The random cluster model on the complete graph via large deviations

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 $q\geq 1$.