标题： 测地线的粗糙度在黎曼流形量子引力中 显示英文标题

标题： Roughness of geodesics in Liouville quantum gravity

摘要： 与Liouville量子引力（LQG）表面相关的度量已经通过一系列最近的研究构建出来，并且已经研究了其相关测地线的几个性质。 在本文中，我们确认了一个民间猜想，即对于所谓的Liouville首次通过渗流（LFPP）参数$\xi$的所有值，LQG测地线的欧几里得豪斯多夫维数严格大于1。 我们从Aizenman和Burchard提出的一般准则推导出这一结论，在我们的情况下，这相当于对LQG测地线在交叉次数上的某些交叉事件的概率的近几何界。 我们通过证明高斯自由场（GFF）的一个特殊正则性属性，获得了这样的界限。 我们还证明了关于LFPP测地线的类似结果。 显示英文摘要

摘要： 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 $\xi$. 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.