  We assume one site measures without a boundary $e^{-\phi(x)}dx/Z$ that
satisfy a log-Sobolev inequality. We prove that if these measures are perturbed
with quadratic interactions, then the associated infinite dimensional Gibbs
measure on the lattice always satisfies a log-Sobolev inequality. Furthermore,
we present examples of measures that satisfy the inequality with a phase that
goes beyond convexity at infinity.
