Sharp tunneling estimates for a double-well model in infinite dimension

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 satisfies 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. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

The study investigates stochastic quantization of a quartic double-well energy functional and derives optimal asymptotics for the exponentially small splitting of the ground state energy in the semiclassical regime. The results demonstrate that the L-2 spectral gap of the one-dimensional stochastic Allen-Cahn equation in finite volume satisfies a Kramers-type formula in the limit of vanishing noise, with tunneling estimates being uniform in dimension. Key estimates show that the constant separating two exponentially small eigenvalues from the rest of the spectrum can be independent of dimension.