The random-cluster model has a uniqueness transition at p_s(q,Δ) on wired Δ-regular trees for all q, yielding near-linear mixing of Glauber dynamics on trees and on random regular graphs when q ≥ C log Δ.
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Uniqueness and Mixing in the Low-Temperature Random-Cluster Model on Trees and Random Graphs
The random-cluster model has a uniqueness transition at p_s(q,Δ) on wired Δ-regular trees for all q, yielding near-linear mixing of Glauber dynamics on trees and on random regular graphs when q ≥ C log Δ.