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We consider a class of weakly asymmetric continuous microscopic growth models with polynomial smoothing mechanisms, general nonlinearities and a Poisson type noise. We show that they converge to the KPZ equation after proper rescaling and re-centering, where the coupling constant depends nontrivially on all details of the smoothing and growth mechanisms in the microscopic model. This confirms some of the predictions in HQ18. The proof builds on the general discretisation framework of regularity structures (EH19), and employs the idea of using the spectral gap inequality to control stochastic objects as developed and systematised in LOTT21, HS24, together with a new observation on structures of the Malliavin derivatives in our situation.
Kong et al. (Sun,) studied this question.