We are interested in this talk by the $L^2$ spectral gap of a large system of strongly coupled particles subject to a double-well potential. This system can be seen as a spatially discrete approximation of the stochastic Allen-Cahn equation on the one-dimensional torus. We prove upper and lower bounds for the leading term of the spectral gap in the small temperature regime with uniform control in the system size, the upper bound being given by an Eyring-Kramers-type formula. Under suitable assumptions on the growth of the system size with respect to the temperature, the asymptotic optimality of the upper bound is also established.
These results can be reformulated in terms of a semiclassical Witten Laplacian in large dimension.
This is a work in collaboration with Giacomo Di Gesù.