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Try to look at Introduction to the Replica Theory of Disordered Statistical Systems by V. Dotsenko. In the following, I've written a possible answer to your question: $$f=-\lim_{N\rightarrow\infty}\frac{1}{\beta N}\mathbb{E}\left[\ln Z_{J}\right]$$ where: \$\mathbb{E}\left[\mathcal{O}\right]=\left(\prod_{\left\{ i,j\right\} ...