Hairer, Martin; Xu, Weijun Large scale limit of interface fluctuation models. (English) Zbl 1453.60120 Ann. Probab. 47, No. 6, 3478-3550 (2019). The authors extend the weak universality of KPZ to weakly asymmetric interface models with general growth mechanisms \(F:\mathbb R\to\mathbb R\) being a \(C^{7+\alpha}\)-smooth symmetric function with sub-polynomial growth of the derivatives at infinity. More precisely, let \(\widehat{\xi}\) be a space-time Gaussian random field with a smooth short range covariance that integrates to \(1\), let \(\xi\) be a space-time white noise and \(\rho:\mathbb R^2\to\mathbb R\) a smooth compactly supported function which integrates to \(1\) (in the space-time) and which is symmetric in the space variable. Let the law of \(\widehat{\xi}\) coincide with the law of \(\xi*\rho\). Roughly speaking, if \[ \partial_th_\varepsilon=\partial^2_xh_\varepsilon+\varepsilon^{-1}F(\varepsilon^\frac{1}{2}\partial_xh_\varepsilon)+\xi_\varepsilon-C_\varepsilon\tag{1} \] holds on the one-dimensional torus \(\mathbf{T}\) where \(\xi_\varepsilon(t,x)=\varepsilon^{-\frac{3}{2}}\widehat{\xi}(t/\varepsilon^2,x/\varepsilon)\), \(C_\varepsilon\) are suitably chosen constants and \(h_\varepsilon(0)\) converges in a suitable sense to some \(h_0\in C^\eta(\mathbf T)\) for a suitable \(\eta<\frac{1}{2}\), then \(h_\varepsilon\) converges in probability in \(C^\eta([0,T];\mathbf T)\) to the Hopf-Cole solution to the KPZ equation \[ \partial_th=\partial^2_xh+a(\partial_xh)^2+\xi,\qquad h(0)=h_0, \] with an explicitly given coupling constant \(a\), for every \(T>0\). In particular, if \[ \partial_th=\partial^2_xh+\sqrt{\varepsilon}F(\partial_xh)+\widehat{\xi}, \] then \(h_\varepsilon(t,x)=\varepsilon^\frac{1}{2}h(t/\varepsilon^2,x/\varepsilon)-C_\varepsilon t\) satisfies (1). 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