论文标题
Liouville量子重力
Geodesics and metric ball boundaries in Liouville quantum gravity
论文作者
论文摘要
最近的作品表明，有一种规范的方法可以为任何参数$γ\ in（0,2）$分配给liouville量子重力（LQG）表面的度量（距离函数）。我们为LQG测量学建立了强大的汇合属性，该物业概括了Angel，Kolesnik和Miermont为Brownian Map证明的结果。使用此属性，我们还为地球学，度量球边界和公制网络W.R.T.的Hausdorff维度建立了零法律。欧几里得或LQG度量。在度量球边界的情况下，我们的结果与Gwynne（2020）的早期工作相结合，为A.S.提供了公式。 Hausdorff的尺寸，即公制球边界的尺寸在整个LQG表面的Hausdorff尺寸上击中固定点时停止了。我们还表明，公制球边界的Hausdorff尺寸是由不在球的任何互补连接组件的边界的点携带的。
Recent works have shown that there is a canonical way to to assign a metric (distance function) to a Liouville quantum gravity (LQG) surface for any parameter $γ\in (0,2)$. We establish a strong confluence property for LQG geodesics, which generalizes a result proven by Angel, Kolesnik and Miermont for the Brownian map. Using this property, we also establish zero-one laws for the Hausdorff dimensions of geodesics, metric ball boundaries, and metric nets w.r.t. the Euclidean or LQG metric. In the case of a metric ball boundary, our result combined with earlier work of Gwynne (2020) gives a formula for the a.s. Hausdorff dimension for the boundary of the metric ball stopped when it hits a fixed point in terms of the Hausdorff dimension of the whole LQG surface. We also show that the Hausdorff dimension of the metric ball boundary is carried by points which are not on the boundary of any complementary connected component of the ball.
