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