LA Probability Forum

UCLA, Math Sciences Room 6627

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, extending work of Bencs, Borbenyi and Csikvary (2023). 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 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 Assumption 1.4 of Basak, Dembo and Sly (2023). 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.

This is joint work with Hao Can. See https://arxiv.org/abs/2503.17636 for the preprint.