  Variational monotone recurrence relations arise in solid state physics as
generalizations of the Frenkel-Kontorova model for a ferromagnetic crystal. For
such problems, Aubry-Mather theory establishes the existence of "ground states"
or "global minimizers" of arbitrary rotation number.
  A nearest neighbor crystal model is equivalent to a Hamiltonian twist map. In
this case, the global minimizers have a special property: they can only cross
once. As a nontrivial consequence, every one of them has the Birkhoff property.
In crystals with a larger range of interaction and for higher order recurrence
relations, the single crossing property does not hold and there can exist
global minimizers that are not Birkhoff.
  In this paper we investigate the crossings of global minimizers. Under a
strong twist condition, we prove the following dichotomy: they are either
Birkhoff, and thus very regular, or extremely irregular and nonphysical: they
then grow exponentially and oscillate. For Birkhoff minimizers, we also prove
certain strong ordering properties that are well known for twist maps.
