论文标题
布朗地图中的大地学：强汇率和几何结构
Geodesics in the Brownian map: Strong confluence and geometric structure
论文作者
论文摘要
我们在布朗图$（\ Mathcal {s}，d，ν）$中研究大地测量，这是均匀随机平面图的Gromov-Hausdorff缩放限制的随机度量测量空间。我们的结果适用于所有的大地测量学，包括在特殊点之间。 首先，我们证明了地球学现象汇合的强烈而定量的形式，该形式指出，除了端点附近以外，在Hausdorff距离中足够接近的任何一对大地测量学都必须彼此重合。 然后，我们表明，任何两个大地测量学的交汇处减去它们的终点的连接，从一个点散发出来的大地测量数量，除了其起点最多是$ 5 $，而连接任何一对点的大地测量线数量为$ 9 $。对于每个$ 1 \ le K \ le 9 $，我们都会获得由$ K $ Geodesics连接的点对的Hausdorff尺寸。对于$ k = 7,8,9 $，这样的对尺寸为零，并且是无限的。此外，我们将（有限的）在$ \ Mathcal {s} $中的任何对点之间的（有限数量）进行分类，直至同构，并为每种情况下的一组端点提供一个维度上限。 最后，我们表明，通过将$ν$ typical点连接的测量线相连，每个大地测量都可以任意地近似。特别是，这对Angel，Kolesnik和Miermont的猜想提供了肯定的答案，即$ \ Mathcal {s} $的地球框架，$ \ Mathcal {s} $中的所有大地测量公司的结合，它们的端点很小，它们的端点具有尺寸，一个地理位置的尺寸。
We study geodesics in the Brownian map $(\mathcal{S},d,ν)$, the random metric measure space which arises as the Gromov-Hausdorff scaling limit of uniformly random planar maps. Our results apply to all geodesics including those between exceptional points. First, we prove a strong and quantitative form of the confluence of geodesics phenomenon which states that any pair of geodesics which are sufficiently close in the Hausdorff distance must coincide with each other except near their endpoints. Then, we show that the intersection of any two geodesics minus their endpoints is connected, the number of geodesics which emanate from a single point and are disjoint except at their starting point is at most $5$, and the maximal number of geodesics which connect any pair of points is $9$. For each $1\le k \le 9$, we obtain the Hausdorff dimension of the pairs of points connected by exactly $k$ geodesics. For $k=7,8,9$, such pairs have dimension zero and are countably infinite. Further, we classify the (finite number of) possible configurations of geodesics between any pair of points in $\mathcal{S}$, up to homeomorphism, and give a dimension upper bound for the set of endpoints in each case. Finally, we show that every geodesic can be approximated arbitrarily well and in a strong sense by a geodesic connecting $ν$-typical points. In particular, this gives an affirmative answer to a conjecture of Angel, Kolesnik, and Miermont that the geodesic frame of $\mathcal{S}$, the union of all of the geodesics in $\mathcal{S}$ minus their endpoints, has dimension one, the dimension of a single geodesic.
