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We study the emergence of the giant component in the random cluster model onthe complete graph, which was first studied by Bollob\'as, Grimmett, andJanson. We give an alternative analysis using a thermodynamic/large deviationsapproach introduced by Biskup, Chayes, and Smith for the case of percolation.In particular, we compute the rate function for large deviations of the size ofthe largest connected component of the random graph for $q\geq 1$.

The random cluster model on the complete graph via large deviations