  We study the non restoration of symmetries with a local order parameter in
field theory at finite temperature. After giving an interpretation of the
phenomenon, we show that hierarchy problems are a necessary condition for its
realization in renormalizable theories. We then use a large N treatment, and
find that high temperature symmetry can stay broken in this limit (in
opposition with a previous result), and further that the running of couplings
reinforces the effect in the simplest model with two scalars.
