SCALING LIMITS OF PLANAR MAPS UNDER THE SMITH EMBEDDING

• 成果类型: Article
• 署名作者: Bertacco, Federico; Gwynne, Ewain; Sheffield, Scott
• 署名单位: Imperial College London; University of Chicago; Massachusetts Institute of Technology (MIT)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/24-AOP1731
• 发表日期: 2025
• 页码: 1138-1196
• 关键词: liouville quantum-gravity; random-walks; CONVERGENCE

摘要

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).