  We study the evolution of the cluster correlation function and its
richness-dependence from z = 0 to z = 3 using large-scale cosmological
simulations. A standard flat LCDM model with \Omega_m = 0.3 and, for
comparison, a tilted \Omega_m = 1 model, TSCDM, are used. The evolutionary
predictions are presented in a format suitable for direct comparisons with
observations. We find that the cluster correlation strength increases with
redshift: high redshift clusters are clustered more strongly (in comoving
scale) than low redshift clusters of the same mass. The increased correlations
with redshift, in spite of the decreasing mass correlation strength, is caused
by the strong increase in cluster bias with redshift: clusters represent higher
density peaks of the mass distribution as the redshift increases. The
richness-dependent cluster correlation function, presented as the
correlation-scale versus cluster mean separation relation, R_0 - d, is found to
be, remarkably, independent of redshift to z <~ 2 for LCDM and z <~ 1 for TCDM
(for a fixed correlation function slope and cluster mass within a fixed
comoving radius). The non-evolving R_0 - d relation implies that both the
comoving clustering scale and the cluster mean separation increase with
redshift for the same mass clusters so that the R_0 - d relation remains
essentially unchanged. The evolution of the R_0 - d relation from z ~ 0 to z ~
3 provides an important new tool in cosmology; it can be used to break
degeneracies that exist at z ~ 0 and provide precise determination of
cosmological parameters.
