  This work considers the two-dimensional Allen-Cahn equation
  $$ \partial_t u = \frac{1}{2}\Delta u + \mathfrak{m}\, u -u^3\;, \quad
u(0,x)= \eta (x)\;, \qquad \forall (t,x) \in [0, \infty) \times \mathbb{R}^{2}
\;, $$ where the initial condition $ \eta $ is a two-dimensional white noise,
which lies in the scaling critical space of initial data to the equation. In a
weak coupling scaling, we establish a Gaussian limit with nontrivial size of
fluctuations, thus casting the nonlinearity as marginally relevant. The result
builds on a precise analysis of the Wild expansion of the solution and an
understanding of the underlying stochastic and combinatorial structure. This
gives rise to a representation for the limiting variance in terms of Butcher
series associated to the solution of an ordinary differential equation.
