Xifeng Su, Rafael de la Llave
A CONTINUOUS FAMILY OF EQUILIBRIA IN FERROMAGNETIC MEDIA ARE GROUND STATES
(186K, pdf)

ABSTRACT.  We show that a foliation of 
equilibria (a continuous family of equilibria 
whose graph covers all the configuration space) 
in ferromagnetic models are ground states. 
The result we prove is very general, and it applies 
to models with long range interactions and many body. 
As an application, 
we consider several models of networks of interacting particles including models of Frenkel-Kontorova type on $\mathbb{Z}^d$ and one-dimensional quasi-periodic media. 
The result above is an analogue of several results in the calculus 
variations (fields of extremals) and in PDE's. Since the models 
we consider are discrete and long range, new proofs need to 
be given. 
We also note that the main hypothesis of our result 
(the existence of foliations of equilibria) is the 
conclusion (using KAM theory) of several recent papers. Hence, we 
obtain that the KAM solutions recently established are minimizers 
when the interaction is ferromagnetic (and transitive).