Weak LQG metrics and Liouville first passage percolation
Weak LQG metrics and Liouville first passage percolation
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DOI：
10.1007/s00440-020-00979-6
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发表时间：
2019-05
影响因子：
2
通讯作者：
Julien Dub'edat;Hugo Falconet;Ewain Gwynne;Joshua Pfeffer;Xin Sun
中科院分区：
文献类型：
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作者：
Julien Dub'edat;Hugo Falconet;Ewain Gwynne;Joshua Pfeffer;Xin Sun
本文定义弱Liouville量子引力度规为一个函数，它以平面高斯自由场为例，输出满足一定自然公理的平面上的度规。我们表明，这些公理是满足任何基本限制的刘维第一次通过渗流。Ding等人证明了这种连续极限的存在（Liouville第一通道渗透的紧密性，2019年。ArXiv电子印刷品，arXiv：1904.08021）。我们还知道，这些公理对于米勒和谢菲尔德（2013-2016）构造的-LQG度量也是满足的。对于任何弱LQG度量，我们得到的时刻范围的直径集以及点到点，集到集，点到集的距离。我们还表明，任何这样的度量是局部bi-Hölder连续的欧几里德度量和计算的最佳Hölder指数在两个方向。最后，我们证明了LQG测地线不能在直线或度量球的边界附近停留很长时间。这些结果被Gwynne和米勒在随后的工作中使用，证明了弱LQG度量对于每个都是唯一的，这反过来又给出了Liouville第一次通过渗流的唯一性极限。然而，我们的大多数结果是新的，即使在特殊的情况下。
For, we define aweak-Liouville quantum gravity(LQG)metricto be a functionwhich takes in an instance of the planar Gaussian free field and outputs a metric on the plane satisfying a certain list of natural axioms. We show that these axioms are satisfied for any subsequential limits of Liouville first passage percolation. Such subsequential limits were proven to exist by Ding et al. (Tightness of Liouville first passage percolation for, 2019. ArXiv e-prints, arXiv:1904.08021 ). It is also known that these axioms are satisfied for the-LQG metric constructed by Miller and Sheffield (2013–2016). For any weak-LQG metric, we obtain moment bounds for diameters of sets as well as point-to-point, set-to-set, and point-to-set distances. We also show that any such metric is locally bi-Hölder continuous with respect to the Euclidean metric and compute the optimal Hölder exponents in both directions. Finally, we show that LQG geodesics cannot spend a long time near a straight line or the boundary of a metric ball. These results are used in subsequent work by Gwynne and Miller which proves that the weak-LQG metric is unique for each, which in turn gives the uniqueness of the subsequential limit of Liouville first passage percolation. However, most of our results are new even in the special case when.