UIBOT: Uniform Infinite Bipolar Triangulation

• UIBOT is the infinite-volume limit of finite bipolar-oriented triangulations, yielding a one-ended directed planar map with its bipolar structure pushed to infinity.
• Its structure is encoded by a bi-infinite lattice walk that, after diffusive scaling, converges to correlated Brownian motion representing √(4/3)-LQG decorated by space-filling SLE₁₂.
• The dual orientation and Busemann function analyses provide key insights into directed geometric exponents and scaling limits relevant for renewal and excursion studies.

Searching arXiv for recent and foundational papers on UIBOT and related bipolar-oriented maps. Uniform Infinite Bipolar-Oriented Triangulation (UIBOT) is the uniform infinite-volume limit of finite bipolar-oriented triangulations, obtained as the Benjamini–Schramm local limit of finite bipolar-oriented triangulations re-rooted at a uniformly chosen edge. It is an infinite, one-ended, locally finite directed planar map whose bipolar structure has its source and sink at infinity, and whose combinatorics are encoded by a bi-infinite lattice walk. In the peanosphere sense, its tree contour functions converge after diffusive scaling to the correlated Brownian coordinates of a $\sqrt{4/3}$-Liouville quantum gravity surface decorated by space-filling $\mathrm{SLE}_{12}$; moreover, the corresponding dual bipolar-oriented triangulation admits a joint scaling limit encoding a second, orthogonal space-filling $\mathrm{SLE}_{12}$ on the same LQG surface (Gwynne et al., 2016).

1. Definition and local-limit construction

A bipolar orientation on a planar map is an acyclic orientation with a unique source and a unique sink in the finite setting. In the conventions used for bipolar-oriented triangulations, the source is the southeast pole and the sink is the northwest pole, and it is convenient to add an extra unoriented edge joining the two poles, thereby splitting the outer face into a southwest pole face and a northeast pole face; this augmented map has a naturally oriented dual map (Gwynne et al., 2016).

For finite triangulations with boundary, the basic model fixes integers $\ell,r \ge 0$ with $\ell+r>0$, takes a triangulation with $n$ edges, southwest boundary length $\ell$, northeast boundary length $r$, and a bipolar orientation whose source is the southeast corner and sink is the northwest corner, and samples uniformly from all such objects. The infinite-volume object relevant for UIBOT is obtained by fixing $\ell=2$ and $r=1$, choosing a root edge uniformly from the edges, and sending $\mathrm{SLE}_{12}$0 in the Benjamini–Schramm topology. The resulting rooted oriented map $\mathrm{SLE}_{12}$1 is the uniform infinite bipolar-oriented triangulation (Gwynne et al., 2016).

This limit has several structural properties. It is infinite, one-ended, and locally finite. Its orientation is almost surely acyclic and has no finite source or sink; equivalently, the poles are pushed to infinity. The two canonical primal trees become infinite trees rooted at infinity, and the discrete interface becomes a bi-infinite exploration path $\mathrm{SLE}_{12}$2 with $\mathrm{SLE}_{12}$3 that visits each edge exactly once. The dual infinite bipolar-oriented triangulation is obtained as the local limit of the duals of the finite augmented maps, rooted at the dual edge crossing the primal root (Gwynne et al., 2016).

Two related terminological points are standard in the literature. First, earlier finite-volume work on bipolar-oriented maps identified the natural infinite-volume encoding on the walk side before the local-limit construction was isolated explicitly; in that sense UIBOT is the infinite object naturally suggested by the bipolar walk bijection (Kenyon et al., 2015). Second, later work on directed geometry takes UIBOT as the canonical infinite random bipolar-oriented triangulation around a typical edge and studies its internal directed metric structure directly (Borga et al., 30 Oct 2025).

2. Walk encoding, trees, and the sewing procedure

The foundational combinatorial mechanism is the Kenyon–Miller–Sheffield–Wilson bijection between bipolar-oriented planar maps and lattice walks. Associated with a bipolar orientation are two canonical spanning trees in the primal map. In one compass convention these are the east-going and west-going trees; in another they are the southeast and northwest trees. The interface between them is a Peano-type exploration path that traverses edges and crosses faces in a canonical order. Recording appropriate tree distances along that interface yields a two-dimensional lattice walk, and conversely a compatible walk reconstructs the map by a sewing procedure (Kenyon et al., 2015).

For bipolar-oriented triangulations, one coordinate convention gives the step set

$\mathrm{SLE}_{12}$4

with each increment occurring with probability $\mathrm{SLE}_{12}$5. Another, equivalent convention gives

$\mathrm{SLE}_{12}$6

again with equal probabilities $\mathrm{SLE}_{12}$7. The difference reflects a choice of coordinates and compass rotation, not a different model. In the finite setting the walk is conditioned to stay in the first quadrant or an equivalent cone and to hit a prescribed endpoint; under this conditioning, the walk is in bijection with a finite bipolar-oriented triangulation with prescribed boundary lengths (Gwynne et al., 2016).

For UIBOT, uniform re-rooting in the finite map corresponds on the walk side to choosing a uniform time and recentering the conditioned walk there. After recentering, finite windows of the conditioned walk converge in total variation to a bi-infinite simple random walk with i.i.d. increments uniform on the triangulation step set. Applying the sewing procedure locally to that bi-infinite walk yields the infinite map $\mathrm{SLE}_{12}$8. Thus UIBOT is characterized by a bi-infinite i.i.d. encoding walk together with the local sewing map from walk increments to faces and edges (Gwynne et al., 2016).

This encoding is the reason UIBOT occupies a particularly rigid position among infinite random planar maps. It is not merely an infinite triangulation with an orientation; it is an infinite triangulation whose bipolar structure, interface, and canonical trees are all encoded by a stationary two-sided walk. A plausible implication is that many local questions on UIBOT reduce to renewal, excursion, or invariance-principle statements for that walk and for auxiliary processes derived from it.

3. Peanosphere scaling limit and the dual orientation

The continuum interpretation of UIBOT is given by the peanosphere, or mating-of-trees, formalism. In this framework, a $\mathrm{SLE}_{12}$9-Liouville quantum gravity cone decorated by an independent whole-plane space-filling $\mathrm{SLE}_{12}$0 is encoded by a correlated two-dimensional Brownian motion $\mathrm{SLE}_{12}$1 with

$\mathrm{SLE}_{12}$2

For bipolar-oriented triangulations the discrete walk has correlation $\mathrm{SLE}_{12}$3, so $\mathrm{SLE}_{12}$4, which gives $\mathrm{SLE}_{12}$5 and $\mathrm{SLE}_{12}$6. The discrete contour functions therefore converge to the Brownian peanosphere coordinates of a $\mathrm{SLE}_{12}$7-LQG surface decorated by space-filling $\mathrm{SLE}_{12}$8 (Gwynne et al., 2016).

For the primal UIBOT, if $\mathrm{SLE}_{12}$9 denotes the encoding walk, the rescaling

$\ell,r \ge 0$0

yields convergence to a correlated Brownian motion with covariance matrix

$\ell,r \ge 0$1

The dual infinite bipolar-oriented triangulation has its own Peano exploration and height pair $\ell,r \ge 0$2. Although the dual walk is not a simple random walk with i.i.d. increments, it admits a functional central limit theorem, and the primal and dual encodings converge jointly: $\ell,r \ge 0$3 in the topology of uniform convergence on compact sets (Gwynne et al., 2016).

The limiting object is a single $\ell,r \ge 0$4-quantum cone equipped with two coupled space-filling $\ell,r \ge 0$5 curves: a north-going curve for the primal orientation and a west-going curve for the dual orientation. In the imaginary-geometry language used in the paper, the second curve travels in a direction perpendicular to the first. This confirms the conjectural picture that a bipolar-oriented triangulation and its dual should jointly converge to two orthogonal $\ell,r \ge 0$6 trees on the same LQG surface (Gwynne et al., 2016).

A recurring point of clarification is that this is convergence in the peanosphere sense, not graph-metric convergence. The contour functions converge, and hence the tree-decorated curve system converges in the mating-of-trees encoding, but this does not identify the scaling limit of the graph metric. For $\ell,r \ge 0$7, including $\ell,r \ge 0$8, the corresponding metric-convergence problem remains open (Gwynne et al., 2016).

4. Directed geometry, Busemann functions, and path exponents

Recent work studies UIBOT as a directed random planar map, focusing on longest and shortest directed paths rather than solely on contour encodings. In this setting one defines a canonical infinite submap $\ell,r \ge 0$9 inside the UIBOT using the root edge together with a rightmost directed path to $\ell+r>0$0 and a leftmost directed path from $\ell+r>0$1. Its boundary vertices $\ell+r>0$2 provide the natural interface along which directed distance to infinity is measured (Borga et al., 30 Oct 2025).

The key object is a directed Busemann function $\ell+r>0$3, normalized by $\ell+r>0$4, defined so that differences $\ell+r>0$5 stabilize as differences of directed distances from $\ell+r>0$6 and $\ell+r>0$7 to sufficiently distant common targets. This construction is carried out for both $\ell+r>0$8, the length of the longest directed path, and $\ell+r>0$9, the length of the shortest directed path. The resulting Busemann increments are independent, stationary on each half-line, and satisfy one-sided monotonicity inequalities reflecting the orientation of the boundary (Borga et al., 30 Oct 2025).

The increments have heavy tails with distinct exponents in the longest- and shortest-path regimes. In the LDP case,

$n$0

while in the SDP case,

$n$1

Accordingly, the rescaled Busemann process converges to stable Lévy limits: in the LDP case, $n$2 converges to a process whose positive-time restriction is a $n$3-stable subordinator up to sign; in the SDP case, $n$4 converges to a $n$5-stable Lévy process with only upward jumps on positive times (Borga et al., 30 Oct 2025).

These one-dimensional limits feed back into the geometry of finite portions of UIBOT. In a typical subset with $n$6 edges, longest directed path lengths are of order $n$7 and shortest directed path lengths are of order $n$8, up to multiplicative constants. Equivalent up-to-constants statements hold for Boltzmann bipolar-oriented triangulations and for the finite KMSW cells $n$9 cut out by the walk encoding. The paper interprets these exponents as scaling dimensions for discretizations of hypothetical $\ell$0-directed LQG metrics (Borga et al., 30 Oct 2025).

This directed-geometric viewpoint materially extends the earlier peanosphere picture. The contour process describes how UIBOT approaches $\ell$1-LQG with $\ell$2; the Busemann theory describes how directed geodesics organize within that discrete geometry and isolates stable processes that are not visible in the basic Brownian mating-of-trees encoding.

5. Relation to finite-volume theory and to adjacent universality classes

The finite-volume precursor to UIBOT is the uniform bipolar-oriented triangulation with fixed boundary lengths and fixed number of edges. The 2015 work on bipolar orientations established the bijection with lattice walks and proved that uniformly random bipolar-oriented planar maps, including triangulations, quadrangulations, $\ell$3-angulations, and mixed face-degree models satisfying suitable analytic conditions, converge in the peanosphere topology to a $\ell$4-LQG sphere decorated by an independent $\ell$5 curve (Kenyon et al., 2015).

That finite-volume theory already identified the natural infinite-volume encoding: a bi-infinite i.i.d. walk on $\ell$6-indexed times. What it did not yet do was construct the local weak limit explicitly as an infinite rooted map or prove the joint primal-dual scaling limit at infinite volume. Those steps were supplied later, producing the object now standardly called UIBOT and clarifying that its dual orientation lives on the same continuum LQG surface in the scaling limit (Gwynne et al., 2016).

UIBOT also sits at the boundary of a broader family of bipolar-oriented random planar maps with heavy-tailed face degrees. In that setting the face-degree weights satisfy $\ell$7 for $\ell$8, the corresponding infinite-volume bipolar-oriented map exists as a Benjamini–Schramm limit, and the rescaled contour functions converge not to Brownian motion but to a correlated pair of $\ell$9-stable Lévy processes. Combined with related continuum work, this identifies the scaling limit with $r$0 on $r$1-LQG, where

$r$2

In this sense, UIBOT is the Brownian, finite-variance endpoint of a larger stable family of bipolar-oriented maps (Kavvadias et al., 2022).

This comparison is conceptually useful. It shows that the Brownian peanosphere limit for UIBOT is not an isolated phenomenon but the finite-variance representative of a continuum of bipolar models. The transition from UIBOT to the heavy-tailed regime replaces correlated Brownian coordinates by correlated stable coordinates and replaces the $r$3 picture by exotic $r$4 processes with $r$5 (Kavvadias et al., 2022).

6. Conceptual status, misconceptions, and open directions

A common point of confusion is the scope of current convergence results. What is rigorously established for UIBOT is the peanosphere scaling limit: the primal and dual height functions converge jointly to continuum boundary-length processes that encode a $r$6-LQG surface decorated by two orthogonal space-filling $r$7 curves. What is not established is graph-metric convergence of UIBOT itself to an LQG metric surface. The available results therefore concern the mating-of-trees encoding, the curve system, and quantum boundary lengths rather than a full metric identification (Gwynne et al., 2016).

Another point is historical rather than mathematical. The 2015 finite-volume work effectively gave the infinite-volume walk encoding and the continuum $r$8-quantum cone interpretation, but did not by itself construct the infinite rooted map as a Benjamini–Schramm limit. The later infinite-volume analysis made that local-limit object explicit and showed that the dual bipolar structure converges jointly with the primal one (Kenyon et al., 2015).

Several open problems now organize the subject. On the peanosphere side, one would like stronger forms of convergence and a fuller understanding of the imaginary-geometry grid generated by multiple flow-line directions. On the directed-geometry side, the construction of continuum $r$9-directed LQG metrics remains open, as do joint scaling limits of Busemann functions, universality of the $\ell=2$0 and $\ell=2$1 exponents across other bipolar-oriented map models, finite-volume directed metric scaling limits, and a Schaeffer-type bijection that would encode directed distances by labels on a tree (Borga et al., 30 Oct 2025).

Taken together, these results place UIBOT at a junction of bijective combinatorics, random planar maps, imaginary geometry, and directed random geometry. Discretely, it is an infinite bipolar-oriented triangulation encoded by a bi-infinite walk and organized by canonical primal and dual trees. Continuum-wise, it is a discrete model for $\ell=2$2-Liouville quantum gravity with space-filling $\ell=2$3, enriched by a dual orthogonal $\ell=2$4 structure and by stable-process phenomena governing directed distances (Gwynne et al., 2016).