论文标题
有向景观的非典型恒星
Atypical stars on a directed landscape geodesic
论文作者
论文摘要
在随机的几何形状中，一个反复出现的主题是，从典型的点部分散发出的任何两个大地测量学都以严格的正距离距离上点，我们将其称为$ 1 $ - 标准之类的点。然而，零的非典型恒星集零集，这种结合失败的点通常是不可数的，并且这些集合的相应的Hausdorff尺寸已被大量研究，用于各种模型，包括有向景观，Liouville量子量子重力和Brownian Map。在本文中，我们考虑了定向的景观 - dauvergne-Ortmann-Virág'18的最后一段渗透的缩放限制 - 并探讨了位于地理位生上的非典型恒星集的Hausdorff维度。我们表明，上述维度几乎肯定等于$ 1/3 $。这与Ganguly-Zhang '22形成鲜明对比的是，在该公司表明该行上的非典型恒星$ \ {x = 0 \} $具有尺寸$ 2/3 $。从$ 2/3 $到$ 1/3 $的尺寸减少了围绕特殊行为的地球围绕环境平滑的定量表现。
In random geometry, a recurring theme is that any two geodesics emanating from a typical point part ways at a strictly positive distance from the above point, and we call such points as $1$-stars. However, the measure zero set of atypical stars, the points where such coalescence fails, is typically uncountable and the corresponding Hausdorff dimensions of these sets have been heavily investigated for a variety of models including the directed landscape, Liouville quantum gravity and the Brownian map. In this paper, we consider the directed landscape -- the scaling limit of last passage percolation as constructed in the work Dauvergne-Ortmann-Virág '18 -- and look into the Hausdorff dimension of the set of atypical stars lying on a geodesic. We show that the above dimension is almost surely equal to $1/3$. This is in contrast to Ganguly-Zhang '22, where it was shown that set of atypical stars on the line $\{x=0\}$ has dimension $2/3$. This reduction of the dimension from $2/3$ to $1/3$ yields a quantitative manifestation of the smoothing of the environment around a geodesic with regard to exceptional behaviour.
