Infinite Schnyder Woods
Louigi Addario-Berry, Emma Hogan, Lukas Michel, Alex Scott
TL;DR
This paper extends the concept of Schnyder woods to infinite triangulations, establishing existence, uniqueness, and convergence properties, and analyzing their structural features in infinite settings.
Contribution
It introduces the notion of Schnyder woods for infinite triangulations and proves their existence, uniqueness, and convergence in various infinite triangulation models.
Findings
Unique maximal Schnyder woods exist for infinite triangulations with finite boundary.
The maximal Schnyder wood of the uniform infinite planar triangulation is a limit of finite cases.
Structural properties of infinite Schnyder woods are characterized.
Abstract
It is well-known that any finite triangulation possesses a unique maximal Schnyder wood. We introduce Schnyder woods of infinite triangulations, and prove there exists a unique maximal Schnyder wood of any infinite triangulation with finite boundary, and of the uniform infinite half-planar triangulation. Furthermore, the maximal Schnyder wood of the uniform infinite planar triangulation is the limit of maximal Schnyder woods of large finite random triangulations. Several structural properties of infinite Schnyder woods are also described.
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Taxonomy
TopicsStochastic processes and statistical mechanics · Computational Geometry and Mesh Generation · Geometric and Algebraic Topology