论文标题
部分可观测时空混沌系统的无模型预测
Localization of the continuum directed random polymer
论文作者
论文摘要
我们认为中间疾病制度中的连续性随机聚合物（CDRP）模型是从$ 1+1 $尺寸定向聚合物的缩放限制。我们表明，对于长度$ t $的点对点聚合物和（0,1）$的任何$ p \，路径上的淬灭密度是$ pt $ a $ pt $的距离，当围绕其随机模式$ \ mathcal {m} _ {m} _ {p，t，t} $在法律上转换为explacit of stricit nofer tate stepry fortial take fortion nontive fortim fortion n而不按任何spection unfty os t the Origin。同样，在长度$ t $的点对上聚合物的情况下，围绕其随机模式$ \ MATHCAL {M} _ {*，T} $的路径端点的淬灭密度在法律上收敛于显式随机密度。限制随机密度与$ e^{ - \ Mathcal {r}_σ（x）} $成比例，其中$ \ nathcal {r}_σ（x）$是具有适当的扩散系数$σ$的两边3D Bessel过程。 In addition, the laws of the random modes $\mathcal{M}_{*,t}$, $\mathcal{M}_{p,t}$ themselves converge in distribution upon $t^{2/3}$ scaling to the maximizer of $\operatorname{Airy}_2$ process minus a parabola and points on the geodesics of the directed landscape 分别。上面所述的我们的本地化结果为民间传说“最喜欢的地区”猜想提供了肯定的案例。我们的证明技术还使我们能够证明KPZ方程的属性，例如Ergodicity和限制最大值的Bessel行为。
We consider the continuum directed random polymer (CDRP) model that arises as a scaling limit from $1+1$ dimensional directed polymers in the intermediate disorder regime. We show that for a point-to-point polymer of length $t$ and any $p\in (0,1)$, the quenched density of the point on the path which is $pt$ distance away from the origin when centered around its random mode $\mathcal{M}_{p,t}$ converges in law to an explicit random density function as $t\to\infty$ without any scaling. Similarly, in the case of point-to-line polymers of length $t$, the quenched density of the endpoint of the path when centered around its random mode $\mathcal{M}_{*,t}$ converges in law to an explicit random density. The limiting random densities are proportional to $e^{-\mathcal{R}_σ(x)}$ where $\mathcal{R}_σ(x)$ is a two-sided 3D Bessel process with appropriate diffusion coefficient $σ$. In addition, the laws of the random modes $\mathcal{M}_{*,t}$, $\mathcal{M}_{p,t}$ themselves converge in distribution upon $t^{2/3}$ scaling to the maximizer of $\operatorname{Airy}_2$ process minus a parabola and points on the geodesics of the directed landscape respectively. Our localization results stated above provide an affirmative case of the folklore "favorite region" conjecture. Our proof techniques also allow us to prove properties of the KPZ equation such as ergodicity and limiting Bessel behaviors around the maximum.
