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

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

2006-11-29

arxiv.org/abs/cond-mat/0611763v1

Physics

Condensed Matter

Disordered Systems and Neural Networks

19 Pages, 6 figures

Scientific paper

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.

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