Long-time behavior for a nonlocal model from directed polymers

TL;DR

This paper studies the long-time behavior of solutions to a nonlocal reaction diffusion equation from directed polymers, revealing different spreading and profile behaviors in various dimensions, including KPZ scaling in 1D and Gaussian convergence in higher dimensions.

Contribution

The authors extend previous results to include nonlocal kernels in 1D, derive exact solutions for the delta kernel, and analyze the asymptotic behavior in higher dimensions, including the construction of a non-Gaussian solution in 2D.

Findings

Solutions in 1D spread with a 2/3 power law consistent with KPZ scaling.

Exact solution profiles are obtained for the delta kernel case.

In dimensions three and higher, solutions converge to Gaussian profiles.

Abstract

We consider the long time behavior of solutions to a nonlocal reaction diffusion equation that arises in the study of directed polymers. 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 $R =\delta$ was considered; in particular, we show that solutions spread according to a $2/3$ power law consistent with the KPZ scaling conjectured for direct polymers. In the special case when $R = \delta$, we find the exact profile of the solution in the rescaled coordinates. We also consider the behavior in higher dimensions. When the dimension is three or larger, we show that the long-time behavior 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.