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In the present work, we study the two-dimensional anisotropic KPZ equation(AKPZ), which is formally given by \begin{equation*} \partial_t h=\tfrac12\Delta h + \lambda ((\partial_1 h)^2)-(\partial_2 h)^2) +\xi\,, \end{equation*}where $\xi$ denotes a space-time white noise and $\lambda>0$ is the so-calledcoupling constant. The AKPZ equation is a {\it critical} SPDE, meaning that notonly it is analytically ill-posed but also the breakthrough path-wisetechniques for singular SPDEs [M. Hairer, Ann. Math. 2014] and [M. Gubinelli,P. Imkeller and N. Perkowski, Forum of Math., Pi, 2015] are not applicable. Asshown in [G. Cannizzaro, D. Erhard, F. Toninelli, arXiv, 2020], the equationregularised at scale $N$ has a diffusion coefficient that divergeslogarithmically as the regularisation is removed in the limit $N\to\infty$.Here, we study the \emph{weak coupling limit} where$\lambda=\lambda_N=\hat\lambda/\sqrt{\log N}$: this is the correct scaling thatguarantees that the nonlinearity has a still non-trivial but non-divergenteffect. In fact, as $N\to\infty$ the sequence of equations converges to thelinear stochastic heat equation \begin{equation*} \partial_t h =\tfrac{\nu_{\rmeff}}{2} \Delta h + \sqrt{\nu_{\rm eff}}\xi\,, \end{equation*} where $\nu_{\rmeff} >1$ is explicit and depends non-trivially on $\hat\lambda$. This is thefirst full renormalization-type result for a critical, singular SPDE whichcannot be linearised via Cole-Hopf or any other transformation.

Weak coupling limit of the Anisotropic KPZ equation