Generalized cluster algorithms for Potts lattice gauge theory

Monte Carlo algorithms, like the Swendsen–Wang and invaded-cluster, sample the Ising and Potts models asymptotically faster than single-spin Glauber dynamics do. By way of a 2-dimensional cellular representation called the plaquette random-cluster model (PRCM), we describe the plaquette Swendsen–Wang and plaquette invaded-cluster algorithms for sampling Potts lattice gauge theory (PLGT), a model obtained from ℤ_q lattice gauge theory by substituting a Potts interaction. Constructing the underlying spaces this way lets us recover an Edwards-Sokal-like coupling of the PRCM and PLGT, a probabilistic tool on which both algorithms rely.

The plaquette Swendsen–Wang algorithm behaves much like its classical analogue, targeting PLGT by alternately sampling random spin assignments on edges, then random subsets of nonfrustrated plaquettes. The plaquette invaded-cluster algorithm targets the same by implementing a stopping condition defined in terms of homological percolation, the emergence of spanning surfaces on the torus. Simulations for ℤ₂ and ℤ₃ lattice gauge theories on the 4-dimensional cubical torus indicate that both generalized algorithms exhibit much faster autocorrelation decay than single-spin dymanics. In addition to algorithmic efficiency, we encode the cell structure and the algorithms' subroutines with basic with basic linear algebra, allowing for fast sampling of PLGT on 4-dimensional tori of linear scale at least 40.