Discrete Complex Analysis and the Convergence of Observables on Orthodiagonal Maps

Orthodiagonal maps are a broad class of discretizations of continuous 2D space that accommodate a notion of discrete complex analysis. They generalize isoradial graphs, which have been widely studied in connection with critical statistical physics in 2D. In this thesis, we resolve several problems related to the behavior of discrete harmonic and holomorphic functions on orthodiagonal maps. The hope is that the tools that we develop for handling discrete harmonic and holomorphic functions will be useful for extending the landmark results of critical 2D statistical physics that are known for isoradial graphs to the more general orthodiagonal setting. 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. In the first part of this thesis, we make this intuition precise by showing that if $\Omega$ is a simply connected domain with four distinguished prime ends $A,B,C,D$ in counterclockwise order and $(\Omega_{n})_{n\geq{1}}$ 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 $\Omega$ 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 $\Omega$. In the second part of this thesis, we extend recent work of Gurel-Gurevich, Jerison and Nachmias and Gwynne and Bou-Rabee by showing that as long as our boundary data is H\"older, we have an explicit polynomial rate of convergence for the Dirichlet problem on orthodiagonal maps to its continuous counterpart. The key idea here is that the convolution of a discrete harmonic function on an orthodiagonal map with a smooth mollifier has small Laplacian and so is "almost harmonic." This also allows us to show that discrete harmonic functions on orthodiagonal maps are Lipschitz in the bulk on a mesoscopic scale.

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