An Eyring–Kramers law for the stochastic Allen–Cahn equation in dimension two

We study spectral Galerkin approximations of an Allen–Cahn equation over the two-dimensional torus perturbed by weak space-time white noise of strength √ ε. We introduce a Wick renormalisation of the equation in order to have a system that is well-defined as the regularisation is removed. We show sharp upper and lower bounds on the transition times from a neighbourhood of the stable configuration −1 to the stable configuration 1 in the asymptotic regime ε → 0. These estimates are uniform in the discretisation parameter N , suggesting an Eyring–Kramers formula for the limiting renormalised stochastic PDE. The effect of the " infinite renormalisation " is to modify the prefactor and to replace the ratio of determinants in the finite-dimensional Eyring– Kramers law by a renormalised Carleman–Fredholm determinant.