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

Sharp Tunneling Estimates for a Double-Well Model in Infinite Dimension