Orthodiagonal Maps, Tilings of Rectangles, and their Convergence to Conformal Maps

TL;DR

The paper develops a comprehensive discrete complex-analytic framework for orthodiagonal rectangle tilings of simply connected planar domains with four distinguished boundary points. By leveraging electrical-network theory, extremal length, and discrete holomorphicity on orthodiagonal maps, it proves precompactness of the associated tiling maps and shows that any subsequential limit is holomorphic. The main result asserts that, for a sequence of increasingly fine orthodiagonal approximations $(G_n,A_n^\\bullet,B_n^\\bullet,C_n^\\bullet,D_n^\\bullet)$ of a conformal rectangle $(\\Omega,A,B,C,D)$, the tiling maps $\\phi_n$ converge uniformly on compacts to the uniformizing conformal map $\\phi$ to the rectangle, with the corresponding discrete extremal lengths converging to the continuous extremal length. The arguments provide explicit modulus-of-continuity bounds in both the continuous and discrete settings and establish two-sided extremal-length estimates, enabling convergence without boundary smoothness assumptions and offering explicit convergence rates analogous to the circle-packing paradigm. The results unify discrete conformal mapping with probabilistic and geometric approaches in 2D, and extend prior discrete-to-continuum convergence results to a broad class of orthodiagonal discretizations while furnishing quantitative bounds.

Abstract

A classic result of Brooks, Smith, Stone and Tutte associates to any finite planar network with distinguished source and sink vertices, a tiling of a rectangle by smaller subrectangles whose aspect ratios are given by the conductances of corresponding edges in the network. This tiling can be viewed as a discrete analogue of the uniformizing conformal map that maps a simply connected domain with four distinguished prime ends to a rectangle, so that the four prime ends are mapped to the four corners of the rectangle. \\ \\ We make this intuition precise by showing that if $Ω$ is a simply connected domain with four distinguished prime ends $A,B,C,D$ in counterclockwise order and $(Ω_{n})_{n\geq{1}}$ is a sequence of orthodiagonal maps with distinguished boundary vertices $A_{n}, B_{n}, C_{n}, D_{n}$ in counterclockwise order, that are finer and finer approximations of $Ω$ with its distinguished boundary points $A,B,C,D$, then the corresponding ``rectangle tiling maps" converge uniformly on compacts to the aforementioned conformal map on $Ω$.

Figures (17)

• Figure 1: A discrete conformal rectangle with its four distinguished boundary arcs.
• Figure 2: An orthodiagonal rectangle with its four distinguished boundary arcs.
• Figure 3: The comb domain $\Omega$ with its four distinguished prime ends and an orthodiagonal approximation $G$ of $\Omega$. Notice that regardless of how fine the mesh of $G$ is, there is no choice of $A^{\bullet}, B^{\bullet}, C^{\bullet}, D^{\bullet}\in{\partial{V^{\bullet}}}$ so that each of the discrete boundary arcs of $(G,A^{\bullet}, B^{\bullet}, C^{\bullet}, D^{\bullet})$ is close to its continuous analogue.
• Figure 4: The tiling map $\phi$ associated with an orthodiagonal rectangle.
• Figure 5: On the left: The subrectangle $\Omega_{x,y}$ of $\Omega$ with its distinguished boundary arcs $N, E, S$ and $W$. On the right: Notice that any path from $N$ to $S$ in $\Omega_{x,y}$ must cross the annulus $\mathcal{A}$.

Theorems & Definitions (44)

• Proposition 2.1
• Proposition 2.2: Dirichlet's Principle
• Proposition 2.3: Thomson's Principle
• Proposition 2.4
• Proposition 2.5
• Proposition 2.6
• Lemma 2.7
• Corollary 2.8
• Proposition 2.9
• Proposition 2.10