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

TL;DR

This paper derives optimal tunneling estimates for a double-well model in infinite dimensions, connecting stochastic quantization, spectral gaps, and semiclassical analysis, with uniform bounds across finite-dimensional approximations.

Contribution

It extends finite-dimensional tunneling estimates to infinite dimensions, providing uniform semiclassical bounds and a Kramers-type formula for stochastic PDEs.

Findings

Optimal asymptotics for ground state energy splitting

Uniform semiclassical estimates in finite-dimensional lattice approximations

Dimension-independent constant for spectral gap estimates

Abstract

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.