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Anomalous diffusion of random walk on random planar maps
Probability Theory and Related Fields ( IF 1.5 ) Pub Date : 2020-07-20 , DOI: 10.1007/s00440-020-00986-7
Ewain Gwynne , Tom Hutchcroft

We prove that the simple random walk on the uniform infinite planar triangulation (UIPT) typically travels graph distance at most $n^{1/4 + o_n(1)}$ in $n$ units of time. Together with the complementary lower bound proven by Gwynne and Miller (2017) this shows that the typical graph distance displacement of the walk after $n$ steps is $n^{1/4 + o_n(1)}$, as conjectured by Benjamini and Curien (2013). More generally, we show that the simple random walks on a certain family of random planar maps in the $\gamma$-Liouville quantum gravity (LQG) universality class for $\gamma\in (0,2)$---including spanning tree-weighted maps, bipolar-oriented maps, and mated-CRT maps---typically travels graph distance $n^{1/d_\gamma + o_n(1)}$ in $n$ units of time, where $d_\gamma$ is the growth exponent for the volume of a metric ball on the map, which was shown to exist and depend only on $\gamma$ by Ding and Gwynne (2018). Since $d_\gamma > 2$, this shows that the simple random walk on each of these maps is subdiffusive. Our proofs are based on an embedding of the random planar maps under consideration into $\mathbb C$ wherein graph distance balls can be compared to Euclidean balls modulo subpolynomial errors. This embedding arises from a coupling of the given random planar map with a mated-CRT map together with the relationship of the latter map to SLE-decorated LQG.

中文翻译:

随机平面图上随机游走的异常扩散

我们证明了均匀无限平面三角剖分 (UIPT) 上的简单随机游走通常在 $n$ 单位时间内最多移动图距离 $n^{1/4 + o_n(1)}$。连同 Gwynne 和 Miller (2017) 证明的互补下界,这表明在 $n$ 步之后步行的典型图距离位移是 $n^{1/4 + o_n(1)}$,正如 Benjamini 所推测的和 Curien (2013)。更一般地说,我们证明了在 $\gamma\in (0,2)$ 的 $\gamma\-Liouville 量子引力 (LQG) 普适性类中的某个随机平面图系列上的简单随机游走---包括跨越tree-weighted maps, bipolar-oriented maps, and mate-CRT maps---通常以$n$单位的时间移动图形距离$n^{1/d_\gamma + o_n(1)}$,其中$d_\ gamma$ 是地图上公制球体积的增长指数,Ding 和 Gwynne (2018) 证明它存在并且仅依赖于 $\gamma$。由于 $d_\gamma > 2$,这表明每个地图上的简单随机游走是次扩散的。我们的证明基于将考虑中的随机平面图嵌入到 $\mathbb C$ 中,其中图距离球可以与欧几里得球模子多项式误差进行比较。这种嵌入源于给定的随机平面图与配对的 CRT 地图的耦合,以及后者的地图与 SLE 装饰的 LQG 的关系。我们的证明基于将考虑中的随机平面图嵌入到 $\mathbb C$ 中,其中图距离球可以与欧几里得球模子多项式误差进行比较。这种嵌入源于给定的随机平面图与配对的 CRT 地图的耦合,以及后者的地图与 SLE 装饰的 LQG 的关系。我们的证明基于将考虑中的随机平面图嵌入到 $\mathbb C$ 中,其中图距离球可以与欧几里得球模子多项式误差进行比较。这种嵌入源于给定的随机平面图与配对的 CRT 地图的耦合,以及后者的地图与 SLE 装饰的 LQG 的关系。
更新日期:2020-07-20
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