Higher period ordered phases on the Bethe lattice

We solve for period-three ordered phases on the infinite-coordination Bethe lattice. The model we have chosen to analyze is the spinless Falicov-Kimball model (although we believe these results should have more general validity). Contrary to the belief of many researchers in the field, the Bethe lattice can support higher period ordered phases even though there is no momentum space'' associated with the lattice. These higher period phases can be rigorously shown to appear at zero temperature and a numerical analysis indicates that their thermodynamic phase transition from the homogeneous, period-two, or higher period ordered phases is generically a first-order transition.