Abstract polymer model and connected coefficients

An abstract polymer model consists of an at most countable set $\mathrm{\Gamma }$ (finite or in bijection with $\mathbb{N}$), a reflexive symmetric relation $\sim$ on $\mathrm{\Gamma }$ (so $\gamma \sim \gamma$ and $\gamma \sim \eta$ implies $\eta \sim \gamma$), and an activity function $z:\mathrm{\Gamma }\to \mathbb{C}$. Its elements are called polymers, and $\gamma \sim \eta$ means that the two polymers are incompatible. A subset $X\subseteq \mathrm{\Gamma }$ is compatible when no two distinct elements of $X$ are related by $\sim$. For real and natural summands, finite-index sums and products are enumeration-independent. For complex summands, define a finite sum or product by any enumeration and repeated addition or multiplication in $\mathbb{C}$; associativity and commutativity make the result independent of the enumeration. The empty sum is $0$ and the empty product is $1$. For each finite $V\subseteq \mathrm{\Gamma }$, define the finite-volume partition polynomial

$Z_V(z):={\sum }_{\begin{array}{c}X\subseteq V\\ X\text{ compatible}\end{array}}{\prod }_{\gamma \in X}{z}_{\gamma },$

where the empty subset contributes $1$.

Write $[n]:=\{1,\dots ,n\}$ and $\left(\genfrac{}{}{0px}{}{[n]}{2}\right):=\{\{i,j\}:i,j\in [n],i\mathrm{\ne }j\}$. A simple graph on $[n]$ is a pair $([n],E)$ with $E\subseteq \left(\genfrac{}{}{0px}{}{[n]}{2}\right)$; a path is a finite vertex list with distinct vertices and each consecutive pair an edge. A graph is connected when every two vertices are joined by a path. A cycle is a vertex list $(v_0,\dots ,v_k)$ with $k\ge 3$, $v_k=v_0$, $v_0,\dots ,v_{k-1}$ distinct, and each consecutive pair an edge; a tree is a connected simple graph with no cycle. For $n\in \mathbb{N}$ with $n\ge 1$, let ${\mathcal{C}}_n$ be the set of connected simple graphs on $[n]$ and let ${\mathcal{T}}_n$ be the set of trees on $[n]$. The set $[n]$ is finite; its power set is finite, so its subset $\left(\genfrac{}{}{0px}{}{[n]}{2}\right)$ is finite; the power set of that edge set is finite as well. Each simple graph on $[n]$ is determined by an edge set in this last power set, so ${\mathcal{C}}_n$ and ${\mathcal{T}}_n$ are finite.

For an ordered tuple $({\gamma }_1,\dots ,{\gamma }_n)\in {\mathrm{\Gamma }}^n$, put $f_{ij}:=-{\mathbf{1}}_{\{{\gamma }_i\sim {\gamma }_j\}}$ and define the connected coefficient

$U_n({\gamma }_1,\dots ,{\gamma }_n):={\sum }_{G\in {\mathcal{C}}_n}{\prod }_{\{i,j\}\in E(G)}{f}_{ij}\mathrm{.}$

For $n=1$, ${\mathcal{C}}_1$ contains the one-vertex graph and the empty edge product equals $1$. These are finite polynomial and coefficient definitions; they assert no convergence.

The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$

Finite, countably infinite, countable, uncountable

The sum ${\sum }_{i\in S}{a}_i$ over a finite index set, and its product form

A finite simple graph is a finite vertex set together with a set of two-element vertex subsets

Simple graphs on an arbitrary vertex set

Walks, paths, connectedness and components in a simple graph on an arbitrary vertex set

Cycles, trees and forests in a simple graph on an arbitrary vertex set

$\mid \mathcal{P}(A)\mid =2^{\mid A\mid }$ for finite $A$

A subset of a finite set is finite, with $\mid B\mid \le \mid A\mid$, and equality holds if and only if $B=A$