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Random-cluster dynamics in \(\mathbb{Z}^2\): rapid mixing with general boundary conditions. (English) Zbl 1434.60273
Summary: The random-cluster model with parameters \((p,q)\) is a random graph model that generalizes bond percolation \((q=1)\) and the Ising and Potts models \((q\geq 2)\). We study its Glauber dynamics on \(n\times n\) boxes \(\Lambda_n\) of the integer lattice graph \(\mathbb{Z}^2\), where the model exhibits a sharp phase transition at \(p=p_c(q)\). Unlike traditional spin systems like the Ising and Potts models, the random-cluster model has non-local interactions. Long-range interactions can be imposed as external connections in the boundary of \(\Lambda_n\), known as boundary conditions. For select boundary conditions that do not carry long-range information (namely, wired and free), Blanca and Sinclair proved that when \(q>1\) and \(p\neq p_c(q)\), the Glauber dynamics on \(\Lambda_n\) mixes in optimal \(O(n^2\log n)\) time. In this paper, we prove that this mixing time is polynomial in \(n\) for every boundary condition that is realizable as a configuration on \(\mathbb{Z}^2\setminus\Lambda_n\). We then use this to prove near-optimal \(\tilde{O}(n^2)\) mixing time for “typical” boundary conditions. As a complementary result, we construct classes of nonrealizable (nonplanar) boundary conditions inducing slow (stretched-exponential) mixing at \(p\ll p_c\).

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
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