# The Dirichlet problem for orthodiagonal maps

## TL;DR

This paper proves that discrete harmonic functions on orthodiagonal maps converge to their continuous counterparts as the mesh size diminishes, extending convergence results to a broader class of planar graphs without regularity constraints.

## Contribution

It establishes convergence of discrete harmonic functions on orthodiagonal maps without regularity assumptions, broadening applicability to random planar maps.

## Key findings

- Convergence of discrete harmonic functions to continuous solutions.
- Applicable to models of random planar maps with orthodiagonal representation.
- Provides effective bounds independent of vertex degree.

## Abstract

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.