Complete graph asymptotics for the Ising and random-cluster models on five-dimensional grids with a cyclic boundary

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
P H Lundow, K Markström

Abstract

The finite-size scaling behavior for the Ising model in five dimensions, with either free or cyclic boundary, has been the subject of a long-running debate. The older papers have been based on ideas from, e.g., field theory or renormalization. In this paper we propose a detailed and exact scaling picture for critical region of the model with cyclic boundary. Unlike the previous papers our approach is based on a comparison with the existing exact and rigorous results for the FK-random-cluster model on a complete graph. Based on those results we predict several distinct scaling regions in an L-dependent window around the critical point. We test these predictions by comparing with data from Monte Carlo simulations and find a good agreement. The main feature which differs between the complete graph and the five-dimensional model with free boundary is the existence of a bimodal energy distribution near the critical point in the latter. This feature was found by the same authors in an earlier paper in the form of a quasi-first-order phase transition for the same Ising model.

References

Feb 1, 1985·Physical Review. B, Condensed Matter·K BinderA P Young
Dec 1, 1985·Physical Review. B, Condensed Matter·J RudnickV Privman
Aug 1, 1996·Physical Review. B, Condensed Matter·G Parisi, J J Ruiz-Lorenzo
Jan 23, 1989·Physical Review Letters·U Wolff
Mar 4, 1996·Physical Review Letters·E Luijten, H W Blöte

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Citations

Mar 20, 2018·Physical Review. E·Wei HuangYoujin Deng
Feb 20, 2021·Physical Review. E·Sheng FangYoujin Deng

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