Ordering and demixing transitions in multicomponent Widom-Rowlinson models.

We use Monte Carlo techniques and analytical methods to study the phase diagram of multicomponent Widom-Rowlinson models on a square 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 between two and six there is a direct transition from the gas phase at z < z d (M) to a demixed phase consisting mostly of one species at z > z d (M) while for M 7 there is an intermediate \crystal phase" for z lying between z c (M) and z d (M). In this phase, which is driven by entropy, particles, independent of species, preferentially occupy one of the sublattices, i.e. spatial symmetry 1 but not particle symmetry is broken. The transition at z d (M) appears to be rst order for M 5 putting it in the Potts model universality class. For large M the transition between the crystalline and demixed phase at z d (M) can be proven to be rst order with z d (M) M ?2+1=M +:::, while z c (M) is argued to behave as cr =M, with cr the value of the fugacity at which the one component hard square lattice gas has a transition, and to be always of the Ising type. Explicit calculations for the Bethe lattice with the coordination number q = 4 give results similar to those for the square lattice except that the transition at z d (M) becomes rst order at M > 2. This happens for all q, consistent with the model being in the Potts universality class.