论文标题
连续体的短期和长时间路径紧密到定向的随机聚合物
Short- and long-time path tightness of the continuum directed random polymer
论文作者
论文摘要
我们考虑点对点连续元素定向的随机聚合物（$ \ Mathsf {CDRP} $）模型，该模型以缩放限制为$ 1+1 $ dimensional dimensional定向聚合物在中等疾病中。我们表明，当$ t \ downarrow 0 $时，$ t $ t $长度$ t $长度$ t $长度$ t $ thength $ t $的退火定律收敛于布朗桥。如果$ t $很大，我们表明点对点$ \ mathsf {cdrp} $的横向波动由$ 2/3 $ endent支配。更确切地说，随着$ t $倾向于无穷大，我们证明了将长度$ t $的点对点$ \ mathsf {cdrp} $ to $ t $缩放长度缩放$ t $时的紧密度，而路径的波动则通过$ t^{2/3} $。 $ 2/3 $的指数很紧，因此，重新定制路径的单点分布会汇聚到定向景观的大地测量学。可以增强这种点的收敛性，以提高过程级模型。我们的短期和长期紧缩结果也扩展到点对上的$ \ mathsf {cdrp} $。在证明我们的主要结果的过程中，我们建立了长期$ \ MATHSF {CDRP} $的定量版本的连续性估算模量，这具有独立的兴趣。
We consider the point-to-point continuum directed random polymer ($\mathsf{CDRP}$) model that arises as a scaling limit from $1+1$ dimensional directed polymers in the intermediate disorder regime. We show that the annealed law of a point-to-point $\mathsf{CDRP}$ of length $t$ converges to the Brownian bridge under diffusive scaling when $t \downarrow 0$. In case that $t$ is large, we show that the transversal fluctuations of point-to-point $\mathsf{CDRP}$ are governed by the $2/3$ exponent. More precisely, as $t$ tends to infinity, we prove tightness of the annealed path measures of point-to-point $\mathsf{CDRP}$ of length $t$ upon scaling the length by $t$ and fluctuations of paths by $t^{2/3}$. The $2/3$ exponent is tight such that the one-point distribution of the rescaled paths converges to the geodesics of the directed landscape. This point-wise convergence can be enhanced to process-level modulo a conjecture. Our short and long-time tightness results also extend to point-to-line $\mathsf{CDRP}$. In the course of proving our main results, we establish quantitative versions of quenched modulus of continuity estimates for long-time $\mathsf{CDRP}$ which are of independent interest.
