Phase Transitions in the Multicomponent Widom-Rowlinson Model and in Hard Cubes on the BCC--Lattice

We use Monte Carlo techniques and analytical methods to study the phase diagram of the M--component Widom-Rowlinson model on the bcc-lattice: there are M species all with the same fugacity z and a nearest neighbor hard core exclusion between unlike particles. Simulations show that for M greater or equal 3 there is a ``crystal phase'' for z lying between z_c(M) and z_d(M) while for z > z_d(M) there are M demixed phases each consisting mostly of one species. For M=2 there is a direct second order transition from the gas phase to the demixed phase while for M greater or equal 3 the transition at z_d(M) appears to be first order putting it in the Potts model universality class. For M large, Pirogov-Sinai theory gives z_d(M) ~ M-2+2/(3M^2) + ... . In the crystal phase the particles preferentially occupy one of the sublattices, independent of species, i.e. spatial symmetry but not particle symmetry is broken. For M to infinity this transition approaches that of the one component hard cube gas with fugacity y = zM. We find by direct simulations of such a system a transition at y_c ~ 0.71 which is consistent with the simulation z_c(M) for large M. This transition appears to be always of the Ising type.