论文标题
随机步行在配偶平面地图和Liouville Brownian Motion上
Random walks on mated-CRT planar maps and Liouville Brownian motion
论文作者
论文摘要
我们证明，在某些随机平面图上随机行走具有自然时间参数化的缩放极限结果。特别是，我们表明，对于$γ\（0,2）$，与参数$γ$的配对crt地图上的随机步行收敛到$γ$ -liouville Brownian Motion，这是Brownian Motion在$γ$ -Liouville量子重力（LQG）表面的Brownian Motion的自然量子时间参数。如果将配合的crt图通过来自SLE / LQG理论或TUTTE嵌入（又称谐波或Barycentric嵌入）的嵌入到平面中，我们的结果将适用。在这两种情况下，融合都相对于曲线上的局部统一拓扑，并且在静止的意义上，即鉴于地图收敛的步行条件定律。 格温（Gwynne），米勒（Miller）和谢菲尔德（Sheffield）（2017）的先前工作表明，在配合crt地图上的随机步行会收敛到布朗运动模量时间参数化。这是显示参数化步行的收敛性的第一项工作。作为独立兴趣的中间结果，我们得出了liouville布朗运动的公理表征，对此，马尔可夫过程的revuz概念起着至关重要的作用。
We prove a scaling limit result for random walk on certain random planar maps with its natural time parametrization. In particular, we show that for $γ\in (0,2)$, the random walk on the mated-CRT map with parameter $γ$ converges to $γ$-Liouville Brownian motion, the natural quantum time parametrization of Brownian motion on a $γ$-Liouville quantum gravity (LQG) surface. Our result applies if the mated-CRT map is embedded into the plane via the embedding which comes from SLE / LQG theory or via the Tutte embedding (a.k.a. the harmonic or barycentric embedding). In both cases, the convergence is with respect to the local uniform topology on curves and it holds in the quenched sense, i.e., the conditional law of the walk given the map converges. Previous work by Gwynne, Miller, and Sheffield (2017) showed that the random walk on the mated-CRT map converges to Brownian motion modulo time parametrization. This is the first work to show the convergence of the parametrized walk. As an intermediate result of independent interest, we derive an axiomatic characterisation of Liouville Brownian motion, for which the notion of Revuz measure of a Markov process plays a crucial role.
