# Rationale behind the 'joint cavity distribution'?

I have a question about equation (17) of this paper: http://arxiv.org/pdf/1009.1635v1.pdf

First, I was hoping that someone could explain how it is arrived at.

Second, I find the notation to be a bit perplexing--I do not quite understand the notation, particularly this "product of sums": $$\prod_{j\not\in\{0,1,2,3\}}\sum_{\sigma_j}e^{-\beta H_{G\backslash \{0\}}}$$ Suppose that we have some term of the product, say $j=n$ for some $n$ not in $\{0,1,2,3\}$. Then the summation is over $\sigma_n$, but what does that mean? $\sigma_n$ corresponds to a particular node in the graph (see Fig. 6), so I do not entirely understand how we can sum over it, or indeed why it would even be desirable to do so since the summand $e^{-\beta H_{G\backslash\{0\}}}$ certainly does not depend on which $\sigma_n$ we choose--it is just a function of the entire graph with one node 'carved out' as specified earlier in the paper.

-