Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs

Understand

For $\Delta \ge 5$ and $q$ large as a function of $\Delta$, we give a detailed picture of the phase transition of the random cluster model on random $\Delta$-regular graphs.

• In particular, we determine the limiting distribution of the weights of the ordered and disordered phases at criticality and prove exponential decay of correlations and central limit theorems away from criticality.
• Our techniques are based on using polymer models and the cluster expansion to control deviations from the ordered and disordered ground states.
• These techniques also yield efficient approximate counting and sampling algorithms for the Potts and random cluster models on random $\Delta$-regular graphs at all temperatures when $q$ is large.