  We prove an asymptotic coupling theorem for the $2$-dimensional Allen--Cahn
equation perturbed by a small space-time white noise. We show that with
overwhelming probability two profiles that start close to the minimisers of the
potential of the deterministic system contract exponentially fast in a suitable
topology. In the $1$-dimensional case a similar result was shown in
\cite{MS88,MOS89}.
  It is well-known that in more than one dimension solutions of this equation
are distribution-valued, and the equation has to be interpreted in a
renormalised sense. Formally, this renormalisation corresponds to moving the
minima of the potential infinitely far apart and making them infinitely deep.
We show that despite this renormalisation, solutions behave like perturbations
of the deterministic system without renormalisation: they spend large stretches
of time close to the minimisers of the (un-renormalised) potential and the
exponential contraction rate of different profiles is given by the second
derivative of the potential in these points.
  As an application we prove an Eyring--Kramers law for the transition times
between the stable solutions of the deterministic system for fixed initial
conditions.
