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 interal 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.
Joint work with Ron Peled.
 
org: GIULIANI Alessandro