## Abstract

We consider the directed mean curvature flow on the plane in a weak Gaussian random environment. We prove that, when started from a sufficiently flat initial condition, a rescaled and recentred solution converges to the Cole–Hopf solution of the KPZ equation. This result follows from the analysis of a more general system of nonlinear SPDEs driven by inhomogeneous noises, using the theory of regularity structures. However, due to inhomogeneity of the noise, the “black box” result developed in the series of works cannot be applied directly and requires significant extension to infinite-dimensional regularity structures. Analysis of this general system of SPDEs gives two more interesting results. First, we prove that the solution of the quenched KPZ equation with a very strong force also converges to the Cole–Hopf solution of the KPZ equation. Second, we show that a properly rescaled and renormalised quenched Edwards–Wilkinson model in any dimension converges to the stochastic heat equation.

Original language | English |
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Pages (from-to) | 1850-1939 |

Number of pages | 90 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 77 |

Issue number | 3 |

Early online date | 3 Oct 2023 |

DOIs | |

Publication status | Published - 31 Mar 2024 |

Externally published | Yes |

### Bibliographical note

Funding Information:The authors would like to thank Ajay Chandra for numerous useful discussions of the renormalisation and coherence in Section 3.2 , as well as Francesco Pedullá for pointing out a mistake in an earlier version of Lemma 7.12 . AG gratefully acknowledges the financial support by the Leverhulme Trust through Hendrik Weber's Philip Leverhulme Prize. MH was supported by the Royal Society through a research professorship. KM was partially supported by NSF grant DMS‐1953859 (transferred to DMS‐2321493).

### Funding

The authors would like to thank Ajay Chandra for numerous useful discussions of the renormalisation and coherence in Section 3.2, as well as Francesco Pedullá for pointing out a mistake in an earlier version of Lemma 7.12. AG gratefully acknowledges the financial support by the Leverhulme Trust through Hendrik Weber's Philip Leverhulme Prize. MH was supported by the Royal Society through a research professorship. KM was partially supported by NSF grant DMS-1953859 (transferred to DMS-2321493). The authors would like to thank Ajay Chandra for numerous useful discussions of the renormalisation and coherence in Section 3.2 , as well as Francesco Pedullá for pointing out a mistake in an earlier version of Lemma 7.12 . AG gratefully acknowledges the financial support by the Leverhulme Trust through Hendrik Weber's Philip Leverhulme Prize. MH was supported by the Royal Society through a research professorship. KM was partially supported by NSF grant DMS‐1953859 (transferred to DMS‐2321493).

Funders | Funder number |
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National Science Foundation | DMS-2321493, DMS‐1953859 |

Leverhulme Trust | |

Royal Society |

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics