Algorithmic overlaps as thermodynamic variables: From local to cluster Monte Carlo dynamics in critical phenomena

Physical Review B: Condensed Matter and Materials Physics. 2026. Vol. 114. No. 1. Article 014101.

Pilé I., Deng Y., Shchur L.

We investigate the spatial overlap of successive spin configurations in Markov chain Monte Carlo simulations using the local Metropolis algorithm and the Swendsen-Wang and Wolff cluster algorithms. We examine the dynamics of these algorithms for models in different universality classes: Ising model, Potts model with three components, and four-state Potts model. The overlap of two successive Wolff clusters reflects critical behavior and can be used as an order parameter for the algorithm's dynamics. In the case of the Swendsen-Wang algorithm, similar behavior is demonstrated by the variance of the overlap of two consecutive lattice configurations, which behaves like an order parameter. Nothing similar is observed for the Metropolis algorithm, where the dynamics in the critical region are determined by the spin-flip frequency, which is equivalent to the acceptance rate. Thus, the critical behavior of the Wolff cluster overlap and of the variance of the configuration overlap in the Swendsen-Wang algorithm are naturally related to the critical behavior of geometric objects—Fortuin-Kasteleyn clusters. Interestingly, in all cases the geometric quantity—the configuration overlap or its variance—reflects the thermodynamics of the phase transition.

Keywords: Potts model, lattice models, Monte Carlo methods, Ising model, 2nd order phase transitions