论文标题
Liouville量子重力中的大地学粗糙度
Roughness of geodesics in Liouville quantum gravity
论文作者
论文摘要
与Liouville量子重力（LQG）表面相关的度量已经通过一系列最近的作品构建，并且已经研究了其相关的大地测量学特性。在当前文章中，我们确认了民间传说的猜想，即对于所谓的Liouville First Pass passage Percolation（LFPP）参数$ξ$，LQG测量学的欧几里得豪斯多夫尺寸大于1。由于Aizenman和Burchard在我们的情况下，我们从一般标准中推断出来，这在我们的情况下等同于几乎几乎几乎几乎几个界限，该杂种在交叉数量中的某些交叉事件的概率。在证明高斯自由场（GFF）的特殊规律性之后，我们使用LQG度量的公理表征获得此类边界。我们还证明了LFPP测量学的类似结果。
The metric associated with the Liouville quantum gravity (LQG) surface has been constructed through a series of recent works and several properties of its associated geodesics have been studied. In the current article we confirm the folklore conjecture that the Euclidean Hausdorff dimension of LQG geodesics is stirctly greater than 1 for all values of the so-called Liouville first passage percolation (LFPP) parameter $ξ$. We deduce this from a general criterion due to Aizenman and Burchard which in our case amounts to near-geometric bounds on the probabilities of certain crossing events for LQG geodesics in the number of crossings. We obtain such bounds using the axiomatic characterization of the LQG metric after proving a special regularity property for the Gaussian free field (GFF). We also prove an analogous result for the LFPP geodesics.
