We prove that the discrete harmonic function corresponding to smooth Dirichlet boundary conditions on orthodiagonal maps, that is, plane graphs having quadrilateral faces with orthogonal diagonals, converges to its continuous counterpart as the mesh size goes to 0. This provides a convergence statement for discrete holomorphic functions, similar to the one obtained by Chelkak and Smirnov for isoradial graphs. We observe that by the double circle packing theorem, any finite, simple, 3-connected planar map admits an orthodiagonal representation. Our result improves the work of Skopenkov and Werness by dropping all regularity assumptions required in their work and providing effective bounds. In particular, no bound on the vertex degrees is required. Thus, the result can be applied to models of random planar maps that with high probability admit orthodiagonal representation with mesh size tending to 0. In a companion paper, we show that this can be done for the discrete mating-of-trees random map model of Duplantier, Gwynne, Miller and Sheffield.

中文翻译：

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正交映射的狄利克雷问题

我们证明了对应于正交映射上的平滑 Dirichlet 边界条件的离散调和函数，即具有正交对角线的四边面的平面图，随着网格大小趋于 0 收敛到其连续对应物。 这提供了离散全纯的收敛陈述函数，类似于 Chelkak 和 Smirnov 为等轴图获得的函数。我们观察到，根据双圆堆积定理，任何有限的、简单的、3 连通的平面图都允许正交表示。我们的结果通过放弃他们工作中所需的所有规律性假设并提供有效的界限，改进了 Skopenkov 和 Werness 的工作。特别是，不需要对顶点度数进行限制。因此，