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  Question on Glimm and Jaffe's exposition of the cluster expansion

+ 3 like - 0 dislike

I am currently going through Glimm and Jaffe's exposition of the Cluster Expansion given in Quantum Physics: A Functional Integral Point of View. I have trouble understanding one claim they make in the "EXAMPLE 2" section just below theorem 18.2.5 (page 364). They define a function $F(\Lambda, s)$ that they claim factorizes at $s = 0$ (this notion is explained in definition 18.2.2).

I am not sure I can see why this last claim is true. Indeed, consider for instance the very simple case $\Gamma_2 = \emptyset$ and $\Gamma = b_2$ where $b_2$ is a lattice edge distant from $b_1$ by at least 2 lattice squares. Then $\Gamma^c = (\mathbf{Z}^2)^* - \{ b_1, b_2 \}$ and $\mathbf{R}^2 - \Gamma^c$ is the disjoint union of a connected component $X_1$ containing $b_1$, a connected component $X_2$ containing $b_2$ and a connected component $X_3$ containing neither $b_1$ nor $b_2$.

Now, if I apply the decomposition (18.2.11) to $F(\Lambda, s)$ with $s = 1$, the terms with $X_2$ and $X_3$ will give $0$ since the covariances appearing in the $Z$'s defining $F$ will coincide when restricted on the domains in question if I am correct. But this implies that $F$ itself is $0$, which I don't understand!

What am I getting wrong?

asked May 8, 2018 in Mathematics by IchKenneDeinenNamen (30 points) [ no revision ]

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