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Improved mixing bounds for the anti-ferromagnetic Potts model on \(\mathbb Z^2\). (English) Zbl 1122.82007

Summary: The authors of this paper consider the anti-ferromagnetic Potts model on the integer lattice \(\mathbb Z^2\). The model has two parameters: \(q\), the number of spins, and \(\lambda = \exp(-\beta)\), where \(\beta\) is “inverse temperature”. It is known that the model has strong spatial mixing if \(q>7\), or if \(q=7\) and \(\lambda=0\) or \(\lambda>1/8\), or if \(q=6\) and \(\lambda=0\) or \(\lambda>1/4\). The \(\lambda=0\) case corresponds to the model in which configurations are proper \(q\)-colourings of \(\mathbb Z^2\). It is shown that the system has strong spatial mixing for \(q\geq 6\) and any \(\lambda\). This implies that Glauber dynamics is rapidly mixing (so there is a fully-polynomial randomised approximation scheme for the partition function), and also that there is a unique infinite-volume Gibbs state. It is also shown that strong spatial mixing occurs for a larger range of \(\lambda\) than was previously known for \(q=3\), \(q=4\) and \(q=5\).

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
68W20 Randomized algorithms
82D40 Statistical mechanics of magnetic materials
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