The Allen-Cahn equation with weakly critical random initial datum

This work considers the two-dimensional Allen-Cahn equation $${\mathrm{\partial }}_{t}u=\frac{1}{2}\mathrm{\Delta }u+\mathfrak{m}u-u^3,u(0,x)=\eta (x),\mathrm{\forall }(t,x)\in [0,\mathrm{\infty })\times {\mathbb{R}}^2,$$ 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.