2024/07/30 by Ilia Binder, Binder, Ilia, David Pechersky +1 · 1 citation
Mathematics · Computer Science · #Mathematics and Applications #Algebraic and Geometric Analysis #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2407.20851
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 (Ωn)_n≥1 is a sequence of orthodiagonal maps with distinguished boundary vertices An, Bn, Cn, Dn 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.