L$, and the other in the high-temperature phase, making sure that $\xi_\infty \ll
L$ to reduce finite-size corrections as much as possible. The quantities we have measured are the internal energy, specific heat, correlation length, average cluster size, susceptibility, and higher order moments. In the critical region this can be done most efficiently by combining the single-cluster update with multi-histogram sampling techniques, and in the high-temperature phase by using improved estimators for measurements.

Improved estimators
-------------------

An improved “cluster estimator” for the spin-spin correlation function in the high-temperature phase, $G({\mbox{\boldmath $x$}}-{\mbox{\boldmath $x$}}') \equiv \langle
\vec{s}({\mbox{\boldmath $x$}}) \cdot  \vec{s}({\mbox{\boldmath $x$}}') \rangle$, is given by [@r10a] $$\tilde{G}({\mbox{\boldmath $x$}}-{\mbox{\boldmath $x$}}') = 3 \frac{V}{|C|} \vec{r} \cdot \vec{s}({\mbox{\boldmath $x$}}) \,\,
\vec{r} \cdot \vec{s}({\mbox{\boldmath $x$}}') \Theta_C({\mbox{\boldmath $x$}}) \Theta_C({\mbox{\boldmath $x$}}'),
\label{eq:3}$$ where $\vec{r}$ is the normal of the mirror plane used in the construction of the cluster of size $|C|$ and $\Theta_C({\mbox{\boldmath $x$}})$ is its characteristic function (=1 if ${\mbox{\boldmath $x$}} \in C$ and 0 otherwise). For the Fourier transform, $\hat{G}({\mbox{\boldmath $k$}}) = \sum_{{\mbox{\boldmath $x$}}} G({\mbox{\boldmath $x$}}) \exp(-i{\mbox{\boldmath $k$}}{\mbox{\boldmath $x$}})$, this implies the improved estimator $$\tilde{\hat{G}}({\mbox{\boldmath $k$}}) = \frac{3}{|C|} \left[ \left( \sum_{{\mbox{\boldmath $x$}} \in C}
\vec{r} \cdot \vec{s}({\mbox{\boldmath $x$}}) \cos {\mbox{\boldmath $k$}} {\mbox{\boldmath $x$}} \right)^2 +
\left( 