Ising and Potts models are an important class of discrete probability distributions which originated from Statistical Physics and since then have found applications in several disciplines. Simulation from these models is a well known challenging problem. In this paper, we propose a class of MCMC algorithms to simulate from both Ising and Potts models, by using auxiliary Gaussian random variables. Our algorithms apply to coupling matrices with both positive and negative entries, thus including Spin Glass models such as the SK and Hopfield model. In contrast to existing methods of a similar flavor, our algorithm can take advantage of the low-rank structure of the coupling matrix, and scales linearly with the number of states in a Potts model. We compare our proposed algorithm to existing state of the art algorithms, such as the Swendsen-Wang and Wolff algorithms for Ising and Potts models on graphs, and the Heat Bath for Spin Glass models. Our comparison takes into account a wide range of coupling matrices and temperature regimes, focusing in particular on behavior at or below the critical temperature. For cold systems, augmenting our algorithm with a tempering scheme yields significant improvements.
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