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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.
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Long-time behaviour for a nonlocal model from directed polymers
Abstract We consider the long time behaviour of solutions to a nonlocal reaction diffusion equation that arises in the study of directed polymers in a random environment. The model is characterized by convolution with a kernel R and an L 2 inner product. In one spatial dimension, we extend a previous result of the authors (arXiv: 2002.02799 ), where only the case R = δ was considered; in particular, we show that solutions spread according to a 2 / 3 power law consistent with the KPZ scaling conjectured for directed polymers. In the special case when R = δ , we find the exact profile of the solution in the rescaled coordinates. We also consider the behaviour in higher dimensions. When the dimension is three or larger, we show that the long-time behaviour is the same as the heat equation in the sense that the solution converges to a standard Gaussian. In contrast, when the dimension is two, we construct a non-Gaussian self-similar solution.
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- PAR ID:
- 10413404
- Date Published:
- Journal Name:
- Nonlinearity
- Volume:
- 36
- Issue:
- 2
- ISSN:
- 0951-7715
- Page Range / eLocation ID:
- 902 to 954
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1088/1361-6544/aca9b3
