Sampling Weighted Perfect Matchings on the Square-Octagon Lattice

We consider perfect matchings of the square-octagon lattice, also known as “fortresses” [16]. There is a natural local Markov chain on the set of perfect matchings that is known to be ergodic. However, unlike Markov chains for sampling perfect matchings on the square and hexagonal lattices, corresponding to domino and lozenge tilings, respectively, the seemingly related Markov chain on the square-octagon lattice appears to converge slowly. To understand why, we consider a weighted version of the problem. As with domino and lozenge tilings, it will be useful to view perfect matchings on the squareoctagon lattice in terms of sets of paths and cycles on a corresponding lattice region; here, the paths and cycles lie on the Cartesian lattice and are required to turn left or right at every step. For input parameters λ and μ, we define the weight of a configuration to be λ|E(σ)|μ|V (σ)|, where E(σ) is the total number of edges on the paths and cycles of σ and V (σ) is the number of vertices that are not incident to any of the paths or cycles in σ. Weighted paths already come up in the reduction from perfect matchings to turning lattice paths, corresponding to the case when λ = 1 and μ = 2. First, fixing μ = 1, we show that there are choices of λ for which the chain converges slowly and another for which it is fast, suggesting a phase change in the mixing time. More precisely, the chain requires exponential time (in the size of the lattice region) when λ < 1/(2 √ e) or λ > 2 √ e, while it is polynomially mixing at λ = 1. Further, we show that for μ > 1, the Markov chainM is slowly mixing when λ < √ μ/(2 √ e) or λ > 2μ √ e. These are the first rigorous proofs explaining why the natural local Markov chain can be slow for weighted fortresses or perfect matchings on the square-octagon lattice.