Directed distances in bipolar-oriented triangulations: exact exponents and scaling limits

TL;DR

This work exactly solves directed distances in the uniform infinite bipolar-oriented triangulation (UIBOT) by constructing a Busemann function that encodes directed distance to infinity along a natural interface. Leveraging the Kenyon–Miller–Sheffield–Wilson (KMSW) bijection, the authors prove that the Busemann function converges, under appropriate rescaling, to a $2/3$-stable Lévy process for the longest directed paths and to a $4/3$-stable Lévy process for the shortest directed paths. They further obtain precise scaling exponents for directed distances in finite models: longest directed paths grow like $n^{3/4}$ and shortest like $n^{3/8}$ in typical size-$n$ submaps, and in Boltzmann maps with right boundary length $r$, $LDP hicksim r^{3/2}$ and $SDP hicksim r^{3/4}$, with corresponding edge-count scaling. Importantly, all arguments are discrete and rely solely on the KMSW encoding and elementary probabilistic tools, not on continuum LQG formalisms, yet the results align with directed LQG expectations for the $ heta=rac{ frac{eta}{2}}{1}$-type parameter that yields solvable directed metrics. These findings establish exact scaling exponents for directed distances in a nontrivial random planar-map model and suggest a concrete directed Liouville quantum gravity framework, with potential extensions to other decorated map classes and to increasing subsequences in pattern-avoiding permutations.

Abstract

We study longest and shortest directed paths in the following natural model of directed random planar maps: the uniform infinite bipolar-oriented triangulation (UIBOT), which is the local limit of uniform bipolar-oriented triangulations around a typical edge. We construct the Busemann function which measures directed distance to $\infty$ along a natural interface in the UIBOT. We show that in the case of longest (resp.\ shortest) directed paths, this Busemann function converges in the scaling limit to a $2/3$-stable Lévy process (resp.\ a $4/3$-stable Lévy process). We also prove up-to-constants bounds for directed distances in finite bipolar-oriented triangulations sampled from a Boltzmann distribution, and for size-$n$ cells in the UIBOT. These bounds imply that in a typical subset of the UIBOT with $n$ edges, longest directed path lengths are of order $n^{3/4}$ and shortest directed path lengths are of order $n^{3/8}$. These results give the scaling dimensions for discretizations of the (hypothetical) $\sqrt{4/3}$-directed Liouville quantum gravity metrics. The main external input in our proof is the bijection of Kenyon-Miller-Sheffield-Wilson (2015). We do not use any continuum theory. We expect that our techniques can also be applied to prove similar results for directed distances in other random planar map models and for longest increasing subsequences in pattern-avoiding permutations.

Figures (18)

• Figure 1: Left: A finite bipolar-oriented triangulation, drawn so that edges are directed from south-west to north-east (i.e. edges can point in the direction highlighted in yellow in the top-left circle). The four colored directed path denote the leftmost/rightmost paths introduced just before Definition \ref{['defn-future-map']}. Right: A portion of the infinite rooted submap $(M_{0,\infty},\lambda_0)$ of the UIBOT introduced in Definition \ref{['defn-future-map']}. Edges on the boundary of $M_{0,\infty}$ lying to the west of the root edge $\lambda_0$ are not considered to be part of $M_{0,\infty}$. See Definition \ref{['defn:planar-map-with-missing-edges']} for a precise definition of planar map with missing edges.
• Figure 2: Left: The blue regions are the sets of points reachable by directed paths started from the origin for the hypothetical directed LQG metric in the cases when $\widetilde{\theta} \in (0,\pi)$, $\widetilde{\theta} =\pi$, $\widetilde{\theta} \in (\pi,2\pi)$, and $\widetilde{\theta} =2\pi$. The colored curves are flow lines (in the sense of Imaginary Geometry ig4) of a whole plane-GFF, with angles $-\pi/2$ and $\widetilde{\theta}-\pi/2$. In particular, they are each SLE$_{\gamma^2}(\gamma^2-2)$ curves. Right: Graph of the possible parameter values for the directed LQG metrics. The parameter value $\widetilde{\theta} = 2\pi$ (dashed horizontal line) corresponds to the undirected LQG metric. When $\widetilde{\theta} \in (0,\pi)$ (green region), both shortest-path and longest-path directed LQG metrics should exist. When $\widetilde{\theta} \in [\pi,2\pi]$ (yellow region), only the shortest-path directed LQG metric should exist. We do not know whether there is any notion of directed LQG metrics when $\widetilde{\theta} > 2\pi$. The red curve corresponds to parameter values where we expect additional solvability. The point $(\sqrt{4/3},\pi/2)$ corresponds to the scaling limit of directed distances in bipolar-oriented planar maps (i.e., the setting considered in this paper). The point $(\sqrt{8/3},2\pi)$ corresponds to the scaling limit of undirected distances on uniform planar maps (i.e., Brownian surfaces). The points $(\sqrt 2 , \pi)$ and $(1,\pi/3)$ correspond to other models which we expect can be analyzed using the techniques of this paper (spanning tree decorated maps and Schnyder-wood decorated maps, respectively); see Section \ref{['sec-other']}. The segment $\{2\}\times (0,\pi)$ corresponds to the $\gamma\to 2$ case, which is closely related to Brownian separable permutons as studied in abbds-separable-lis (see Remark \ref{['remark-critical-metric']}).
• Figure 3: Left: Last passage percolation on $\mathbbm Z^2$. A portion of $\mathbbm Z^2$, rotated counterclockwise by 45 degrees, is shown. A reference point $(\overline x, \overline y)$ is shown in blue. The grey labels associated with each vertex $(x,y)$ are the weights $\omega_{x,y}$, while the purple labels are the last-passage times $\operatorname{LPT}(x,y)$ introduced in \ref{['eq:LPPFPP-defn']}. The labels on the bottom of the picture are the $\operatorname{LPT}(x,y)$ times of the corresponding green vertices renormalized so that the $\operatorname{LPT}$ time for the squared violet vertex is zero. The orange paths are $\operatorname{LPT}$-geodesics, i.e. directed paths that realize the $\operatorname{LPT}$ times. Middle: A bipolar-oriented triangulation with its vertices labeled by their $\operatorname{LDP}$ distance to the sink vertex, as introduced in Definition \ref{['def-ldp']}. The orange paths are $\operatorname{LDP}$ geodesics from each vertex to the sink. Right: A portion of the infinite rooted submap $(M_{0,\infty},\lambda_0)$ of the UIBOT introduced in Definition \ref{['defn-future-map']}. The boundary vertices are labeled in purple by the values of the Busemann function $\mathcal{X}$ for $\operatorname{LDP}$ introduced in Theorem \ref{['thm-busemann']}. These labels are the analogue of the bottom labels in the picture on the left when one sends the blue reference point $(\overline x, \overline y)$ to infinity (recall that the limiting labels depend on the direction $\xi$ in which we send $(\overline x, \overline y)$ to infinity compared to the squared violet vertex). The orange paths are $\operatorname{LDP}$ geodesics from the boundary vertices to the sink "at infinity" in the sense of Definition \ref{['def-infinite-geo']}. The two models in the middle and right picture are a natural version of last-passage percolation on planar maps, one in finite and one in infinite volume.
• Figure 4: Left: An oriented planar triangulation with missing edges. The missing edges are the dotted (non-oriented) edges on the external face. The upper-left/lower-left/upper-right/lower-right boundaries of the triangulation on the left are shown in darkred/red/lightblue/blue. Note that the upper-right and lower-left boundaries are formed by missing edges. Right: The first bipolar-oriented triangulation $\mathfrak m$ is the map on the left picture where we fixed an orientation of the missing edges (turning them into non-missing edges) to obtain a bipolar orientation. The second map $\mathfrak m^{{\operatorname{b}}}$ is a boundary-channeled bipolar-oriented triangulation (Definition \ref{['def-reverse-map']}). The trees $T_{UL}(\mathfrak m)$ and $T_{LR}(\mathfrak m)$ (resp. $T_{UL}(\mathfrak m^{{\operatorname{b}}})$ and $T_{LR}(\mathfrak m^{{\operatorname{b}}})$) introduced in the description of the inverse KMSW procedure (below Lemma \ref{['lem-kmsw-bdy']}) are shown in red and blue in both maps. The corresponding Peano paths are shown in gold. Note that the right-most branch of $T_{UL}(\mathfrak m)$ has two edges that correspond to the first two right-boundary edges of $\mathfrak m$, while the right-most branch of $T_{UL}(\mathfrak m^{\operatorname{b}})$ has four edges that correspond to the four (and so all) right-boundary edges of $\mathfrak m^{\operatorname{b}}$.
• Figure 5: A Boltzmann bipolar-oriented triangulation $M(l,r)$ with left boundary length $l=3$ and right boundary length $r=4$. In orange, the leftmost $\operatorname{LDP}$ geodesic $P$ from the source to the sink, as introduced in Definition \ref{['def-geodesic']}. The map $M^L$ (resp. $M^R$) is the submap of $M(l,r)$ consisting of the vertices, edges, and faces of $M(l,r)$ in the lightblue (resp. lightgreen) region including the vertices and edges which lie on the orange leftmost ${\operatorname{XDP}}$ geodesic $P$. Note that the lightblue map $M^L$ does not contain any other geodesic from the source to the sink other than $P$, while the lightgreen map $M^R$ contains additional geodesics. By Lemma \ref{['lem-lr-ind']}, the submaps $M^L$ and $M^R$ are conditionally independent given the length of $P$.

Theorems & Definitions (174)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 1.9
• Definition 1.10