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i was reading Runnels' paper on cayley tree where he has squared the partition function of a cyley tree to get that of two exactly similar trees. why square? why not add the two partition functions to get that of the whole system? im an undergraduate student, may be this is a very dumb question.

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Ultimately, it's because the probability of two independent events is the product of the two separate events, i.e. $P(X_1,X_2) = P(X_1)P(X_2)$. In the language of statistical mechanics, the partition function gives a probability $$ P(X) \propto \exp(-\beta\,H(X)). $$ So the joint partition function is $$ Z_{1,2}(\beta) = \sum_{\rm configs} \exp(-\beta\,H(X_1,X_2))\,. $$ The statement of independence means that their probabilities separate but the statement of their identical nature means they have the same distribution, i.e.: $$ H(X_1,X_2) = H(X_1) + H(X_2) $$ Hence, \begin{align} Z_{1,2}(\beta) &= \sum_{\rm configs}\exp(-\beta\,H(X_1))\exp(-\beta\,H(X_2))\\ &= \sum \exp(-\beta\,H(X_1))\sum\exp(-\beta\,H(X_2))\\ &= Z_1(\beta)Z_2(\beta) \end{align}

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