We use very large cosmological N--body simulations to obtain accurate predictions for the two-point correlations and power spectra of mass-limited samples of galaxy clusters. We consider two currently popular cold dark matter (CDM) cosmogonies, a critical density model ($tau$CDM) and a flat low density model with a cosmological constant ($Lambda$CDM). Our simulations each use $10^9$ particles to follow the mass distribution within cubes of side $2h^{-1}$Gpc ($tau$CDM) and $3h^{-1}$Gpc ($Lambda$CDM) with a force resolution better than $10^{-4}$ of the cube side. We investigate how the predicted cluster correlations increase for samples of increasing mass and decreasing abundance. Very similar behaviour is found in the two cases. The correlation length increases from $r_0=12$ -- 13$h^{-1}$Mpc for samples with mean separation $d_{rm c}=30h^{-1}$Mpc to $r_0=22$-- 27$h^{-1}$Mpc for samples with $d_{rm c}=100h^{-1}$Mpc. The lower value here corresponds to $tau$CDM and the upper to $Lambda$CDM. The power spectra of these cluster samples are accurately parallel to those of the mass over more than a decade in scale. Both correlation lengths and power spectrum biases can be predicted to better than 10% using the simple model of Sheth, Mo & Tormen (2000). This prediction requires only the linear mass power spectrum and has no adjustable parameters. We compare our predictions with published results for the APM cluster sample. The observed variation of correlation length with richness agrees well with the models, particularly for $Lambda$CDM. The observed power spectrum (for a cluster sample of mean separation $d_{rm c}=31h^{-1}$Mpc) lies significantly above the predictions of both models.
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