The distribution function
PL(s) of the local order parameter
s in finite blocks of size
Ld is studied for Ising models for dimensionalities
d=2,3,and4 by Monte Carlo methods. A real-space renormalization group based on phenomenological scaling yields fairly accurate results for rather small
L (e.g., the standard exponents $\ensuremath{\beta}$ and $\ensuremath{\nu}$ for
d=3 are found as $\frac{2\ensuremath{\beta}}{\ensuremath{\nu}}=1.03\ifmmode\pm\else\textpm\fi{}0.01$, $\frac{1}{\ensuremath{\nu}}=1.60\ifmmode\pm\else\textpm\fi{}0.05$). The method can easily be generalized to arbitrary Hamiltonians, including spin dimensionalities
n>1.
We present some exact results on percolation properties of the Ising model, when the range of the percolating bonds is larger than the nearest neighbours. We show that the phase diagram for next-nearest neighbour percolation can be exactly obtained from the nearest-neighbour case, which implies that the percolation threshold, , is still equal to the Ising critical temperature . In addition, we present Monte Carlo calculations of the finite size behaviour of the correlated resistor network defined on the Ising model. The thermal exponent, t, of the conductivity that follows from it is found to be . We observe no corrections to scaling in its finite size behaviour.
We obtain the critical parameters for the site-percolation problem on the square lattice to a high degree of accuracy (comparable to that of series expansions) by using a Monte Carlo position-space renormalization-group procedure directly on the site-occupation probability. Our method involves calculating recursion relations using progressively larger lattice rescalings,
b. We find smooth sequences for the value of the critical percolation concentration
pc(b) and for the scaling powers
yp(b) and
yh(b). Extrapolating these sequences to the limit $b\ensuremath{\rightarrow}\ensuremath{\infty}$ leads to quite accurate numerical predictions. Further, by considering other weight functions or "rules" which also embody the essential connectivity feature of percolation, we find that the numerical results in the infinite-cell limit are in fact "rule independent." However, the actual fashion in which this limit is approached does depend upon the rule chosen. A connection between extrapolation of our renormalization-group results and finite-size scaling is made. Furthermore, the usual finite-size scaling arguments lead to independent estimates of
pc and
yp. Combining both the large-cell approach and the finite-size scaling results, we obtain ${y}_{p}=0.7385\ifmmode\pm\else\textpm\fi{}0.0080$ and ${y}_{h}=1.898\ifmmode\pm\else\textpm\fi{}0.003$. Thus we find ${\ensuremath{\alpha}}_{p}=\ensuremath{-}0.708\ifmmode\pm\else\textpm\fi{}0.030$, ${\ensuremath{\beta}}_{p}=0.138(+0.006,\ensuremath{-}0.005)$, ${\ensuremath{\gamma}}_{p}=2.432\ifmmode\pm\else\textpm\fi{}0.035$, ${\ensuremath{\delta}}_{p}=18.6\ifmmode\pm\else\textpm\fi{}0.6$, ${\ensuremath{\nu}}_{p}=1.354\ifmmode\pm\else\textpm\fi{}0.015$, and $2\ensuremath{-}{\ensuremath{\eta}}_{p}=1.796\ifmmode\pm\else\textpm\fi{}0.006$. The site-percolation threshold is found for the square lattice at ${p}_{c}=0.5931\ifmmode\pm\else\textpm\fi{}0.0006$. We note that our calculated value of ${\ensuremath{\nu}}_{p}$ is in considerably better agreement with the proposal of Klein et al. that ${\ensuremath{\nu}}_{p}=\frac{\mathrm{ln}\sqrt{3}}{\mathrm{ln}(\frac{3}{2})}\ensuremath{\cong}1.3548$, than with den Nijs' recent conjecture, which predicts ${\ensuremath{\nu}}_{p}=\frac{4}{3}$. However, our results cannot entirely rule out the latter possibility.
The critical behaviour of the spin- Ising model with a transverse field at zero temperature in triangular and square lattices is investigated through a finite-size rescaling transformation. This technique enables the critical properties of infinite systems to be calculated from the scaling properties of systems of finite size, in a generalisation of the so-called phenomenological renormalisation group. The critical field and correlation length exponent calculated using systems of fairly small sizes have reasonable accuracy, but calculations on larger clusters are limited by computer time and storage.