Title: Scaling limits of planar maps under the Smith embedding Show Chinese title

Title: Smith嵌入下的平面地图的尺度极限

Abstract: The Smith embedding of a finite planar map with two marked vertices, possibly with conductances on the edges, is a way of representing the map as a tiling of a finite cylinder by rectangles. In this embedding, each edge of the planar map corresponds to a rectangle, and each vertex corresponds to a horizontal segment. Given a sequence of finite planar maps embedded in an infinite cylinder, such that the random walk on both the map and its planar dual converges to Brownian motion modulo time change, we prove that the a priori embedding is close to an affine transformation of the Smith embedding at large scales. By applying this result, we prove that the Smith embeddings of mated-CRT maps with the sphere topology converge to $\gamma$-Liouville quantum gravity ($\gamma$-LQG). Show Chinese abstract

Abstract: 具有两个标记顶点的有限平面地图的史密斯嵌入（可能在边上有电导率），是将该地图表示为有限圆柱体上矩形镶嵌的一种方式。 在此嵌入中，平面地图的每条边对应一个矩形，每个顶点对应一条水平线段。 给定嵌入到无限圆柱体中的平面地图序列，使得地图及其平面对偶上的随机游走都收敛于时间改变后的布朗运动，我们证明了在大尺度下，先验嵌入接近于史密斯嵌入的仿射变换。 通过应用此结果，我们证明了与球面拓扑相关的 mated-CRT 图的地图的史密斯嵌入收敛于$\gamma$-Liouville 量子引力 ($\gamma$-LQG)。