The Allen–Cahn equation with weakly critical random initial datum

This work considers the two-dimensional Allen–Cahn equation \(\begin{aligned} \partial _t u = \frac{1}{2}\Delta u + \mathfrak {m}\, u -u^3, \quad u(0,x)= \eta (x), \qquad \forall (t,x) \in [0, \infty ) \times {\textbf {R}}^{2}, \end{aligned}\) $$\begin{array}{c}\hfill {\partial }_{t}u=\frac{1}{2}\mathrm{\Delta }u+\mathfrak{m}u-u^3,u(0,x)=\eta (x),\forall (t,x)\in [0,\infty )\times {\mathbf{R}}^2,\end{array}$$ where the initial condition \( \eta \) $\eta$ 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.