A condition for long-range order in discrete spin systems
We present a new condition for the existence of long-range order in
discrete spin systems, which emphasizes the role of entropy and high
dimension. The condition applies to all symmetric nearest-neighbor discrete
spin systems with an internal symmetry of `dominant phases'. Specific
applications include a proof of Kotecky's conjecture (1985) on
anti-ferromagnetic Potts models, a strengthening of results of
Lebowitz-Gallavotti (1971) and Runnels-Lebowitz (1975) on Widom-Rowlinson
models and of Burton-Steif (1994) on shifts of finite type, and a
significant extension of results of Engbers-Galvin (2012) on random graph
homomorphisms on the hypercube.
No background in statistical physics will be assumed and all terms will be
explained thoroughly. Joint work with Ron Peled.