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 $40$.

翻译：蒙特卡洛算法（如Swendsen-Wang算法和入侵聚类算法）对Ising模型和Potts模型的采样速度渐近快于单自旋Glauber动力学。本文通过一种称为plaquette随机团簇模型的二维胞腔表示，将这两种算法推广至Potts格点规范场的采样。入侵聚类算法通过实现一个由同调渗流（即环面上跨越曲面的涌现）定义的停止条件，专门针对临界状态下的Potts格点规范场。针对立方四维环面上$\mathbb Z(2)$和$\mathbb Z(3)$格点规范场的模拟表明，两种广义算法的自相关衰减速度均远快于单自旋动力学，且能够在线性尺度至少为40的四维环面上实现高效采样。