Random cluster models on random graphs
Van Hao Can, Remco van der Hofstad
TL;DR
The paper develops a cohesive framework connecting random cluster models on locally tree-like graphs to two-spin Ising models with vertex-dependent fields, and proves that the Bethe partition function provides a universal lower bound for the graph pressure. Central to the approach is a rank-2 approximation that rewrites the random cluster partition function as a two-spin model, allowing a direct link to extended Ising models with fields B_v = k d_v + h. The authors establish existence and explicit forms for the thermodynamic limit of the pressure in several regimes, characterize a first-order phase transition for q>2 along a calculable critical curve, and show quenched and annealed pressures coincide for random d-regular graphs, validating Bethe predictions in this setting. They also extend the analysis to extended Ising models with fixed-sign external fields, proving a thermodynamic limit via a message-passing representation. Collectively, these results provide rigorous connections between RC/ Potts/Ising models on random graphs, Bethe-type bounds, and phase diagrams, with implications for statistical mechanics and related computational problems.
Abstract
On locally tree-like random graphs, we relate the random cluster model with external magnetic fields and $q\geq 2$ to Ising models with vertex-dependent external fields. The fact that one can formulate general random cluster models in terms of two-spin ferromagnetic Ising models is quite interesting in its own right. However, in the general setting, the external fields are both positive and negative, which is mathematically unexplored territory. Interestingly, due to the reformulation as a two-spin model, we can show that the Bethe partition function, which is believed to have the same pressure per particle, is always a {\em lower bound} on the graph pressure per particle. We further investigate special cases in which the external fields do always have the same sign. The first example is the Potts model with general external fields on random $d$-regular graphs. In this case, we show that the pressure per particle in the quenched setting agrees with that of the annealed setting, and verify \cite[Assumption 1.4]{BasDemSly23}. We show that there is a line of values for the external fields where the model displays a first-order phase transition. This completes the identification of the phase diagram of the Potts model on the random $d$-regular graph. As a second example, we consider the high external field and low temperature phases of the system on locally tree-like graphs with general degree distribution.