  This article is devoted to the analysis of the weak rates of convergence of
schemes introduced by the authors in a recent work, for the temporal
discretization of the stochastic Allen-Cahn equation driven by space-time white
noise. The schemes are based on splitting strategies and are explicit. We prove
that they have a weak rate of convergence equal to $\frac12$, like in the more
standard case of SPDEs with globally Lipschitz continuous nonlinearity. To deal
with the polynomial growth of the nonlinearity, several new estimates and
techniques are used. In particular, new regularity results for solutions of
related infinite dimensional Kolmogorov equations are established. Our
contribution is the first one in the literature concerning weak convergence
rates for parabolic semilinear SPDEs with non globally Lipschitz
nonlinearities.
