Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with {1 ≤ q ≤ 4}
Abstract
This article studies the planar Potts model and its random-cluster representation. We show that the phase transition of the nearest-neighbor ferromagnetic q-state Potts model on Z^2 is continuous for {q in {2,3,4}}, in the sense that there exists a unique Gibbs state, or equivalently that there is no ordering for the critical Gibbs states with monochromatic boundary conditions. The proof uses the random-cluster model with cluster-weight {q ≥ 1} (note that q is not necessarily an integer) and is based on two ingredients:
The fact that the two-point function for the free state decays sub-exponentially fast for cluster-weights {1≤ q≤ 4}, which is derived studying parafermionic observables on a discrete Riemann surface.- Publication:
-
Communications in Mathematical Physics
- Pub Date:
- January 2017
- DOI:
- 10.1007/s00220-016-2759-8
- arXiv:
- arXiv:1505.04159
- Bibcode:
- 2017CMaPh.349...47D
- Keywords:
-
- Mathematics - Probability;
- Mathematical Physics;
- 60K35;
- 82B20;
- 82B27
- E-Print:
- 66 pages, 15 figures