  We study spectral Galerkin approximations of an Allen--Cahn equation over the
two-dimensional torus perturbed by weak space-time white noise of strength
$\sqrt{\varepsilon}$. 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 $\varepsilon \to 0$. 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.
