Dynamics for the Random-Cluster Model

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Proves a comparison inequality showing Chayes–Machta global updates and single-edge heat‑bath mix within polynomial factors; analyzes the mean‑field complete graph and gives a full mixing picture: for 1<q≤2 mixing is polylogarithmic (Θ(n log n scaling of the chain), at q>2 there is an exponential slowdown in a window λL<λ<λR and fast O(n log n) mixing off that window, with a power‑law boundary case. On finite boxes of Z^d establishes a tight heat‑bath bound ~n^2 log n using drift for the largest cluster, monotone/identity couplings, Alexander’s exponential decay and size‑homogeneous boundary conditions.

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1. 0:00Random cluster model (Fortuin & Kasteleyn, 1969)
2. 0:59Motivation
3. 2:41Random cluster model in infinite graphs
4. 3:50Phase transition
5. 16:01Meanfield mixing Interpretation of results
6. 20:09Phase 1 Independent evolution
7. 25:00Proofs: Phase 2 - Activation coupling
8. 28:48Boundary conditions examples
9. 29:52Spatial Mixing (SM)
10. 34:09Outline of proof of Theorem 3
11. 36:17Proof of Lemma 2 SM implies fast mixing (cont.)
12. 39:55Proof of Lemma 3
13. 41:27Speed of propagation of disagreements: Proof
14. 42:47Open problems