Cluster expansion

Definition

Let $\varphi \colon \R^d \to \R \cup \set{+\infty}$ be an even measurable pair potential that is bounded below, let $\beta > 0$, and let $f$ be its Mayer f-function, with $f_{ij} = f(x_i - x_j)$. For a bounded region $\Lambda \subset \R^d$ and an activity $z > 0$, the grand canonical partition function is $$\Xi_\Lambda(z) = \sum_{n \ge 0} \frac{z^n}{n!} \int_{\Lambda^n} \prod_{1 \le i < j \le n} (1 + f_{ij}) \, dx_1 \cdots dx_n.$$ For $n \ge 1$, let $\mathcal{C}_n$ be the set of connected graphs with vertex set $\set{1, \dots, n}$. The cluster expansion (Mayer expansion) of $\ln \Xi_\Lambda$ is the power series $$\ln \Xi_\Lambda(z) = \sum_{n \ge 1} \frac{z^n}{n!} \int_{\Lambda^n} \sum_{G \in \mathcal{C}_n} \prod_{\set{i, j} \in E(G)} f_{ij} \, dx_1 \cdots dx_n.$$ It holds as an identity of formal power series in $z$.