  We discuss Dyson's argument that the vacuum is unstable under a change g^2 ->
- g^2, in the context of lattice gauge theory. For compact gauge groups, the
partition function is well defined at negative g^2, but the average plaquette P
has a discontinuity when g^2 changes sign. This reflects a change of vacuum
rather than a loss of vacuum. In addition, P has poles in the complex g^2
plane, located at the complex zeros of the partition function (Fisher's zeros).
We discuss the relevance of these singularities for lattice perturbation
theory. We present new methods to locate Fisher's zeros using numerical values
for the density of state in SU(2) and U(1) pure gauge theory. We briefly
discuss similar issues for O(N) nonlinear sigma models where the local
integrals are also over compact spaces.
