A mating-of-trees approach for graph distances in random planar maps
A mating-of-trees approach for graph distances in random planar maps
复制标题
DOI：
10.1007/s00440-020-00969-8
复制
发表时间：
2017-11
影响因子：
2
通讯作者：
Ewain Gwynne;N. Holden;Xin Sun
中科院分区：
文献类型：
--
作者：
Ewain Gwynne;N. Holden;Xin Sun
本文介绍了一类随机平面映射的估计的一般证明技术，这些映射属于LQG通用性类。我们考虑的随机平面映射族是那些可以通过树的配对双射以i.d增量的二维随机漫步来编码的映射族，包括均匀无限平面三角剖分（UIPT;）；以及由可以放在地图上的不同生成树（）、双极方向（）或施奈德森林（）的数量加权的平面地图。利用我们的技术，我们证明了上述随机平面映射族中图距离的估计。特别地，我们得到了与Watabiki （Prog theory Phys supl . 1:1 - 17, 1993）对lqg的Hausdorff维数的预测相一致的图距离球的基数的非平凡上界和下界，并且我们建立了图中某些距离的指数的存在性。我们方法的基本思想是将给定的随机平面映射与匹配的crt映射进行比较——一个由相关的二维布朗运动构建的随机平面映射——使用编码行走形式和用于构建匹配的crt映射的布朗运动的强耦合（Zaitsev in ESAIM Probab Stat 2:41 - 108,1998）。这使我们能够从我们在之前的工作中（使用连续统理论）证明的配对crt图中的图距离估计中推断出m中的图距离估计。在特殊情况下，我们从已知的upt结果中推断出配对crt映射的估计值。本文的论点没有直接使用SLE/LQG，可以在不了解这些对象的情况下阅读。
We introduce a general technique for proving estimates for certain random planar maps which belong to the-Liouville quantum gravity (LQG) universality class for. The family of random planar maps we consider are those which can be encoded by a two-dimensional random walk with i.i.d. increments via a mating-of-trees bijection, and includes the uniform infinite planar triangulation (UIPT;); and planar maps weighted by the number of different spanning trees (), bipolar orientations (), or Schnyder woods () that can be put on the map. Using our technique, we prove estimates for graph distances in the above family of random planar maps. In particular, we obtain non-trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the Watabiki (Prog Theor Phys Suppl 114:1–17, 1993) prediction for the Hausdorff dimension of-LQG and we establish the existence of an exponent for certain distances in the map. The basic idea of our approach is to compare a given random planar mapMto amated-CRT map—a random planar map constructed from a correlated two-dimensional Brownian motion—using a strong coupling (Zaitsev in ESAIM Probab Stat 2:41–108, 1998) of the encoding walk forMand the Brownian motion used to construct the mated-CRT map. This allows us to deduce estimates for graph distances inMfrom the estimates for graph distances in the mated-CRT map which we proved (using continuum theory) in a previous work. In the special case when, we instead deduce estimates for the-mated-CRT map from known results for the UIPT. The arguments of this paper do not directly use SLE/LQG, and can be read without any knowledge of these objects.