Complete graph asymptotics for the Ising and random cluster models on 5D grids with cyclic boundary

Abstract

The finite size scaling behaviour for the Ising model in five dimensions, with either free or cyclic boundary, has been the subject for 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 identify 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.

13 pages, 14 figures

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• The scaling window of the 5D Ising model with free boundary conditions
• Complete graph and Gaussian fixed point asymptotics in the five-dimensional Fortuin-Kasteleyn Ising model with periodic boundaries
• Geometric scaling behaviors of the Fortuin-Kasteleyn Ising model in high dimensions
• Percolation effects in the Fortuin-Kasteleyn Ising model on the complete graph