2018/07/23 by Antonio Blanca, Reza Gheissari, Blanca, Antonio +3
Mathematics · Physics and Astronomy · #60K35 #82B20 #82C20 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.MP #math.PR #msc:60K35 #msc:82B20 #msc:82C20
paper · pdf · doi:10.48550/arxiv.1807.08722
40 pages, 6 figures
arxiv created 2019/05/06 · arxiv updated 2019/05/07
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≥ 2). We study its Glauber dynamics on n× n boxes Λn of the integer lattice graph \mathbb Z2, where the model exhibits a sharp phase transition at p=pc(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 Λn, 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≠ pc(q), the Glauber dynamics on Λn mixes in optimal O(n2 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 Z2 ∖ Λn. We then use this to prove near-optimal O(n2) mixing time for "typical'' boundary conditions. As a complementary result, we construct classes of non-realizable (non-planar) boundary conditions inducing slow (stretched-exponential) mixing at p≪ pc(q).