  We consider the stochastic quantization of a quartic double-well energy
functional in the semiclassical regime and derive optimal asymptotics for the
exponentially small splitting of the ground state energy. Our result provides
an infinite-dimensional version of some sharp tunneling estimates known in
finite dimensions for semiclassical Witten Laplacians in degree zero. From a
stochastic point of view it proves that the $L^2$ spectral gap of the
stochastic one-dimensional Allen-Cahn equation in finite volume satisifies a
Kramers-type formula in the limit of vanishing noise. We work with
finite-dimensional lattice approximations and establish semiclassical estimates
which are uniform in the dimension. Our key estimate shows that the constant
separating the two exponentially small eigenvalues from the rest of the
spectrum can be taken independently of the dimension.
