Physics – Condensed Matter – Disordered Systems and Neural Networks
Scientific paper
2004-07-07
phys. stat. sol. (b) 241, 2109 (2004)
Physics
Condensed Matter
Disordered Systems and Neural Networks
9 pages, 6 .eps figures, contribution to the Festschrift for Michael Schreiber's 50th birthday
Scientific paper
10.1002/pssb.200404783
Some properties of $d$-dimensional disordered models with long-range random hopping amplitudes are investigated numerically at criticality. We concentrate on the correlation dimension $d_2$ (for $d=2$) and the nearest level spacing distribution $P_c(s)$ (for $d=3$) in both the weak ($b^d \gg 1$) and the strong ($b^d \ll 1$) coupling regime, where the parameter $b^{-d}$ plays the role of the coupling constant of the model. It is found that (i) the extrapolated values of $d_2$ are of the form $d_2=c_db^d$ in the strong coupling limit and $d_2=d-a_d/b^d$ in the case of weak coupling, and (ii) $P_ (s)$ has the asymptotic form $P_c(s)\sim\exp (-A_ds^{\alpha})$ for $s\gg $, with the critical exponent $\alpha=2-a_d/b^d$ for $b^d \gg 1$ and $\alpha=1+c_d b^d$ for $b^d \ll 1$. In these cases the numerical coefficients $A_d$, $a_d$ and $c_d$ depend only on the dimensionality.
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