We introduce a general technique for proving estimates for certain random planar maps which belong to the $$\gamma $$ γ -Liouville quantum gravity (LQG) universality class for $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) . 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; $$\gamma =\sqrt{8/3}$$ γ = 8 / 3 ); and planar maps weighted by the number of different spanning trees ( $$\gamma =\sqrt{2}$$ γ = 2 ), bipolar orientations ( $$\gamma =\sqrt{4/3}$$ γ = 4 / 3 ), or Schnyder woods ( $$\gamma =1$$ γ = 1 ) 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 $$\gamma $$ γ -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 map M to a mated-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 for M and the Brownian motion used to construct the mated-CRT map. This allows us to deduce estimates for graph distances in M from 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 $$\gamma =\sqrt{8/3}$$ γ = 8 / 3 , we instead deduce estimates for the $$\sqrt{8/3}$$ 8 / 3 -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.

中文翻译：

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随机平面图中图距离的树交配方法

我们引入了一种通用技术来证明某些随机平面图的估计值，这些图属于 $$\gamma $$ γ -Liouville 量子引力 (LQG) 普适性类，用于 $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) 。我们考虑的随机平面图族是那些可以通过具有 iid 增量的二维随机游走通过树的交配双射进行编码的，并且包括均匀无限平面三角剖分（UIPT; $$\gamma =\sqrt {8/3}$$ γ = 8 / 3 ); 和由不同生成树的数量加权的平面图（ $$\gamma =\sqrt{2}$$ γ = 2 ），双极方向（ $$\gamma =\sqrt{4/3}$$ γ = 4 / 3 ) 或施奈德森林 ( $$\gamma =1$$ γ = 1 ) 可以放在地图上。使用我们的技术，我们证明了上述随机平面图系列中图距离的估计。特别是，我们获得了与 Watabiki (Prog Theor Phys Suppl 114:1–17, 1993) 对 $$\gamma $$ γ -LQG 的 Hausdorff 维的预测一致的图距离球的基数的非平凡上下界和我们确定地图中某些距离的指数的存在。我们方法的基本思想是使用强耦合（Zaitsev in ESAIM Probab Stat 2:41– 108, 1998) M 的编码游走和用于构建配对 CRT 映射的布朗运动。这使我们能够从我们在之前的工作中证明（使用连续统理论）的配对 CRT 图中的图距离估计值推断出 M 中图距离的估计值。在 $$\gamma =\sqrt{8/3}$$ γ = 8 / 3 的特殊情况下，我们推导出 $$\sqrt{8/3}$$ 8 / 3 -mated-CRT 映射的估计值来自 UIPT 的已知结果。本文的论点不直接使用 SLE/LQG，无需了解这些对象即可阅读。