Joint scaling limit of a bipolar-oriented triangulation and its dual in the peanosphere sense

TL;DR

This work proves a joint scaling limit for the height-function encodings of a uniform infinite bipolar-oriented triangulation and its dual, showing convergence to two correlated Brownian motions that encode the same $\\sqrt{4/3}$-LQG cone decorated by two independent space-filling SLE$_{12}$ curves traveling in perpendicular directions. The approach combines discrete bijections between bipolar-oriented maps and lattice walks with the peanosphere framework (mating-of-trees) for LQG, and a continuum decomposition based on imaginary geometry. The results establish a concrete peanosphere limit for the joint primal-dual setting and identify a precise normalization (factor $3$ between dual and primal Brownian motions) and the Brownian correlation structure, including a special case where $\\kappa' = 12$ and $\\theta = 0$ yielding a local-time constant $c=2$. This work lays groundwork for connections between LQG, SLE, and combinatorial permutation models such as Baxter permutons, broadening the scope of peanosphere universality across map ensembles.

Abstract

Kenyon, Miller, Sheffield, and Wilson (2015) showed how to encode a random bipolar-oriented planar map by means of a random walk with a certain step size distribution. Using this encoding together with the mating-of-trees construction of Liouville quantum gravity (LQG) due to Duplantier, Miller, and Sheffield (2014), they proved that random bipolar-oriented planar maps converge in the scaling limit to a $\sqrt{4/3}$-LQG surface decorated by an independent SLE$_{12}$ in the peanosphere sense, meaning that the height functions of a particular pair of trees on the maps converge in the scaling limit to the correlated planar Brownian motion which encodes the SLE-decorated LQG surface. We improve this convergence result by proving that the pair of height functions for an infinite-volume random bipolar-oriented triangulation and the pair of height functions for its dual map converge jointly in law in the scaling limit to the two planar Brownian motions which encode the same $\sqrt{4/3}$-LQG surface decorated by both an SLE$_{12}$ curve and the ``dual'' SLE$_{12}$ curve which travels in a direction perpendicular (in the sense of imaginary geometry) to the original curve. This confirms a conjecture of Kenyon, Miller, Sheffield, and Wilson (2015). Our paper is the starting point of recent works connecting LQG and random permutons such as the Baxter permuton.

Figures (9)

• Figure 1: Left: A bipolar-oriented triangulation $(G , \mathcal{O})$ embedded into the plane in such a way that direction of every edge is upwards. Middle: one can associate an east-going tree $\mathcal{T}^E$ (shown in red) with $(G , \mathcal{O})$ by cutting each edge other than the leftmost edge leading upward from each vertex at the point where it meets that vertex; and a west-going tree $\mathcal{T}^W$ (shown in blue) by cutting each edge other than the rightmost edge leading downward from each vertex at the point where it meets that vertex. Right: the exploration path $\lambda'$ (shown in green) associated with $(G , \mathcal{O})$, which traces the interface between the trees, starting at the southeast pole and ending at the northwest pole.
• Figure 2: The dual map $(\widetilde{G} , \widetilde{\mathcal{O}})$ (orange) associated with a bipolar-oriented triangulation $(G , \mathcal{O})$ and its associated trees. Note that the northeast and southwest poles of $(\widetilde{G} ,\widetilde{\mathcal{O}})$ correspond to the northeast and southwest pole faces in the augmented version of $G$. The extra edge added to $G$ to form the augmented map is shown as a dashed line. The edges of the dual map are oriented leftwards, i.e. from northeast to southwest. The trees associated with $(\widetilde{G} ,\widetilde{\mathcal{O}})$ are the north-going tree (purple) and the south-going tree (green).
• Figure 3: An illustration of the peanosphere construction of wedges. Let $Z=(L_t,R_t)_{t\in\mathbbm{R}}$ be a correlated two-dimensional Brownian motion. Let $\phi:\mathbbm{R}\to(0,1)$ be an increasing, continuous and bijective function, and for any $t\in(0,1)$ define $L_t' :=\phi(L_{\phi^{-1}(t)})$ and $R_t' :=\phi(L_{\phi^{-1}(t)})$. The left figure shows $R'$ and $C- L'$, where $C$ is a constant chosen so large that the two graphs do not intersect. We draw horizontal lines above the graph of $C- L'$ and below the graph of $R'$, in addition to vertical lines between the two graphs, and then we identify points which lie on the same horizontal or vertical line segment. We also identify all points on the boundary of the square. As explained in wedges, it follows from Moore's theorem moo25 the resulting object is a topological sphere. The sphere is decorated with a space-filling path $\eta'$ where $\eta'(t)$ for $t \in \mathbbm{R}$ is the equivalence class of $(\phi(t),L'_t)$. The pushforward of Lebesgue measure on $\mathbbm{R}$ induces an area measure $\mu$ on the sphere. The resulting structure, i.e. the topological sphere with the curve $\eta'$ and the measure $\mu$, is called a peanosphere. It is shown in wedges that the peanosphere has a canonical embedding into $\mathbbm{C}$ where the pushforward of $\mu$ encodes a LQG surface known as the $\gamma$-quantum cone and $\eta'$ is an independent space-filling $\mathop{\mathrm{SLE}}\nolimits_{\kappa'}$, $\kappa'=16/\gamma^2$. The right part of the figure shows a subset of the SLE-decorated LQG surface, where the green region corresponds to points that are visited by $\eta'$ before some time $t_0$. The two trees are embeddings of the trees with contour functions $L$ and $R$, respectively, such that $L$ (resp. $R$) encode the quantum boundary length of the left (resp. right) frontier of $\eta'$. If $Z=(L_t,R_t)_{t \in [0,1]}$ was a Brownian excursion we would obtain a finite volume LQG surface decorated with an independent space-filling SLE by a similar procedure wedgessphere-constructions.
• Figure 4: The purple curve shows the north-going flow line $\lambda^N$. The process $\mathcal{L}$ (resp. $\mathcal{R}$) gives the height in the blue west-going (resp. red east-going) tree. The light blue (resp. orange) thick lines represent the stopping times $N_k^E$ (resp. $N_k^W$), and the time $N_1^E=0$ is shown in dashed light blue.
• Figure 5: The thick black edges correspond to times of type (a) and (b), respectively, as defined in the proof of Lemma \ref{['prop:discrete-excursions']}. The north-going flow line $\lambda^N$ is shown in purple, and the red (resp. blue) edges are part of the east-going (resp. west-going) tree.

Theorems & Definitions (76)

• Remark 1.1
• Remark 1.5
• Remark 1.6
• Theorem 1.7
• Remark 1.8
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3