#### Computational Science Technical Note CSTN-036

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

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K. A. Hawick and H. A. James

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Archived November 2006

**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

**Full Document Text:**
arXiv PDF version; local PDF version.

**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}
}

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