Generalized cluster algorithms for Potts lattice gauge theory

TL;DR

This paper extends cluster algorithms to efficiently sample Potts lattice gauge theories using a cellular representation, demonstrating faster autocorrelation decay and improved sampling efficiency in high-dimensional models.

Contribution

It introduces generalized cluster algorithms for Potts lattice gauge theory based on a cellular model, enabling faster sampling at criticality compared to traditional methods.

Findings

Algorithms show much faster autocorrelation decay.

Efficient sampling on 4D tori of size at least 40.

Applicable to $ ext{Z}_2$ and $ ext{Z}_3$ gauge theories.

Abstract

Monte Carlo algorithms, like the Swendsen-Wang and invaded-cluster, sample the Ising and Potts models asymptotically faster than single-spin Glauber dynamics do. Here, we generalize both algorithms to sample Potts lattice gauge theory by way of a $2$-dimensional cellular representation called the plaquette random-cluster model. The invaded-cluster algorithm targets Potts lattice gauge theory at criticality by implementing a stopping condition defined in terms of homological percolation, the emergence of spanning surfaces on the torus. Simulations for $\mathbb Z_2$ and $\mathbb Z_3$ lattice gauge theories on the cubical $4$-dimensional torus indicate that both generalized algorithms exhibit much faster autocorrelation decay than single-spin dynamics and allow for efficient sampling on $4$-dimensional tori of linear scale at least $40$.