Circle packing and Riemann uniformization of random triangulations in an ergodic scale-free environment

Abstract

We prove that embedded infinite plane triangulations in ergodic scale-free environments are close to their circle packing and Riemann uniformization embedding on a large scale, as long as suitable moment and connectivity conditions are satisfied. Ergodic scale-free environments were earlier considered by Gwynne, Miller and Sheffield (2018) in the context of the invariance principles for random walk, and they arise naturally in the study of random planar maps and Liouville quantum gravity.

Figures (8)

• Figure 1: An illustration of five cells in a cell configuration $\mathcal{H}$. The cells in $\mathcal{H}$ may not be simply connected and could cross from each other. We allow that the green/blue and red/purple cells to touch each other while not being connected by an edge in $\mathcal{E}\mathcal{H}$.
• Figure 2: Two adjacent vertices $u,v$. The Dubejko conductance of the edge $uv$ is defined to be $\mathfrak{c}(u,v) = \frac{|o_{a}-o_{b}|}{|o_u-o_v|}=\frac{\sqrt{r_ur_v}}{r_u+r_v}\cdot(\sqrt{\frac{r_{w_1}}{r_{w_1}+r_u+r_v}}+\sqrt{\frac{r_{w_2}}{r_{w_2}+r_u+r_v}})$.
• Figure 3: Left: A flower with 8 petals. Right: An illustration for the setup in the proof of Lemma \ref{['lem:circle']} for $d=4$.
• Figure 4: Construction of the path $P_{H,H'}$ in the proof of Proposition \ref{['prop:inv-circle']} when $r_{H'}\leq 0.01d^{-1}r_{H}$. Here for $j=2,...,6$, $r_{H_j}<0.01d^{-1}r_H$ while $0.01d^{-1}r_H\leq r_{H_1}<r_H$.
• Figure 5: An illustration of the definition of $M(S;a)$ and $M(S)$. For simplicity here we assume that two cells are adjacent if they have nonempty intersection. The cells in cyan are those intersecting $\mathring{S}^a$, and their neighbors are in yellow. We also include the cells that are separated from infinity from the yellow and cyan cells, which are colored in green. The surface $M(S;a)\subseteq M(\mathcal{H})$ are formed by vertices for these cells on $M(\mathcal{H})$ as well as the edges connecting these vertices, and the surface $M(S)$ is the largest surface of the form $M(S;a)$ such that for each vertex on $M(S;a)$, its cell is contained within $S$.

Theorems & Definitions (120)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Definition 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 2.1: HeSchramm-hy-pa
• Definition 2.2