  We study the $L^2$ spectral gap of a large system of strongly coupled
diffusions on unbounded state space and 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 is given by an
Eyring-Kramers-type formula. The lower bound is proven to hold also for the
logarithmic Sobolev constant. We establish a sufficient condition for the
asymptotic optimality of the upper bound and show that this condition is
fulfilled under suitable assumptions on the growth of the system size. Our
results can be reformulated in terms of a semiclassical Witten Laplacian in
large dimension.
