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

TL;DR

This paper develops an exact scaling theory for the 5D Ising model with cyclic boundary conditions, using comparisons with complete graph results, and confirms predictions through Monte Carlo simulations, revealing a bimodal energy distribution near criticality.

Contribution

It introduces a detailed and exact finite size scaling framework for the 5D Ising model with cyclic boundary, based on rigorous comparisons with complete graph models.

Findings

Good agreement between theoretical predictions and Monte Carlo data.

Identification of distinct scaling regions near the critical point.

Discovery of bimodal energy distribution indicating quasi-first order transition.

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.