Uniform Infinite Half-Planar Triangulation

• UIHPT is a critical infinite planar triangulation with a bi-infinite boundary that arises as the local limit of large uniform triangulations.
• The construction employs a layer-by-layer peeling process and a domain Markov property to ensure translation invariance and robust combinatorial structure.
• UIHPT underpins key advances in random geometry and statistical mechanics, providing insights into scaling limits, recurrence, and percolation phenomena.

The Uniform Infinite Half-Planar Triangulation (UIHPT) is a canonical random infinite planar map with a bi-infinite simple boundary, which arises as the local weak limit of large uniform planar triangulations with a macroscopic boundary. The UIHPT is the critical member in the one-parameter family of translation-invariant, domain Markov half-planar triangulation laws, and features universal metric, probabilistic, and combinatorial properties that mirror the geometry of generic two-dimensional random surfaces with an infinite boundary.

1. Construction and Characterization

Let $\mathcal{T}_{n,m}$ denote the set of rooted type-2 planar triangulations of a simple boundary of length $m$ with $n$ interior vertices, rooted on a boundary edge. The UIHPT is constructed as the Benjamini–Schramm local limit:

$$(T_{n,m},\text{root}) \xrightarrow[n\to\infty,\, m\to\infty]{\mathrm{loc}} H$$

where $H$ is the random infinite planar map with a bi-infinite simple boundary. The law $\mu_H$ of the UIHPT is uniquely characterized by:

• Weak limit construction: As above.
• Domain Markov property: For any finite simply-connected triangulation $Q$ glued along its boundary as a “root segment,”

$$\mathbb{P}[Q\subset H \text{ at the root}] = 6^{\#V_{\mathrm{int}}(Q)}\, 9^{-\#F(Q)}$$

and given $Q$, the remainder $H\setminus Q$ is an independent UIHPT. The boundary of $H$ is a bi-infinite path, and the law is translation-invariant along this boundary, invariant under shifting the root edge.

2. Combinatorial Structure and Layer Decomposition

The UIHPT possesses a powerful “layer-by-layer” decomposition: starting from the boundary, one recursively peels successive “hulls” parallel to the boundary. Specifically, one defines a sequence $H_0 = H$, and for each $i \ge 0$:

• $S_i = \partial H_i$ is the current bi-infinite boundary.
• $L_{i+1}$ is the hull of all faces of $H_i$ incident to $S_i$, i.e., the union of those faces with all finite components enclosed.
• $H_{i+1} = H_i \setminus L_{i+1}$.

This yields:

$H = \bigcup_{i\ge 1} L_i$

Each layer $L_i$ decomposes into an alternating sequence of i.i.d. finite “blocks”: finite simply-connected holes bordered from below by a boundary segment of random length $B \sim t^{-3/2}$, filled with independent Boltzmann triangulations of a $(B+2)$-gon with weight $2/27$. All layers above the initial boundary are i.i.d., except for the root-containing block, which is size-biased.

3. The One-Parameter Family and Phase Structure

Angel–Ray (Angel et al., 2013, Ray, 2013) established a one-parameter family $\{H^{(3)}_\alpha: 0 \leq \alpha < 1\}$ of translation-invariant, domain–Markov half-planar triangulation laws, with peeling probabilities completely determined by

$$\alpha = H^{(3)}_\alpha(\text{triangle at root has internal third vertex})$$

and an explicit function $\beta(\alpha)$ ensuring normalization. The critical point occurs at $\alpha_c = 2/3$, corresponding to Boltzmann weight $q = \alpha_c \beta(\alpha_c) = 2/27$. The UIHPT is $H^{(3)}_{2/3}$ and exhibits a universal boundary-attachment probability decay:

$$p_i = H^{(3)}_{2/3}(\text{root triangle spans }i \text{ boundary edges}) \sim c\,i^{-5/2}$$

Parameters $\alpha < 2/3$ give subcritical, more tree-like maps with thinner boundaries, while $\alpha > 2/3$ yields supercritical (hyperbolic-type) maps with exponential boundary and volume growth.

Parameter $\alpha$	Boundary Growth	Volume Growth	Random Walk	Geometry Type
$\alpha<2/3$	Bounded	$r^2$	Highly recurrent	Tree-like
$\alpha=2/3$	$r^2$	$r^4$	Recurrent, subdiffusive	Parabolic
$\alpha>2/3$	Exponential	Exponential	Transient	Hyperbolic

4. Metric, Probabilistic, and Percolative Properties

In the critical UIHPT ($\alpha=2/3$), the graph hulls $B_r$ of radius $r$ satisfy:

• Boundary length: $|\partial B_r| \sim r^2$
• Volume: $|B_r| \sim r^4$
• Anchored expansion constant and Cheeger constant are both zero: $\mathit{i}_E^* = 0$.
• The distance from the root of a simple random walk after $n$ steps scales as $n^{1/4}$, indicating sub-diffusive behavior; spectral dimension $d_s = 2$ with return probabilities $p_{2n}(\rho, \rho) \sim n^{-1+o(1)}$.

For Bernoulli site percolation, the critical threshold is $p_c = 1/2$. At $p=1/2$, all clusters are finite almost surely, with cluster size tail $P(|C(0)| > n) \sim n^{-1/2}$, and interface exponents in agreement with the KPZ relations.

5. Full-Plane Extension, Recurrence, and Resistance Estimates

A key feature is the full-plane “layered” extension: new i.i.d. layers can be glued “below” the boundary to construct an infinite triangulation $M = \bigcup_{i \in \mathbb{Z}} L_i$, where the original UIHPT sits as the “upper half.” This construction enables transferring methods from the stationarily rooted full-plane UIPT and facilitates analysis of random walk recurrence.

Angel–Ray (Angel et al., 2016) proved that simple random walk on the UIHPT (and its full-plane extension) is almost surely recurrent. The proof utilizes the following steps:

• Finite approximations: Subgraphs induced by finitely many layers approximate the infinite map locally.
• Star-tree transform: Replace each vertex of degree $d$ by a balanced binary tree, bounding maximal degree and assigning conductances $w_e = d$ to edges.
• Circle packing: Embed the bounded-degree transformed map via the Circle Packing Theorem. The Ring Lemma provides scale control for the radii.
• Electrical resistance bounds: Uniformly positive resistance across large Euclidean balls is established, ultimately implying recurrence by Rayleigh monotonicity.

The overall resistance from a vertex to outside a Euclidean disk of radius $r$ grows at least as fast as $\log r$; precise exponents remain open, with conjectured growth of $\log R$.

6. Structural and Enumerative Aspects

Combinatorial enumeration is rooted in Tutte-style formulae for the number of triangulations of a polygon with given perimeter and area; the critical threshold for the Boltzmann partition function is at $q = 2/27$. Explicit formulas for block-size distributions and face-boundary configurations arise through generating function manipulations. The spatial Markov property governs how the law behaves under peeling operations, and the repeated i.i.d. structure underlies many statistical properties.

In the context of infinite Schnyder woods (Addario-Berry et al., 10 Nov 2025), it is established that the UIHPT almost surely admits a unique maximal Schnyder wood: a 3-orientation and edge-colouring satisfying strong local and boundary conditions. Peeling-based algorithms adapted from the finite case build this maximal structure layer by layer; the monochromatic components form one-ended infinite trees except for one subgraph (blue) which decomposes into a single one-ended tree and infinitely many finite trees rooted at boundary points.

7. Further Developments and Applications

The UIHPT serves as a universal local limit for large uniform triangulations with boundary, underpinning models of two-dimensional quantum gravity, statistical physics on random geometry, and scaling limits towards continuum structures such as the Brownian half-plane. The layer decomposition method is used in first-passage percolation analyses and in the paper of random metric spaces. Conditioning on an infinite critical percolation cluster produces the Incipient Infinite Cluster (IIC), which is explicitly describable as the gluing of independent hulls along random necklaces, providing a tractable laboratory for critical phenomena in random geometry (Richier, 2017).

Extensions of these constructions apply to half-planar quadrangulations (HUIPQ) and to hyperbolic analogues, where analogous layered decompositions yield divergent geometric and probabilistic behaviors (e.g., positive speed of random walk, transience in the hyperbolic case). Open problems include sharp resistance growth exponents and scaling limits for the UIHPT, as well as the structural features of embedded random walks and percolation interfaces.

---

In summary, the Uniform Infinite Half-Planar Triangulation is a critical, translation-invariant, domain Markov infinite random triangulation built as the universal scaling limit of large planar triangulations with boundary, with a symmetric layer decomposition, explicit recurrence properties, and deep structural connections to combinatorics, probability, and statistical mechanics.