Scaling limits of planar maps under the Smith embedding

Bertacco Federico, Imperial College London

Over the past few decades, there has been significant progress in the study of scaling limits of random planar maps. In this talk, I will provide motivation for this problem and then focus on the scaling limits of (random) planar maps under the Smith embedding. This embedding is described by a tiling of a finite cylinder by rectangles, where each edge of the map corresponds to a rectangle, and each vertex corresponds to a horizontal segment. I will argue that when considering a sequence of finite planar maps embedded in an infinite cylinder and satisfying a suitable invariance principle assumption, the a priori embedding is close to an affine transformation of the Smith embedding at larger scales. By applying this result, I will prove that the Smith embeddings of mated-CRT maps with the sphere topology converge to LQG. This is based on joint work with Ewain Gwynne and Scott Sheffield.

Area: IS4 - Stochastic Geometry (Jacopo Borga)

Keywords: Planar maps, Scaling limit, LQG

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