Complete graph asymptotics for the Ising and random-cluster models on five-dimensional grids with a 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 FKrandom-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. * Electronic address: per.hakan.lundow@math.umu.se † Electronic address: klas.markstrom@math.umu.se 1 arXiv:1408.2155v1 [cond-mat.stat-mech]

The scaling window of the 5D Ising model with free boundary conditions

“…Aizenman's conjectured scaling L 2d/3 came from the fact that the cyclic boundary case was expected to behave like the critical Erdős-Renyi random graph, where the maximum cluster has size N 2/3 , where N is the number of vertices, which would be L d for the lattice. Again, there are detailed rigorous results [18] concerning the random cluster model on the complete graph and in [19] it was demonstrated that Monte Carlo data for the FKmodel with q = 2 and d = 5 with cyclic boundary gives a detailed agreement with the scaling for the complete graph. In particular, the size of the largest cluster has the same scaling for both graphs for several different rescaled coupling ranges.…”

Percolation effects in the Fortuin-Kasteleyn Ising model on the complete graph

“…[4] proved that for the q = 2 case, the size of the largest cluster scales as C 1 ∼ V 3/4 within the Ising critical window given by 1 − p/p c = O(V −1/2 ), where p c = 2/V is the critical point and V is the number of vertices on CG. The CG asymptotics for C 1 was later observed numerically on five-dimensional tori [5].…”

The Finite-Size Scaling Study of Five-Dimensional Ising Model

“…where γ = 1, β = 1/2 and δ = 3 for d = 5 [29,30,41,42]. The log-log plots of M (L) at T = T c (L, h) versus h for h in the interval 0 ≤ h ≤ 0.0025 yield to 1/δ(L) (Fig.…”

“…In addition, the four-dimensional ferromagnetic Ising model solution is approximated by using Creutz cellular automaton algorithm with nearest neighbor interactions and near the critical region [14][15][16][17][18][19][20][21][22][23]. The algorithm of approximating finite size behavior of ferromagnetic Ising model is extended to higher dimensions [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. It is established that the algorithm has been powerful in terms of providing the values of static critical exponents near the critical region in four and higher dimensions with nearest neighbor interactions [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]…”