Infinite Schnyder Woods

TL;DR

The paper extends Schnyder woods to infinite triangulations, establishing a unique maximal Schnyder wood for infinite triangulations with finite boundary and for the UIHPT, and proving that the UIHPT's maximal Schnyder wood arises as the weak limit of those from large finite triangulations (with fixed boundary). It introduces a Schnyder peeling process and a two-stage approach using Schnyder wood segments and strips to overcome nonlocality in the UIHPT, enabling a global colouring and orientation. It then characterizes the resulting monochromatic forests (yellow, red, blue) and their infinite-path structures, and proves maximality/uniqueness via leftmost walks and boundary-layer analysis. The work lays groundwork for understanding how canonical straight-line embeddings and geodesic structures extend to infinite maps, and it highlights open questions about UIHPT-limit convergence and winding behavior within the Schnyder framework.

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.

Figures (15)

• Figure 1: The structure of the unique monochromatic paths starting from a vertex chosen uniformly at random from the maximal Schnyder wood of a large random triangulation. See also \ref{['fig:tutte_embedding_big', 'fig:most_zoom']} on fig:tutte_embedding_big,fig:most_zoom, respectively.
• Figure 2: Edges incident with internal vertices satisfy the condition indicated in (a), edges incident with the root edge satisfy the condition indicated in (b), and edges incident with non-root boundary vertices satisfy the condition in (c).
• Figure 3: A maximal Schnyder wood of a triangulation $T' \in \cT_6^1$ constructed by adding $v_b$ as a neighbour to every boundary vertex of a triangulation $T \in \cT_3^3$.
• Figure 4: The first non-trivial Schnyder peeling step on a triangulation $T$. Initially the root edge is $(v_1, v_2)$; the boundary of the unexplored region is indicated in bold. The first chord $(v_0, v_c) = (v_0, v_4)$ anticlockwise from $(v_0, v_1)$ in the unexplored region is selected and coloured yellow, and all remaining neighbours of $v_0$ are also explored and coloured, splitting the unexplored region in two. The left unexplored region $T_\ell$ is rooted at the new root $(v_0, v_c)$ while the right unexplored region $T_r$ retains the original root $(v_1,v_2)$.
• Figure 5: Two distinct Schnyder woods of the half-plane triangular lattice in which every finite directed cycle is oriented clockwise. The left Schnyder wood is the one produced by the Schnyder chiselling algorithm that we later define in \ref{['sec:structure']}, and contains no right-directed paths. The right Schnyder wood contains right-directed paths.

Theorems & Definitions (121)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1
• Lemma 2: one-endedness
• Lemma 3: translation invariance
• Theorem 2.1: Schnyder, schnyder1990
• Lemma 4
• proof