The Allen-Cahn equation with weakly critical random initial datum

This work considers the two-dimensional Allen-Cahn equation $\begin{array}{c}{\partial }_{t}u=\frac{1}{2}\Delta u+mu-u^3,u(0,x)=\eta \left(x\right),\forall (t,x)\in [0,\infty )\times R^2,\end{array}$ 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.