  We study the invariant measure of the one-dimensional stochastic Allen-Cahn
equation for a small noise strength and a large but finite system. We endow the
system with inhomogeneous Dirichlet boundary conditions that enforce at least
one transition from -1 to 1. (Our methods can be applied to other boundary
conditions as well.) We are interested in the competition between the energy
that should be minimized due to the small noise strength and the entropy that
is induced by the large system size.
  Our methods handle system sizes that are exponential with respect to the
inverse noise strength, up to the critical exponential size predicted by the
heuristics. We capture the competition between energy and entropy through upper
and lower bounds on the probability of extra transitions between -1 and 1.
These bounds are sharp on the exponential scale and imply in particular that
the probability of having one and only one transition from -1 to +1 is
exponentially close to one. In addition, we show that the position of the
transition layer is uniformly distributed over the system on scales larger than
the logarithm of the inverse noise strength.
  Our arguments rely on local large deviation bounds, the strong Markov
property, the symmetry of the potential, and measure-preserving reflections.
