# Ising Model Scaling Behaviour on z-Preserving Small-World Networks

## K. A. Hawick and H. A. James

### Abstract

We have investigated the anomalous scaling behaviour of the Ising model on small-world networks based on 2- and 3-dimensional lattices using Monte Carlo simulations. Our main result is that even at low $p$, the shift in the critical temperature $\Delta T_c$ scales as $p^{s}$, with $s \approx 0.50$ for 2-D systems, $s \approx 0.698$ for 3-D and $s \approx 0.75$ for 4-D. We have also verified that a $z$-preserving rewiring algorithm still exhibits small-world effects and yet is more directly comparable with the conventional Ising model; the small-world effect is due to enhanced long-range correlations and not the change in effective dimension. We find the critical exponents $\beta$ and $\nu$ exhibit a monotonic change between an Ising-like transition and mean-field behaviour in 2- and 3-dimensional systems.

Keywords: Ising; scaling; small-world

Citation Information: BiBTeX database for CSTN Notes.

BiBTeX reference:

@TECHREPORT{CSTN-036,
author = {K. A. Hawick and H. A. James},
title = {Ising Model Scaling Behaviour on z-Preserving Small-World Networks},
institution = {Information and Mathematical Sciences, Massey University},
year = {2006},
number = {{arXiv.org Condensed Matter: cond-mat/0611763}},
month = {February},
series = {arXiv.org Condensed Matter: cond-mat/0611763}
}


[ CSTN Index | CSTN BiBTeX ]