\le e^{2\beta B}$ bound this by $h(I,J)$. At $I=\{1\}$, $J=\{2,\dots ,n\}$, $g(I,J)=U_n$ and every forest in ${\mathcal{F}}_I(J)$ is a spanning tree. This proves the pointwise tree bound.

In the Prüfer encoding, repeatedly delete the least-labelled leaf and record its neighbor; after $n-2$ deletions this gives a word of length $n-2$ over $\{1,\dots ,n\}$. Step 2.2 guarantees a leaf other than the root at every stage with at least two vertices, and deleting a leaf leaves a tree until the final vertex pair. At each stage a vertex occurs in the remaining word exactly once for each incident edge deleted before that vertex; it is recorded when a neighboring leaf is removed, and the vertex itself is deleted once it becomes a leaf, except for the final two vertices. Thus its occurrence count is its current degree minus one, and absence from the word is equivalent to being a current leaf. Conversely, from any such word, repeatedly join the least label absent from the remaining word to its first symbol and delete that label and symbol, then join the last two labels. Adding a new vertex with one edge to a tree makes the new vertex a leaf and preserves connectedness and acyclicity; induction on the word length therefore shows this constructs a tree. In the resulting tree the least absent label is a leaf and the first symbol is its neighbor, so encoding reverses every decoding step. The procedures give a bijection with all $n^{n-2}$ words. Integrating the pointwise tree bound of step 3.1 and using monotonicity, homogeneity and finite additivity [F8, F9] with the tree integral from step 2.2 yields $\int \mathrm{\mid }{U}_n(0,x_2,\dots ,x_n)\mathrm{\mid }d{x}_2\cdots d{x}_n\le e^{2\beta Bn}{n}^{n-2}{C}_{\beta }^{n-1}$.