Dual Monte Carlo and Cluster Algorithms

TL;DR

This paper explores the development of cluster algorithms through probability theory, providing a unified framework, explicit assumptions, and new algorithmic variants for classical and quantum models.

Contribution

It introduces a probability-theoretic perspective on cluster algorithms, deriving existing algorithms and proposing new variants like a free cluster version for the random Ising model.

Findings

Rederived the Swendsen-Wang algorithm from probability principles

Presented a loop algorithm for quantum $S=1/2$ models

Proposed a free cluster version of the Swendsen-Wang replica method

Abstract

We discuss the development of cluster algorithms from the viewpoint of probability theory and not from the usual viewpoint of a particular model. By using the perspective of probability theory, we detail the nature of a cluster algorithm, make explicit the assumptions embodied in all clusters of which we are aware, and define the construction of free cluster algorithms. We also illustrate these procedures by rederiving the Swendsen-Wang algorithm, presenting the details of the loop algorithm for a worldline simulation of a quantum $S=$ 1/2 model, and proposing a free cluster version of the Swendsen-Wang replica method for the random Ising model. How the principle of maximum entropy might be used to aid the construction of cluster algorithms is also discussed.