TY - JOUR

T1 - The Dirichlet problem for orthodiagonal maps

AU - Gurel-Gurevich, Ori

AU - Jerison, Daniel C.

AU - Nachmias, Asaf

N1 - Publisher Copyright:
© 2020 Elsevier Inc.

PY - 2020/11/18

Y1 - 2020/11/18

N2 - 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 [4] for isoradial graphs. We observe that by the double circle packing theorem [3], any finite, simple, 3-connected planar map admits an orthodiagonal representation. Our result improves the work of Skopenkov [41] and Werness [47] 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 [21], we show that this can be done for the discrete mating-of-trees random map model of Duplantier, Gwynne, Miller and Sheffield [15,25].

AB - 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 [4] for isoradial graphs. We observe that by the double circle packing theorem [3], any finite, simple, 3-connected planar map admits an orthodiagonal representation. Our result improves the work of Skopenkov [41] and Werness [47] 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 [21], we show that this can be done for the discrete mating-of-trees random map model of Duplantier, Gwynne, Miller and Sheffield [15,25].

KW - Circle packing

KW - Planar map

KW - Random walk

UR - http://www.scopus.com/inward/record.url?scp=85090479045&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2020.107379

DO - 10.1016/j.aim.2020.107379

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AN - SCOPUS:85090479045

SN - 0001-8708

VL - 374

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 107379

ER -