arXiv:1408.2155 · Physical Review E

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

P. H. Lundow, Klas Markström

Published Feb 11, 2015• 9 citations

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.

DOI: 10.1103/physreve.91.022112

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