Finite size scaling of the 5D Ising model with free boundary conditions

Abstract

There has been a long running debate on the finite size scaling for the Ising model with free boundary conditions above the upper critical dimension, where the standard picture gives a $L^2$ scaling for the susceptibility and an alternative theory has promoted a $L^{5/2}$ scaling, as would be the case for cyclic boundary. In this paper we present results from simulation of the far largest systems used so far, up to side $L=160$ and find that this data clearly supports the standard scaling. Further we present a discussion of why rigorous results for the random-cluster model provides both supports the standard scaling picture and provides a clear explanation of why the scalings for free and cyclic boundary should be different.Comment: 7 pages, 11 figure

The length of self-avoiding walks on the complete graph

“…In Cases (iii) and (iv), Eq. (20) shows that √ nη n converges to a finite limit, of α and 0, respectively. From Lemma (2.2) we then obtain lim n→∞ Q(n, nλ n ) = Φ (α), β = 1/2,…”

“…Cases (i) and (ii) then follow by combining with (16) and (20), respectively. In Cases (iii) and (iv), Eq.…”

Geometric Explanation of Anomalous Finite-Size Scaling in High Dimensions

“…It is well-known [3] that models of critical phenomena typically possess an upper critical dimension, d c , such that in dimensions d ≥ d c , their thermodynamic behaviour is governed by critical exponents taking simple mean-field values [4]. In contrast to the simplicity of the thermodynamic behaviour, however, the theory of FSS in dimensions above d c is surprisingly subtle, and remains the subject of ongoing debate [5][6][7][8][9][10][11][12]. We will show here that such subtleties can be explained in a simple way, by taking an appropriate geometric perspective.…”

“…In this Letter, we apply a geometric approach to reexamine a long-standing debate concerning the FSS of the n-vector model with d > d c [5][6][7][8][9][10][11][12]. The majority of this debate has focused on the boundary-dependent FSS of the ferromagnetic Ising model when d > d c ; particularly on the case d = 5.…”

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

“…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]…”