We prove that a convex phase may be perturbed into a non-convex phase prese
rving the spectral gap properties of the unbounded spin system with nearest
neighbour interaction associated to this potential. The proof is based on
Heifer's method hat reduces the spectral properties of the unbounded spin s
ystem to some uniform spectral gap of the one-dimensional phase. We then ma
ke: use of Hardy's criterion for Poincare inequalities on the real line to
construct our examples. (C) 2001 Academic Press.