A mating-of-trees approach for graph distances in random planar maps

We introduce a general technique for proving estimates for certain random planar maps which belong to the $\gamma$-Liouville quantum gravity (LQG) universality class for $\gamma \in (0,2)$. The family of random planar maps we consider are those which can be encoded by a two-dimensional random walk with i.i.d.\ increments via a mating-of-trees bijection, and includes the uniform infinite planar triangulation (UIPT; $\gamma=\sqrt{8/3}$); and planar maps weighted by the number of different spanning trees ($\gamma=\sqrt 2$), bipolar orientations ($\gamma=\sqrt{4/3}$), or Schnyder woods ($\gamma=1$) that can be put on the map. Using our technique, we prove estimates for graph distances in the above family of random planar maps. In particular, we obtain non-trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the Watabiki (1993) prediction for the Hausdorff dimension of $\gamma$-LQG and we establish the existence of an exponent for certain distances in the map. The basic idea of our approach is to compare a given random planar map $M$ to a mated-CRT map---a random planar map constructed from a correlated two-dimensional Brownian motion---using a strong coupling (Zaitsev, 1998) of the encoding walk for $M$ and the Brownian motion used to construct the mated-CRT map. This allows us to deduce estimates for graph distances in $M$ from the estimates for graph distances in the mated-CRT map which we proved (using continuum theory) in a previous work. In the special case when $\gamma=\sqrt{8/3}$, we instead deduce estimates for the $\sqrt{8/3}$-mated-CRT map from known results for the UIPT. The arguments of this paper do not directly use SLE/LQG, and can be read without any knowledge of these objects.


Overview
A planar map is a graph embedded in the plane, viewed modulo orientation-preserving homeomorphisms 1 . Random planar maps are a natural model of discrete random surfaces and are of fundamental importance in mathematical physics, probability, and combinatorics. Many interesting random planar maps converge in various topologies to γ-Liouville quantum gravity (LQG) surfaces -a family of continuum random fractal surfaces -with the parameter γ ∈ (0, 2] depending on the random planar map model. Such random planar maps are said to belong to the γ-LQG universality class. The 8/3-LQG universality class contains uniform random planar maps, including uniform triangulations, quadrangulations, and uniform maps with unconstrained face degree. The γ-LQG universality class for γ = 8/3 contains random planar maps sampled with probability proportional to the partition function of some statistical mechanics model on the map, e.g., the uniform spanning tree (γ = √ 2), bipolar orientations (γ = 4/3), the Ising model (γ = √ 3), or the Schnyder wood (γ = 1). It is expected (and in some cases known) that the universality class is affected only by changing the statistical mechanics model, not by changing the microscopic features of the planar map such as constraints on face or vertex degree.
The goal of this paper is to set up a general framework for proving estimates for a certain family of random planar maps in the γ-LQG universality class for γ ∈ (0, 2) and apply this framework to obtain estimates for graph distances on these maps (see Section 1.4 for precise statements). The family of planar maps we consider are those which can be encoded by a two-dimensional random walk with i.i.d. increments via a so-called mating-of-trees bijection. It includes the uniform infinite planar triangulation (UIPT) as well as infinite-volume limits of planar maps sampled with probability proportional to the number of different spanning trees, bipolar orientations, or Schnyder woods that can be put on the map.
Using a strong coupling of the encoding walk with Brownian motion [KMT76,Zai98], we will obtain a coupling of each map in our family with the γ-mated-CRT map-a specific random planar map in the γ-LQG universality class constructed from a correlated two-dimensional Brownian motion (see Section 1.2.1 for a precise definition). The mated-CRT map, in turn, is directly connected to SLE and LQG, so can be studied using continuum theory (we will not use this theory directly in the present paper, however).
To estimate graph distances in our family of random planar maps, we will show that in the above coupling the mated-CRT map and the other map are roughly isometric up to a polylogarithmic factor with high probability (Theorem 1.5). We will then deduce estimates for distances in the other map from the estimates for distances in the mated-CRT map which were proven in [GHS19]. See Figure 1 for a schematic illustration of our approach.
In particular, for each of the maps in our family, we obtain upper and lower bounds for the cardinality of a graph-metric ball consistent with the prediction of Watabiki [Wat93] for the dimension of γ-LQG (Theorem 1.6); and establish the existence of an exponent χ which describes various distances in the planar map and which, at least for γ ≤ √ 2, we expect be the scaling exponent for the graph distance needed to get a non-trivial scaling limit in the Gromov-Hausdorff topology (Theorem 1.7). To our knowledge, our results constitute the first non-trivial bounds for distances for planar maps weighted by spanning trees, bipolar orientations, or Schnyder woods. Since distances in uniform maps are already well understood, in the case when γ = 8/3 we instead obtain sharper bounds for distances in the 8/3-mated-CRT map than those in [GHS19] (Theorem 1.8).
The results of this paper can also be used to prove other results for random planar maps. For example, our coupling theorems are used in [GM17,GH18] to solve several open problems concerning the simple random walk on the UIPT and on the other random planar maps considered in this paper (including computing the exponents for the return probability and graph distance displacement of the walk). Our results are also used in [DG18] to relate the metric ball volume exponent for random planar maps to various exponents related to LQG -arising from the Liouville heat kernel, Liouville graph distance, and Liouville first passage percolation -and to obtain new bounds for these exponents. More generally, since there is a wide range of tools available for studying mated-CRT maps (due to their connection to SLE/LQG) we expect that the techniques developed in this paper could potentially be useful whenever one needs to prove estimates for random planar maps which can be encoded by mating-of-trees bijections.
In the rest of this section, we provide some background on distances in random planar maps, define mated-CRT maps, introduce some notation, and state our main results. Section 2 contains the core of our argument, where we apply the strong coupling theorem of [Zai98] to compare graph distances for mated-CRT maps and other random planar maps. In Section 3 we review the mating-of-trees bijections for several particular random planar maps, then use the results of Section 2 to deduce our main results (since the bijections for the various maps are defined in slightly different ways, we need to treat the maps individually). In Section 4, we discuss some open problems.

SLE and LQG theory
Distance bounds for mated-CRT maps [GHS19] Correlated 2d Brownian motion Distance bounds for other random planar maps Mating-oftrees theorem [DMS14] Mated-CRT map definition 2d random walk Bijective encoding Strong coupling [KMT76,Zai98] Strong coupling (Theorem 1.5) Figure 1: Schematic illustration of how the results of this paper fit together with other works (blue indicates what is done in the present work). As explained in Section 1.2.1, the mated-CRT map can be constructed from a two-dimensional correlated Brownian motion, or equivalently from a γ-LQG surface decorated by a space-filling SLE κ for κ = 16/γ 2 due to the results of [DMS14]. In [GHS19], we proved estimates for distances in mated-CRT maps using a combination of Brownian motion and SLE/LQG techniques. In the present paper, we transfer these estimates to other random planar map models which can be bijectively encoded by a two-dimensional random walk with i.i.d. increments. This is done by coupling the encoding walk with the Brownian motion used to define the mated-CRT map using [Zai98]; and arguing that in such a coupling, the other random planar map and the mated-CRT map are roughly isometric up to a polylogarithmic factor.

Mating-of-trees bijections and mated-CRT maps
Many random planar map models in the γ-LQG universality class for γ ∈ (0, 2) can be encoded by a two-dimensional random walk via a bijection of mating-of-trees (a.k.a. peanosphere) type. The reason for the name is that these bijections can be interpreted as gluing together two discrete trees (corresponding to the two coordinates of a two-dimensional random walk) to construct a planar map decorated by a space-filling ("peano") curve.
The first mating-of-trees bijection is that of Mullin [Mul67] (explained more explicitly in [Ber07b]), which encodes a random spanning tree-decorated planar map by means of a nearest-neighbor random walk in Z 2 .
Sheffield's hamburger-cheeseburger bijection [She16] (which is a re-formulation of a bijection in [Ber07b]) is a generalization of Mullin's bijection which encodes a planar map decorated by the critical Fortuin Kasteleyn (FK) model [FK72] with parameter q ∈ (0, 4). There are also mating-of-trees bijections for bipolar-oriented planar maps [KMSW19], uniform random triangulations decorated by site percolation [Ber07a,BHS18], random planar maps decorated by a Schnyder wood [LSW17], and random planar maps decorated by an activity-weighted spanning tree [GKMW18]. For γ ∈ (0, 2), the γ-mated-CRT map is a random planar map in the γ-LQG universality class, constructed by means of a semi-continuous variant of a mating-of-trees bijection with a correlated two-dimensional Brownian motion in place of a random walk. Suppose γ ∈ (0, 2) and Z = (L, R) is a two-sided, twodimensional Brownian motion with variances and covariances Var(L t ) = Var(R t ) = |t| and Cov(L t , R t ) = − cos(πγ 2 /4)|t|, ∀t ∈ R.
We define the mated-CRT map G to be the graph with vertex set Z, with two vertices x 1 , x 2 ∈ Z with (1.2) Note that (1.2) holds in particular if |x 1 − x 2 | = 1. The vertices x 1 and x 2 are connected by two edges if both conditions in (1.2) hold but |x 1 − x 2 | > 1. See Figure 2 for an illustration of this definition and an explanation of why G is a planar map (in fact, a triangulation). 2 Z L C − R Z Figure 2: Left: To construct the mated-CRT map G geometrically, one can draw the graph of L (red) and the graph of C − R (blue) for some large constant C > 0 chosen so that the parts of the graphs over some time interval of interest do not intersect. One then divides the region between the graphs into vertical strips (boundaries shown in orange) and identifies each strip with the horizontal coordinate x ∈ Z of its rightmost point. Vertices x 1 , x 2 ∈ Z are connected by an edge if and only if the corresponding strips either share a boundary segment (i.e., |x 1 − x 2 | = 1) or are connected by a horizontal line segment which lies under the graph of L or above the graph of C − R. One such segment is shown in green in the figure for each pair of vertices for which this latter condition holds. Right: One can draw the graph G in the plane by connecting two vertices x 1 , x 2 ∈ Z by an edge by an arc above (resp. below) the blue line if the corresponding strips are connected by a horizontal segment above (resp. below) the graph of L (resp. C − R); and connecting each pair of consecutive vertices of Z by an edge. This gives G a planar map structure under which it is a triangulation. Note that the two pictures do not correspond to the same mated-CRT map realization.
The mated-CRT map is a discretized mating of the continuum random trees (CRT's) [Ald91a,Ald91b,Ald93] associated with L and R. As we will see in Section 3, this adjacency condition (1.2) is an exact continuum analogue of the condition for two vertices to be adjacent in terms of the encoding walk in various discrete mating-of-trees bijections.
It follows from [DMS14, Theorem 1.9] (and is explained in more detail in [GHS19]) that the mated-CRT map is directly connected to SLE-decorated Liouville quantum gravity. We will not directly use this relationship here, but we briefly mention it for the sake of context. Suppose h is the distribution corresponding to a γ-quantum cone (a type of γ-LQG surface) and η is an independent whole-plane space-filling SLE κ from ∞ to ∞ for κ = 16/γ 2 , as constructed in [MS17,Sections 1.2.3 and 4.3]. If we parametrize η : R → C in such a way that the γ-LQG mass of η([s, t]) equals t − s whenever s < t, then two vertices x 1 , x 2 ∈ Z of G are joined by an edge in G, i.e., (1.2) holds, if and only if the "cells" η([x 1 − 1, x 1 ]) and η([x 2 − 1, x 2 ]) intersect along a non-trivial connected boundary arc (this adjacency graph of cells is sometimes called the structure graph associated with (h, η)). This gives an embedding of the mated-CRT map into C which can be analyzed using SLE and LQG estimates.
Using a combination of the above relationship between mated-CRT maps and SLE/LQG and Brownian motion techniques, the paper [GHS19] proves a number of estimates for graph distances in the mated-CRT map G, valid for all γ ∈ (0, 2). This paper will transfer these estimates to other random planar maps. We also mention the recent paper [GMS17], which proves that the mated-CRT map under the so-called Tutte embedding converges to γ-LQG by showing that it is close to the a priori embedding (coming from SLE-decorated LQG) discussed above. We will not need the results of this latter paper here.

Graph distances in random planar maps
To put our results in context, we provide here some background about graph distances in random planar maps.
Uniformly random planar maps (i.e., those in the 8/3-LQG universality class) can be analyzed by means of the Schaeffer bijection [Sch97] and variants thereof (e.g., [BDFG04]), which, unlike mating-of-trees bijections, describe graph distances in a simple way. In particular, for a number of different types of uniformly random planar maps (including k-angulations for k = 3 or k ≥ 4 even) it is known that if we sample a map of size n and re-scale graph distances by n −1/4 then the re-scaled metric converges in law with respect to the Gromov-Hausdorff topology to the Brownian map [Le 13, Mie13], a continuum metric measure space constructed via a continuum analogue of the Schaeffer bijection. It is shown in [MS15,MS16a,MS16b] that the Brownian map is equivalent, as a metric measure space, to a certain type of 8/3-LQG surface. Hence the metric space structure of uniformly random planar maps is well understood.
For random planar maps in the γ-LQG universality class for γ = 8/3, however, the graph distance is not at all well understood. In this case there is no known bijective encoding analogous to the Schaeffer bijection which describes distances in a simple way. However, a candidate for the Gromov-Hausdorff limit, the so-called γ-LQG metric has recently been constructed in [GM19].
In fact, even the basic exponents for distances in random planar maps outside the 8/3-LQG universality class are unknown. Such exponents include the exponent for the diameter of a random planar map of size n or the exponent for the cardinality of a metric ball of radius n in an infinite-volume random planar map (which is expected to be its reciprocal). This latter exponent should be the same as the Hausdorff dimension d γ of the γ-LQG metric. It was predicted by Watabiki in [Wat93] that The log γ −1 for small enough γ, whereas (1.3) gives 2/d γ = 1 − O γ (γ 2 ). It follows from the results of [DZZ18,DG18] that one has an analogous estimate for the γ-mated CRT map when γ is small, which then extends to other random planar maps (in particular, the biased bipolar-oriented maps in 4 below) using the results of the present paper. This then contradicts (1.3) for random planar maps. It is not clear whether (1.3) is true in the regime when γ is not close to 0. See [GHS19,DG16,DG18] for further discussion of Watabiki's prediction and the problem of computing d γ . 5

Basic notation
We write N for the set of positive integers and N 0 = N ∪ {0}. For a, b ∈ R with a < b, we define the discrete intervals [a, b] Z := [a, b] ∩ Z and (a, b) Z := (a, b) ∩ Z.
If a and b are two quantities depending on a variable x, we write a = O x (b) (resp. a = o x (b)) if a/b remains bounded (resp. tends to 0) as x → 0 or as x → ∞ (the regime we are considering will be clear from the context). We write a = o ∞

Notation for graphs
For a planar graph G, we write V(G), E(G), and F(G), respectively, for the set of vertices, edges, and faces of G, respectively. We for the degree of v (i.e., the number of edges with v as an endpoint).
A path in G is a function P : [0, n] Z → V(G) for some n ∈ N such that P (i) ∼ P (i + 1) for i = 0, . . . , n − 1.
The length |P | of P is the integer n.
For sets A, B consisting of vertices and/or edges of G, we write dist(A, B; G) for the graph distance from A to B in G, i.e., the length of the shortest path from a vertex in A or the endpoint of an edge in A to a vertex or an endpoint of an edge in B. If A = {x} and/or B = {y} is a singleton, we drop the set brackets.
For a set A consisting of vertices and/or edges of G and r > 0, we define the graph metric ball B r (A; G) to be the subgraph of G consisting of the set of vertices of G which lie at graph distance at most r from A and the set of edges of G which join two such vertices (the set brackets are omitted if A = {x} is a singleton).
We write diam(A; G) := max x,y∈V(A) dist(x, y; G) for the graph-distance diameter of A with respect to the graph metric on G. We abbreviate diam(G) := diam(G; G). so we can apply the two-dimensional version of the KMT strong coupling theorem [KMT76,Zai98] to couple Z with the correlated Brownian motion Z of (1.1).
We emphasize that all of our results can be extended to any other random planar map model which can be encoded by a mating-of-trees bijection wherein the encoding walk can be strongly approximated by Brownian motion-the models listed above are just the most natural ones for which we presently know this to be the case. Section 2 considers a larger (but perhaps less natural) class of random planar map models for which slightly stronger versions of our results hold. See Section 4 for open problems and additional discussion concerning expanding the set of models for which the results of this paper apply.
As in Section 1.2.1, in each of the above four cases we let Z = (L, R) be a correlated two-dimensional Brownian motion correlation − cos(πγ 2 /4), as in (1.1), and we let G be the mated-CRT map constructed from Z as in (1.2).
Just below, we will state a coupling result (Theorem 1.5) which allows us to compare M and G. The strong coupling theorem [Zai98] only allows us to compare random walk and Brownian motion on a finite time interval, so to state our result we need to describe finite (approximate) subgraphs of M and G which correspond to finite time intervals. We start by defining such graphs for G.
Definition 1.1. For n ∈ N we write G n for the subgraph of G whose vertex set is [−n, n] Z and whose edge set consists of all of the edges of G between two such vertices (G n is called G 1 | [−n,n] in [GHS19]). We also write ∂G n for the subgraph of G n consisting of all vertices of G n which are connected by an edge to a vertex of G \ G n ; and all edges of G n which join two such vertices.
We now discuss the analogue of Definition 1.1 for the map M . In each of the above situations, the corresponding bijection gives for each n ∈ N a planar map M n associated with the random walk increment Z| [−n,n] Z . The maps M n are defined slightly different in each of the four cases and will be defined precisely in Section 3.
The map M n is not necessarily a subgraph of M (see, however, Remark 1.2), but there is a distinguished boundary 4 ∂M n and an "almost" inclusion function i.e., a function from V(M n ) ∪ E(M n ) to V(M ) ∪ E(M ). The injectivity of ι n on M n \ ∂M n means that we can canonically identify M n \ ∂M n with a subgraph of M . We emphasize that ι n is not injective on all of M n in case 2: see Remark 1.2. The map M n possesses a canonical root vertex which is mapped to v by ι n , and which (by a slight abuse of notation) we identify with v.
To set up a correspondence between the vertex sets V(M n ) and V(G n ) = [−n, n] Z , we will define in Section 3 for each n ∈ N functions φ n : V(M n ) → [−n, n] Z and ψ n : [−n, n] Z → V(M n ), with φ n (v) = 0 and ψ n (0) = v, (1.5) where v denotes the initial endpoint of e 0 . Roughly speaking, the vertex ψ n (i) corresponds to the ith step of the walk Z| [−n,n] Z in the bijective construction of (M, e 0 , T ) from Z and φ n is "close" to being the inverse of ψ n . However, the construction of M n from Z| [−n,n] Z does not set up an exact bijection between [−n, n] Z and the vertex set of M n , so the functions φ n and ψ n are neither injective nor surjective. See Figure 3 for an illustration of the above definitions.
Remark 1.2. In cases 1 and 3 through 5 above, each of the planar maps M n can be canonically identified with a subgraph of M and one can take ι n to be the identity map. Moreover, there are functions φ : V(M ) → Z and ψ : Z → V(M ) for which φ n = φ| V(Mn) and ψ n = ψ| [−n,n] Z . However, in case 2 M n cannot be identified with a subgraph of M since it may be be the case that one or more pairs of vertices or edges of M n get identified together to a single edge of M at a later step of the bijection; see Figure 11. Remark 1.3. For each of the random planar maps considered here, for i ∈ Z the translated encoding walk j → Z j+i − Z i has the same law as Z, so encodes rooted planar map with the same law as (M, e 0 ), decorated by a statistical mechanics model. It is easy to see from the definitions of the bijections, as reviewed in Section 3, that the random planar map encoded by the translated walk is the same as M but with a different choice of root edge which we denote by λ(i) (the statistical mechanics model may not be the same). The function λ : Z → E(M ) is a bijection in cases 2 through 5 and is two-to-one in case 1. It is tempting to try to define M n to be the subgraph of M whose edge set is λ([−n, n] Z ) and whose vertex set consists of the endpoints of edges in λ([−n, n] Z ). This definition does not work in general, however, since the resulting graph is not necessarily connected in cases 3 and 4 and the resulting graph may have pairs of vertices which get identified at a later step in case 2, as in Remark 1.2. However, with our definitions of M n one can check that in each of cases 1 through 4 above, (1.6) In the case 5 of the Schnyder wood-decorated map, (1.6) does not hold with our definitions since we treat such maps as a special case of bipolar-oriented maps for the sake of convenience (see Section 3.3.4). However, we expect that one can use the bijection from [LSW17] to give a different definition of M n for which (1.6) holds. Figure 3: Illustration of the map M n , the "almost inclusion" map ι n : M n → M , and the maps φ n : V(M n ) → V(G n ) and ψ n : V(G n ) → V(M n ). Note that two edges of ∂M n get identified when we apply ι n . Theorem 1.5 asserts that φ n and ψ n are rough isometries up to polylogarithmic factors.
The main result which allows us to compare the graph metrics on M and G is Theorem 1.5 just below, which says that these two maps are roughly isometric (up to a polylogarithmic factor), in the following sense.
Definition 1.4. Let (X 1 , d 1 ) and (X 2 , d 2 ) be metric spaces. For a, b, c > 0, a map φ : X 1 → X 2 is called a rough isometry with parameters (a, b, c) if and for each z ∈ X 2 , there is an x ∈ X 1 such that , the map φ n of (1.5) is a rough isometry from M n to G n (each equipped with their respective graph distances) with parameters a = C(log n) 4 , b = 2, and c = C(log n) 4 ; and the map ψ n is a rough isometry from G n to M n with these same values of a, b, and c.
In cases 1 and 3 through 5 above, Theorem 1.5 will be used to deduce estimates for the graph metric on M from the estimates for the graph metric on the mated-CRT map G from [GHS19] (in case 2, these estimates for M are weaker than the ones obtained from a Schaeffer-type bijection or from peeling). We now state the results we can obtain for the graph metric on M . Our first main result concerns the cardinality of graph metric balls in M , and is a planar map analogue of [GHS19, Theorem 1.10].
Theorem 1.6. In each of the settings listed above, the following is true. Let Then for each u > 0, there exists c = c(u) > 0 such that with probability 1 − O n (n −c ), we have the following bounds for the number of vertices in graph metric balls of M centered at the root edge: (1.10) As noted in [GHS19], it is expected that typically #V(B n (e 0 ; M )) = n dγ +on(1) , where d γ is the Hausdorff dimension of the γ-LQG metric (in fact, it can be proven that this is the case by combining [DG18, Theorem 1.6] and [GP19c, Corollary 1.7], both of which were established after this paper). Hence Theorem 1.6 gives upper and lower bounds for d γ . See Figure 4 for a graph of these bounds and a table giving their values for various special planar map models.
The papers [DG18,GP19a] build on the results of the present paper to prove new upper and lower bounds for #V(B n (e 0 ; M )) which are sharper than those of Theorem 1.6 for most values of γ. For example, the bound in the case of spanning-tree weighted maps is improved to 3.550408 ≤ d √ 2 ≤ 3.63299 [GP19a, Corollary 2.5]. The proof of these new bounds relies crucially on the results of the present paper both to transfer from the mated-CRT map to other random planar maps and to obtain that the volume growth exponent for the 8/3-mated-CRT map is 4 (Theorem 1.8), which is used in combination with the monotonicity (in γ) of various exponents to obtain the new bounds.  This graph was produced using Mathematica, and also appears in [GHS19]. Right: Table of the exact and approximate values of the the known upper/lower bounds and the Watabiki prediction for several special planar map models.
In [GHS19, Theorem 1.12], it is shown that for γ ∈ (0, 2), there exists an exponent χ = χ(γ) > 0 which can be defined as the limit log n (1.11) and which satisfies (1.12) Our next result shows that this same exponent χ also describes distances in the other random planar maps considered above.
Theorem 1.7. Suppose we are in one of the settings listed above and let χ be as in (1.11) (for our given choice of γ). For u > 0 we have (with the notation o ∞ n (n) as in Section 1.3), and P diam(M n ) ≥ n χ−u ≥ n −on(1) . (1.14) In particular, for each p > 0, (1.15) We note that (1.14) does not tell us that diam(M n ) ≥ n χ−on(1) with probability tending to 1 as n → ∞, although we expect this to be the case. In the special case of spanning-tree weighted planar maps (γ = √ 2), a much stronger version of this statement is proven in [GP19b, Theorem 3.1].
In case 2, certain exponents for graph distances in the UIPT are already known (see, e.g., [Ang03]), so we can deduce estimates for graph distances in the 8/3-mated-CRT map from our coupling result Theorem 1.5. For example, we get the correct exponent for the cardinality of a metric ball.

A stronger coupling theorem
Our proof of Theorem 1.5 yields a stronger statement than just the existence of rough isometries between the maps G n and M n , which we state here. such that with probability 1 − O n (n −A ), the following is true (with φ n and ψ n as in (1.5)).

For each
is hit by a total of at most C(log n) 7 of the paths P M i1,i2 for 3. We have dist(ψ n (φ n (v)), v; M n ) ≤ C(log n) 4 for each v ∈ V(M n ) and dist(φ n (ψ n (i)), i; G n ) ≤ C(log n) 4 for each i ∈ [−n, n] Z .
Theorem 1.9 is strictly stronger than Theorem 1.5. Indeed, Theorem 1.9 trivially implies Theorem 1.5 (we record this fact as Lemma 1.10 below for the sake of reference). However, the upper bounds on the total number of the paths P G v1,v2 or P M i1,i2 which hit a given vertex in Theorem 1.9 are not implied by Theorem 1.5. These bounds will be important in [GM17]. Proof. The second condition from Definition 1.4 with c = C(log n) 4 for either φ n or ψ n is immediate from condition 3 of Theorem 1.9, so we just need to check the first condition (concerning the amount by which each of φ n and ψ n distort distances).

Comparison of graph metric balls
In order to use Theorem 1.5 (or Theorem 1.9) to compare graph metric balls in M and G, we need to make sure that such balls are contained in M n and G n , respectively, with high probability. This is the purpose of the present subsection. In particular, we will establish the following lemma, which is sufficient for our purposes.
Lemma 1.11. Suppose we are in one of the five settings listed at the beginning of Section 1.4. For each A > 0, there exists K = K(A) > 0 such that with probability at least 1 − O n (n −A ), we have B n (0; G) ⊂ G n K and the map ι n K of (1.4) restricts to a graph isomorphism from B n (v; M n K ) to B n (v; M ).
To prove Lemma 1.11 we will need the following lemma.
Lemma 1.12. Suppose m < n are positive integers such that and the same holds with R in place of L. Then the map φ n of (1.5) satisfies (1.22) In each of the five cases we consider, Lemma 1.12 is an easy consequence of the definitions in Section 3. We will check the lemma separately for each case in the appropriate subsection of Section 3.
Proof of Lemma 1.11. In light of (1.4), we only need to find K = K(A) as in the statement of the lemma such that (1.23) By [GHS19, Corollary 3.2], there exists K > 2 such that with probability at least 1 − O n (n −A ), (1.24) We will now deduce from (1.24), Lemma 1.12, and Theorem 1.9 that after possibly increasing K, we also . Since (1.24) implies the analogous statement with K 2 in place of K, this will imply (1.23) with K 2 in place of K.
Recall that each of the two coordinates of Z is a one-dimensional random walk started at zero with i.i.d. increments having an exponential tail at ∞. Just below, we will explain using basic random walk estimates that if K is chosen to be sufficiently large, in a manner depending only on A, then with probability at least and the same holds with R in place of L. One way to justify this is as follows. Let B be a standard linear Brownian motion. Using the reflection principle to estimate the running minima of B, one gets that if K is chosen to be sufficiently large, in a manner depending only on A, then with probability at least 1 − O n (n −A ), (1.27) (1.28) Henceforth assume that (1.24), (1.27), and (1.28) all hold, which happens with probability at least (1.29) By (1.28), it therefore follows that for large enough n, . This gives the lemma statement with K 2 in place of K.
Remark 1.13. All of the results in this and the previous two subsections remain true with M replaced by its dual map M * . One can even use the same coupling in Theorems 1.5 and 1.9 for both M and M * simultaneously. This is because one can formulate the bijections used in this paper in terms of M * rather than M , and then use similar arguments to the ones in Section 3 to transfer from Theorem 2.1 below to estimates for M * instead of estimates for M . Similar considerations hold if, instead of M * , we consider, e.g., the so-called radial quadrangulation Q = Q(M ) whose vertex set is V(M ) ∪ V(M * ) with two vertices joined by an edge if and only if they correspond to a face of M and a vertex on the boundary of this face; or the dual map Q * of Q.

Comparing distances via strong coupling
In this section, we prove a variant of Theorem 1.9 which will be the main technical input in the proofs of our main theorems. This theorem compares the mated-CRT map G to a random planar map H constructed from a general two-sided random walk in R 2 with i.i.d. increments. Roughly speaking, this planar map is constructed via a simpler bijection than the ones corresponding to the four cases in Section 1.4, where the functions φ n and ψ n of (1.5) can be taken to be restrictions of globally defined inverse bijections φ : V(H) → Z and ψ : Z → V(H), so the vertex set of H can be identified with Z = V(G). As we will explain in Section 3, for several particular choices of Z the graph H is a close approximation of one of the random planar maps considered in Section 1.4.
Let Z = (L, R) : Z → R 2 be a two-sided two-dimensional random walk normalized so that Z 0 = 0. We assume that the increments Z j − Z j−1 of Z are i.i.d. with mean-zero; that there is a constant c > 0 such that the increment distribution satisfies (2.1) and that the walk is truly two-dimensional in the sense that the correlation ρ := Corr(L j − L j−1 , R j − R j−1 ) belongs to (−1, 1). We note that the hypothesis (2.1) is precisely the condition needed to apply the strong coupling result [Zai98, Theorem 1.3].
We define an infinite random planar map H = H(Z) via the following discrete analogue of the formula (1.2) defining the mated-CRT map. The vertex set of H is Z, and for i 1 , i 2 ∈ Z with i 1 < i 2 we declare that i 1 and i 2 are connected by an edge in H if and only if either (The arguments in this subsection are robust with respect to small modifications of (2.2); see Remark 2.5).
As in Definition 1.1, for n ∈ N we define H n to be the subgraph of H whose vertex set is [−n, n] Z , with two vertices connected by an edge in H n if and only if they are connected by an edge in H. Let γ ∈ (0, 2) be chosen so that ρ = − cos(πγ 2 /4), so that the Brownian motion Z of (1.1) for this choice of γ has correlation ρ.
Theorem 2.1. For each n ∈ N and each A > 0, there is a C > 0 depending on A and the particular law of Z and a coupling of Z with the Brownian motion Z (and thereby the mated-CRT map G) such that for each n ∈ N, the following is true with probability at least of at most C(log n) 6 of the paths P G i1,i2 . Moreover, the same is true with H and G interchanged. In particular, 3) The proof of Theorem 2.1 proceeds by coupling Z and Z using the two-dimensional variant of the KMT coupling theorem [Zai98, Theorem 1.3] then comparing the adjacency conditions 1.2 and (2.2). See Figure 5 for an illustration. The formula (2.2) is unaffected if we rescale each of the coordinates L and R by a (possibly different) constant, so we can assume without loss of generality that these coordinates are normalized so that Var(L j − L j−1 ) = Var(R j − R j−1 ) = 1.
By the multi-dimensional strong coupling theorem [Zai98, Theorem 1.3] (the higher dimensional analogue of [KMT76]), there are constants b 0 , b 1 > 0, depending only on the law of the increments of Z, and a coupling of Z with Z such that By the Chebyshev inequality, (2.4) implies that there is a constant C 0 > 0, depending on A and the law of the increments of Z, such that except on an event of probability O n (n −A ) we have max j∈[−n,n] Z |Z j − Z j | ≤ C 0 log n. Henceforth fix such a coupling and constants C 1 , C 2 ≥ C 0 ∨ 1 to be chosen later in a manner depending only on γ, A, and the law of the increments of Z.
We will show that the conditions in the statement of Theorem 2.1 are satisfied on an event E n depending on Z and Z (c.f. Lemma 2.4). In particular, we let E n = E n (C 0 , C 1 , C 2 ) be the event that the following is true.

For each pair of integers
and the same holds with R in place of L.

5) holds and j
belongs to the set in (2.6) is at most C 2 (log n) 6 ; and the same holds with R in place of L.
The condition (2.5) says that [i 1 , i 2 ] is in some sense "close" to being an excursion interval for L, in the sense that the minimum of L t − L i1 over this interval is not much less than zero and the difference |L i2 − L i1 | is small. The condition (2.6) requires the L does not get close to zero too many times during any such interval. See also Figure 5. Before checking that the conditions in the theorem statement are satisfied on E n , we show that E n occurs with high probability.
Lemma 2.2. If the constants C 1 , C 2 ≥ C 0 ∨ 1 are chosen sufficiently large, in a manner depending only on γ, A, and the law of Z, then For the proof of Lemma 2.2, we will need the following elementary lemma about Brownian motion.

Lemma 2.3. Let B be a standard linear Brownian motion and for
For each c > 0, there are constants a 0 , a 1 > 0 depending only on c such that for r ≥ 1 and s > 0, By the strong Markov property and Brownian scaling, there is a p = p(c) ∈ (0, 1) such that for each k ∈ N, Hence for s > 0, P[K ≥ s] ≤ (1 − p) s , whence (2.8) holds.
Proof of Lemma 2.2. Condition 1 holds except on an event of probability O n (n −A ) by our choice of coupling and condition 2 holds except on an event of sub-polynomial probability in n by the Gaussian tail bound, the reflection principle, and a union bound over all i ∈ [−n, n] Z .
By Lemma 2.3 applied with r = 6C 1 log n, c = 7/6, and s = 1 36 C 1 log n, there is a constant a 1 = a 1 (γ) > 0 such that for each i 1 , i 2 ∈ [−n, n] Z with i 1 ≤ i 2 and each choice of C 1 > 0, the probability of that (2.5) holds but (2.6) fails is at most O n (n −a1C1 ). If we choose C 1 sufficiently large, then by a union bound over all such pairs (i 1 , i 2 ), we find that condition 3 holds except on an event of probability O n (n −A ). Now we turn our attention to condition 4, which will also be obtained using Lemma 2.3.
If also j belongs to the set in (2.6) for this choice of (i 1 , i 2 ), then also inf t∈[j−1,j] (L t − L i1 ) ≤ 7C 1 log n. If these properties are satisfied and furthermore condition 2 in the definition of E n holds, then L j differs from inf t∈[j−1,j] L t by at most log n so |L j − L i1 | ∨ |L j − L i2 | is bounded above by a universal constant times C 1 log n. Hence, for a possibly larger universal constant K, (2.10) By Lemma 2.3, applied to each of the Brownian motions t → L t+j − L j and t → L −t+j − L j and with s equal to a large enough constant times log n and r = KC 1 log n, there is a constant C 2 = C 2 (A) > 0 such for each fixed j ∈ Z, the probability that the number of pairs (i 1 , i 2 ) ∈ Z 2 for which (2.10) holds is larger than . By a union bound over all j ∈ [−n, n] Z , we see that the probability that condition 2 holds but condition 4 fails is at most O n (n −A ). Combining the four preceding paragraphs shows that (2.7) holds.
Lemma 2.4. Let C 1 , C 2 ≥ C 0 ∨ 1 be the constants from Lemma 2.2. There is a constant C ≥ 1, depending only on C 1 , C 2 , and C 0 such that if the event E n defined just above Lemma 2.2 occurs, then the first part of the conclusion of Theorem 2.1 is satisfied. That is, of the paths P G i1,i2 . Moreover, the same is true with H and G interchanged.
Proof. Throughout the proof we assume that E n occurs.
First consider a pair of vertices i 1 , i 2 ∈ [−n, n] Z with i 1 < i 2 and i 1 ∼ i 2 in H. We will construct a path to be the length-1 path in G from i 1 to i 2 . We will construct P G i1,i2 in the case when L i1−1 ∨ L i2 < min j∈[i1,i2−1] Z L j ; the construction when this holds with R in place of L is similar. By conditions 1 and 2 in the definition of E n and since (2.11) By condition 3 in the definition of E n , the set has cardinality at most C 3 1 (log n) 3 . By (1.2), any two consecutive elements of the set (2.12) are connected by an edge in G. Hence this set is connected in G. Since the set (2.12) contains i 1 and i 2 , the vertices i 1 and i 2 can be connected by a path P G i1,i2 contained in this set which has length at most C 3 1 (log n) 3 .
Using condition 4 in the definition of E n , we see that each j ∈ [−n, n] Z is contained in at most C 2 (log n) 6 of the sets (2.12) for i 1 < i 2 such that i 1 ∼ i 2 in H, so each such j is hit by at most C 2 (log n) 6 of the paths P G i1,i2 . This gives the statement of the lemma for adjacent vertices in H.
We next prove the reverse relationship. Assume i 1 , i 2 ∈ [−n, n] Z with i 1 < i 2 and i 1 ∼ i 2 in G. We will construct a path P H i1,i2 in H n from i 1 to i 2 with length at most C 3 1 (log n) 3 . The argument is similar to the one for adjacent vertices of H given above.
By (1.2), the condition that or the same holds with R in place of L. Assume without loss of generality that we are in the former setting. By (2.13) and condition 2 in the definition of E n , (2.14) By this and condition 1 in the definition of E n , (2.16) By (2.14) and condition 3 in the definition of E n , the set on the right side of (2.16) has cardinality at most C 3 1 (log n) 3 . By (2.15) the set on the left side in (2.16) contains i 1 and i 2 . If j 1 and j 2 are consecutive elements of the left set in (2.16) then either j 2 = j 1 + 1 or so by (2.2) j 1 and j 2 are connected by an edge in H. Hence i 1 and i 2 can be connected by a path P H i1,i2 in H with length at most C 3 1 (log n) 3 which is contained in the set on the left side of (2.16).
Using condition 4 in the definition of E n , we see that each j ∈ [−n, n] Z is contained in at most C 2 (log n) 6 of the sets on the right side of (2.16) for i 1 < i 2 such that (2.14) holds, so each such j is contained in at most C 2 (log n) 6 of the paths P H i1,i2 . Consequently, the statement of the lemma holds with C = C 2 ∨ C 3 1 .
Proof of Theorem 2.1. The first statement of the lemma (concerning the existence of paths satisfying the desired properties) is immediate from Lemmas 2.2 and 2.4. To deduce (2.3) from this statement, suppose i 1 , i 2 ∈ [−n, n] Z and let P be a H n -geodesic from i 1 to i 2 . By concatenating paths of length at most C(log n) 3 in G n between the vertices of G n corresponding to the vertices traversed by P , we obtain a path of length at most C(log n) 3 | P | in G n from i 1 to i 2 , which gives the lower bound in (2.3). We similarly obtain the upper bound in (2.3).
Remark 2.5. The arguments in this subsection still work almost verbatim if we slightly modify the adjacency condition (2.2), e.g., by inserting ±1 in various places. The reason for using (2.2) is that this particular adjacency condition is closely connected to the mating-of-trees bijections for spanning tree-decorated maps and for site percolation on the UIPT (see Sections 3.1 and 3.2). In the case of bipolar-oriented and Schnyder wood-decorated maps (Section 3.3), we will apply Theorem 2.1 in the case when H is defined with (2.2) replaced by (3.15) below.

Combinatorial arguments
In this section we review the bijective encodings of each of the random planar maps M listed in Section 1.4 by means of a certain two-dimensional random walk. We then compare M to the random planar map H constructed in Section 2 from this same two-dimensional random walk; and deduce the strong coupling result Theorem 1.9 from this and Theorem 2.1. Our other main results will be easy consequences of Theorem 1.9 and the results of [GHS19].
Since the comparison between M and H relies on the fine geometric properties of the bijection, each of the cases needs to be treated separately. The reader may wish to read only one of the subsections of this section to get a general idea of the sort of arguments involved.
We start in Section 3.1 by treating the simplest case-that of the infinite spanning tree-decorated map (case 1), which is encoded by a simple random walk on Z 2 via the Mullin bijection. We review this encoding, show how the map H from Section 2 arises from the bijection (Proposition 3.3) then prove our main results in this case. In Section 3.2, we treat the case of site percolation on the UIPT (case 2) by relating the bijection from [Ber07a,BHS18] to the Mullin bijection. In Section 3.3, we treat the case of the uniform infinite bipolar-oriented map (case 3) using the bijection of [KMSW19]. Along the way, we check carefully that this infinite map exists and is the local limit of the finite bipolar-oriented maps considered in [KMSW19] (unlike in the other cases, the existence of this local limit has not previously been established rigorously). We then treat the case of more general bipolar-oriented maps (case 4) and deduce the case of Schnyder-wood decorated maps (case 5) from the result for bipolaroriented maps plus a bijection relating Schnyder wood with a special type of bipolar-oriented maps [FFNO11]. Section 3.3 can be read independently of Sections 3.1 and 3.2.
Several places in this section we will use the following notion of submap of a planar map.
The boundary of M is the set of v ∈ V(M ) and e ∈ E(M ) which is incident to a face in F ∞ .

Mullin bijection
The first mating-of-trees bijection to be discovered is the Mullin bijection [Mul67], which encodes a spanningtree decorated map by a nearest-neighbor walk in Z 2 . This bijection is explained in more detail in [Ber07b], and is also equivalent to the p = 0 case of Sheffield's hamburger-cheeseburger bijection [She16]. Here we will review the infinite-volume version of the Mullin bijection, which is also explained in [Che17, GMS19,BLR17] in the more general setting of the hamburger-cheeseburger bijection with arbitrary p ∈ [0, 1]. See Figure 6 for an illustration. Let (M, e 0 , T ) be the uniform infinite spanning-tree decorated map, which is the Benjamini-Schramm [BS01] local limit of uniformly random triples consisting of a planar map with an oriented root edge and a distinguished spanning tree. This infinite-volume limit is shown to exist in [She16,Che17].
Let M * be the dual map of M and let T * be the dual spanning tree of T , so that E(T * ) is the set of edges of M * which do not cross edges of T . Also let Q = Q(M ) be the radial quadrangulation, whose vertex set is V(M ) ∪ V(M * ), with two vertices of Q connected by an edge if and only if they correspond to a face of M , i.e., a vertex of M * , and a vertex of M incident to that face (the number of edges is equal to the multiplicity of the vertex as a prime end on the boundary of the face). We declare that the root edge of Q is the edge e 0 ∈ E(Q) whose primal endpoint coincides with the initial endpoint of e 0 and which is the first edge in clockwise order after e 0 with this property.
Each face of Q is crossed diagonally by an edge of M and an edge of M * , exactly one of which belongs to T ∪ T * . Hence the graph Q ∪ T ∪ T * is a triangulation with the same vertex set as Q. Let T be the planar dual of this triangulation, so that T is the adjacency graph of triangles of Q ∪ T ∪ T * , with two triangles considered adjacent if they share an edge. We declare that the root edge of T is the edge of T which crosses e 0 , oriented so that the primal (resp. dual) endpoint of e 0 is to its left (resp. right).
There is a unique path t : Z → V(T ) which hits each triangle (vertex) of T exactly once; hits the initial and terminal points of the root edge of T at times 0 and 1, respectively; and does not cross T or T * in the sense that t(i) and t(i − 1) share an edge belonging to Q for each i ∈ Z. according to whether the other triangle which shares this same edge of T is hit by t before or after time i. The other coordinate of Z is defined symmetrically. Bottom: The steps of Z corresponding to this segment of t if we assume that each of the exterior triangles (i.e., those with dotted edges) is hit by t before each of the non-dotted triangles.
boundary edges are in Q). There is one other triangle of T with this same edge of T on its boundary. If this triangle is hit by t before (resp. after) time i, we set Z i − Z i−1 = (1, 0) (resp. (−1, 0)). Symmetrically, if t(i) has a boundary edge in T * and the other triangle of T sharing this boundary edge is hit before (resp. after) time i, we set Z i − Z i−1 = (0, 1) (resp. (0, −1)). Then Z has the law of a standard nearest-neighbor simple random walk in Z 2 .
We next state and prove a proposition which will allow us to transfer from the results of Section 2 to results for the infinite spanning-tree weighted map. For the statement, we will use the following definition.
Definition 3.2. Let G and G be graphs and let f : V(G) → V(G ). We say that f is a graph isomorphism modulo multiplicity if f is a bijection and two distinct vertices v, w ∈ V(G) are connected by at least one edge in G if and only if f (v) and f (w) are connected by at least one edge in G (i.e., adjacency, but not necessarily the number of edges between two vertices, is preserved).
Since we are primarily interested in graph distances, there is no difference for our purposes between a true graph isomorphism and a graph isomorphism modulo multiplicity. However, some of the graphs we consider naturally have multiple edges whereas the graphs H from Section 2 by definition do not, so we often get isomorphisms modulo multiplicity instead of true isomorphisms. Proposition 3.3. The function i → t(i) is an isomorphism modulo multiplicity from the graph H of Section 2 with the walk Z as above and adjacency defined as in (2.2); to the graph of triangles T .
Proof. For i ∈ Z, the triangle t(i) has two edges in Q which are shared by t(i − 1) and t(i + 1), respectively, and one edge in either T or T * . Let e i be this third edge. For each edge e of T , there are precisely two values of i ∈ Z for which e i = e, and the corresponding triangles t(i) are adjacent in T . The ordering of the edges of the tree T obtained from i → e i by ignoring the values of i for which e i ∈ T * is the same as the contour (depth-first) ordering of T : indeed, this follows immediately from the definition of the path of triangles i → t(i). The same holds with T * in place of T . By combining this with the above definitions of L and R, we find that L and R coincide with the contour functions of the trees T and T * [Le 05, Section 1] except that they have extra constant steps (which do not affect the trees). From this, it follows that for i 1 < i 2 , the condition that e i1 = e i2 is equivalent to the adjacency condition (2.2) for H. Since the three neighbors of t(i) in T are t(i − 1), t(i + 1), and the other triangle with e i on its boundary, we obtain the statement of the lemma.

The inverse construction and the sewing procedure
In the above discussion, we only explained how to produce the walk Z from the decorated map (M, e 0 , T ). We can also go in the reverse direction and produce (M, e 0 , T ) from Z by first constructing T from Z via (2.2) (see Proposition 3.3) then constructing (M, e 0 , T ) from T . Note that this uses the fact that we can tell which edges of the dual map Q ∪ T ∪ T * of T belong to each of Q, T , and T * since edges of Q correspond to edges of T between consecutive vertices. We emphasize that the procedure for constructing (M, e 0 , T ) works for any bi-infinite walk Z = (L, R) in Z 2 with nearest neighbor steps such that not just a.s. We refer to [Che17, Section 4.1] for more details on the inverse bijection. The construction of (M, e 0 , T ) from Z is uniquely characterized by an inductive relationship between certain submaps of the triangulation Q ∪ T ∪ T * , called the sewing procedure, which we now describe (see The edges of the tree T (resp. T * ) are precisely the edges of Q ∪ T ∪ T * which are not crossed by t and which lie to the left (resp. right) of the path t. It follows from the inverse bijection described above [She16,Che17] that, given a walk Z with i.i.d. increments there a.s. exists a unique tuple (M, e 0 , T ) which can be constructed via the sewing procedure just described.

Proofs of main theorems in the case of spanning-tree decorated maps
In this subsection we prove our main results in case 1, when γ = √ 2 and (M, e 0 ) is an infinite-volume spanning-tree weighted planar map. Throughout, we define the quadrangulation Q, the graph of triangles T , and the bijective path t : Z → V(T ) as in Section 3.1.1. We also let Z = (L, R) be the Brownian motion as in (1.1) for γ = √ 2, so that L and R are independent; and we let G be the associated mated-CRT map.
Our first task is to define the maps M n and the functions φ n and ψ n from Section 1.4 in this setting. In keeping with Remark 1.2, we will first define functions φ : V(M ) → Z and ψ : Z → V(M ). For v ∈ V(M ), we choose φ(v) ∈ Z to be the integer with the smallest absolute value for which the triangle t(φ(v)) ∈ V(T ) has the vertex v on its boundary (in the case of a tie, we choose i to have a positive sign). For i ∈ Z, we define Proof of Theorems 1.5 and 1.9 in case 1. In light of Lemma 1.10, we only need to prove Theorem 1.9. We first define a high-probability event on which we will show that the conditions in the theorem statement are satisfied. Let C 0 > 0 be the constant from Theorem 2.1 for our given choice of A and for Z the simple random walk on Z 2 . Recall from Proposition 3.3 that the corresponding graph H from that proposition is isomorphic to the graph of triangles T via i → t(i). Hence Theorem 2.1 shows that with probability at least 1 − O n (n −A ), the following is true (where here, in obvious notation, we define T n to be the graph t(H n )). 3 ; and each j ∈ [−n, n] Z is contained in at most C 0 (log n) 6 of the paths P G i1,i2 .
(B) For each i 1 , i 2 ∈ [−n, n] Z with i 1 ∼ i 2 in G, there is a path P T i1,i2 from t(i 1 ) to t(i 2 ) in T n with | P T i1,i2 | ≤ C 0 (log n) 3 ; and each j ∈ [−n, n] Z is contained in at most C 0 (log n) 6 of the paths P T Henceforth assume that conditions (A), (B), and (C) above are all satisfied. We will check the conditions of the theorem statement in order.
Proof of condition 1. Suppose v 1 , v 2 ∈ V(M n ) with v 1 ∼ v 2 . We will construct a path P G v1,v2 in G n between φ(v 1 ) and φ(v 2 ).
By the definition of E(M n ), either the edge of M n joining v 1 and v 2 lies on the boundary of a triangle of T n or this edge is contained in the union of two triangles of T n . In either case, we can find i 1 , i 2 ∈ [−n, n] Z such that v 1 is a vertex of t(i 1 ), v 2 is a vertex of t(i 2 ), and the triangles t(i 1 ) and t(i 2 ) are either identical or they share an edge (i.e., they are adjacent in T ).
By the definition of φ, the triangle t(φ(v 1 )) also has v 1 as a vertex. The set of triangles in t([−n, n] Z ) with v 1 as a vertex is connected in T n . Consequently, there exists a path in T n from t(i 1 ) to t(φ(v 1 )) with length at most the total number of triangles of T with v on their boundaries. Similar considerations hold with v 2 in place of v 1 . By the preceding paragraph and condition (C) above, there is a d ≤ 2C 1 log n and integers j 0 , . . . , j d ∈ For each such k, let P G j k−1 ,j k be the path from j k−1 to j k afforded by condition (A) above and let P G v1,v2 be the concatenation of all d of these paths. Then P G v1,v2 is a path from φ(v 1 ) to φ(v 2 ) in G n with length at most Each of the triangles t(j k ) for k ∈ [0, d] Z has either v 1 or v 2 on its boundary. Therefore, for each j ∈ [−n, n] Z the number of pairs (v 1 , v 2 ) ∈ [−n, n] Z × [−n, n] Z for which v 1 ∼ v 2 in M and j is one of the corresponding j k 's is at most the sum of the degrees of the (at most two) vertices of M on the boundary of t(j), which is at most 2C 1 log n by condition (C) above.
Consequently, each of the paths P G j,j for j, j ∈ [−n, n] Z with t(j) ∼ t(j ) in T n is a sub-path of at most 2C 0 log n of the paths P G v1,v2 for v 1 , v 2 ∈ V(M ). Since the total number of the P G j,j 's which hit each specified vertex of G n is at most C 0 (log n) 6 , the total number of the P G v1,v2 's which hit each such vertex is at most 2C 0 C 1 (log n) 7 . This gives condition 1 in the theorem statement with C = 2C 0 C 1 .
Proof of condition 2. Suppose i 1 , i 2 ∈ [−n, n] Z with i 1 ∼ i 2 in G. Let P = P T i1,i2 be the path from t(i 1 ) to t(i 2 ) in t([−n, n] Z ) as in condition (B) above. We will construct a path P M i1,i2 in M n between the vertices ψ(i 1 ) and ψ(i 2 ) of M .
Set v 0 = ψ(i 1 ), let v | P | = ψ(i 2 ), and for k ∈ [1, | P | − 1] Z let v k be a vertex of the triangle P (k) which belongs to V(M n ). For k ∈ [1, | P |] Z , the triangles P (k − 1) and P (k) share an edge. This implies that the vertices v k−1 and v k can be connected by a path of length at most 2 in M n which hits only vertices of P (k − 1) or P (k) (see Figure 8). If we let P M i1,i2 be the concatenation of these | P | paths, then P M i1,i2 is a path from v 1 to v 2 in M n with length at most 2| P | ≤ 2C 0 (log n) 3 .
It remains to bound the number of paths P M i1,i2 for i 1 , i 2 ∈ [−n, n] Z with i 1 ∼ i 2 in G which hit a fixed vertex v of M n . Indeed, if P M i1,i2 hits v then the above construction shows that P T i1,i2 must hit one of the triangles of T which has v as a vertex. By condition (C), the number of such triangles is at most 2C 1 log n. By condition (B) above, the total number of the paths P M i1,i2 which hit each such triangle is at most C 0 (log n) 6 . Therefore, v is hit by at most 2C 0 C 1 (log n) 7 of the P M i1,i2 's, which gives condition 2 in the theorem statement with C = 2C 0 C 1 .
Proof of condition 3. Since the vertices v and ψ(φ(v)) lie on the boundary of the same triangle of T by definition, we obviously have dist(ψ(φ(v)), v; M n ) ≤ 1. On the other hand, the triangles t(i) and t(φ(ψ(i))) share a common vertex of M , so by condition (C) the T n -graph distance between these triangles is at most C 1 log n. By considering a path in T n between these two triangles and applying condition (A) to the successive pairs of triangles which it hits, we get dist(φ(ψ(i)), i; G n ) ≤ C 0 C 1 (log n) 4 .
Proof of Lemma 1.12 in case 1. See Figure 9 for an illustration of the proof. Recall that the adjacency graph of triangles T is isomorphic to the graph H of (2.2) via i → t(i). Furthermore, the condition on the minimum of L (resp. R) in (2.2) holds if and only if t(i 1 ) and t(i 2 ) share an edge of T (resp. T * ). From this, we infer that if the random walk condition (1.21) and its analog for R hold, then there exists i + , j + ∈ [m + 1, n] Z and i − , j − ∈ [−n, −m − 1] Z such that t(i + ), t(i − ) share an edge of T and t(j + ), t(j − ) share an edge of T * .  . Figure 10: Illustration of the sewing procedure for the measure-preserving map between percolation-decorated triangulations and Kreweras walks (i.e., walks with steps a, b, c). The black arrows indicate that we glue e h k−1 to an adjacent edge, i.e., the two edges are identified. See the main text for the precise description. Vertices are colored gray if their color is not yet determined, while they are colored black if their color has been determined, so they are either blue or red, but the color cannot be determined from the figure.

Bijection for site percolation on the UIPT
In this subsection we review the infinite-volume version of the bijection between site-percolated triangulations of type II (no self-loops, but multiple edges allowed) and Kreweras walks-those with steps in {(1, 0), (0, 1), (−1, −1)}-which was first described in [BHS18]. This bijection is based on an earlier bijection by Bernardi [Ber07a] between such walks and trivalent maps decorated by a depth-first search tree. We emphasize that the only part of [BHS18] needed here is the definition of the bijection in [BHS18, Section 2].
Let (M, e 0 ) be an instance of the uniform infinite planar triangulation (UIPT) of type II, rooted at a directed edge e 0 . Let P be an instance of critical site percolation on the map, i.e., P associates each v ∈ V(M ) with either red or blue, uniformly and independently at random. In [BHS18, Section 2.5] (based on [Ber07a]) it was proved that (M, e 0 , P ) can a.s. be encoded by a bi-infinite walk with increments a = (1, 0), b = (0, 1), c = (−1, −1) 5 . Conversely, a bi-infinite walk Z = (Z k ) k∈Z with Z 0 = 0 and steps a, b, c chosen uniformly and independently at random, a.s. encodes an instance of (M, e 0 , P ). Throughout this section we let (∆Z k ) k∈N denote the steps of Z, i.e., ∆Z k := Z k − Z k−1 ∈ {a, b, c}.
As in the case of the Mullin bijection (see Section 3.1.2), the construction of (M, e 0 , P ) from the walk Z is uniquely characterized by a sewing procedure, which we now describe. Each k ∈ Z corresponds to a map M −∞,k with a percolation configuration P −∞,k on the inner vertices and a marked edge e h k . Again the sewing procedure describes how to obtain (M −∞,k , P −∞,k , e h k ) from (M −∞,k−1 , P −∞,k−1 , e h k−1 ) by observing the step ∆Z k of the walk. The map M rooted at e 0 is the local limit of M −∞,k rooted at e 0 as k goes to ∞ in the Benjamin-Schramm sense. We will see that each k can be associated with a unique element λ(k) ∈ V(M ) ∪ F(M ). More precisely, we have λ(k) ∈ V(M ) if ∆Z k = c and λ(k) ∈ F(M ) if ∆Z k ∈ {a, b}. The root edge e 0 of the map is e h −1 . For any k ∈ Z the map M −∞,k has an infinite left frontier and an infinite right frontier. Note that the sewing procedure is only an inductive procedure, but, as for the Mullin bijection, it can be shown that given a walk Z with i.i.d. increments there is a.s. a unique collection of triples (M −∞,k , P −∞,k , e h k ) such that (M −∞,k , P −∞,k , e h k ) is obtained from (M −∞,k−1 , P −∞,k , e h k−1 ) via the sewing procedure. We refer to [BHS18, Section 2.5] for a more complete description, and to Figure 10 for an illustration.
Given M −∞,k−1 and e h k−1 , we will now explain how to construct M −∞,k via the sewing procedure. See Figure 10. If ∆Z k = a (resp. ∆Z k = b), we add a triangle to the edge e h k−1 . We let the new head e h k be the right (resp. left) side of the triangle. We define λ(k) ∈ F(M ) to be the triangle we added. Next consider the case ∆Z k = c. For an edge e on the frontier of the map we say that a face f is behind e if it is the unique face of the map for which e is on its boundary. Let e L (resp. e R ) be the edge on the frontier of M −∞,k−1 immediately to the left (resp. right) of e h k−1 , and let f L (resp. f R ) be the face behind e L (resp. e R ). red (resp. blue), and we define λ(k) ∈ V(M ) to be this vertex. We see from this description that the function L (resp. R) represents the net change in the length of the left (resp. right) frontier of the map, relative to the length at time 0. We remark that M −∞,k is not a submap of M , since for every step ∆Z k = c in the sewing procedure we identify two edges and two vertices, and vertices and edges which are identified in the sewing procedure at times > k are not identified in M −∞,k . However, as remarked above, there are natural maps from the set of vertices (resp. inner faces, edges) of M −∞,k to the set of vertices (resp. faces, edges) of M −∞,k+1 and M . For faces this map is injective, while this is not the case for vertices and edges. We will often identify vertices (resp. faces, edges) of the various maps when this identification is well-defined. We remark that for k ∈ Z, the edge λ(k) of Remark 1.3 is the edge of M corresponding 6 to e h k .
Given n ∈ N we now define the map M n introduced in Section 1.4. We will let M n be a particular  in the sewing procedure. Define E(M n ) = E 1 ∪ E 2 . We let ι n : M n → M be the function which sends each vertex, face, and edge of M n to the corresponding vertex, face, or edge of M . We will define the mappings φ n and ψ n of (1.5) in (3.4) just below.

Proofs of main theorems in the UIPT case
To prove the theorems from Section 1 in case 2, we will make small local changes in the map M and the walk Z defined above and observe that the resulting map and walk are related by Mullin's bijection as described in Section 3.1. In particular, as we will see, if we replace each of the c steps of the walk Z by a −a step followed by a −b step and each of the corresponding "gluing steps" in the sewing procedure by a step which adds a certain subgraph consisting of two triangles to the map, then the resulting modified walk and planar map are related via Mullin's bijection (Section 3.1.1). This will allow us to reduce the proof of Theorem 1.9 in the UIPT case to the Mullin bijection case, which we have already treated. It is also possible to treat case 2 directly, without reference to the Mullin bijection, but the argument given here is shorter and simpler. We first define a modified walk Z with steps in {−a, −b, a, b} by replacing each step c in Z by one step −a followed by one step −b. More precisely, Z and the maps α n and α n describing the associated time Figure 12: Illustration of α : Z → Z, which describes the time change of Z relative to Z.  changes are defined as follows. First define α : Z → Z and α : Z → Z by See Figure 12 for if Z α(k) = c and α(k) = α(k + 1).
Since the walk Z has nearest-neighbor steps, it determines an infinite triangulation M (analogous to the triangulation Q ∪ T ∪ T * from Section 3.1.1), a map t : Z → F( M ), and an infinite triangulations with boundary M −∞,n for n ∈ N (which is a submap of M ) via the Mullin bijection sewing procedure described in Section 3.1.2. More precisely, for steps ∆ Z k ∈ {a, b} the sewing procedure is as before and t(k) is the triangle which we add. If ∆Z k = −a (resp. ∆Z k = −b) we instead add a triangle t(k) adjacent to e h k−1 , we glue the right (resp. left) side of t(k) to the edge immediately to its right (resp. left) on the frontier, and we define e h k to be the (only) edge of t(k) which is still on the frontier of the map.
We define M n be the (finite) submap of M −∞,n consisting of the vertices and edges on the boundaries of the 2n + 1 triangles added at times in [−n, n] Z , as in Section 3.1.3.
Given k ∈ [−n, n] Z such that ∆Z k = c, let D k be the submap of M n of boundary length 4 consisting of the two faces t n ( α n (k)) and t n ( α n (k) + 1) (which correspond to consecutive steps of the Z of the form −a and −b) and the vertices and edges on their boundaries. We say that we contract D k if we modify the map M n by the following procedure. (See Figure 13 for an illustration.) Let e 1 , e 2 , e 3 , e 4 denote the edges on the boundary of D k in counterclockwise order, such that e 1 = e h αn(k)−1 and e 3 = e h αn(k)+1 . A contraction means that we remove the two faces t n ( α n (k)) and t n ( α n (k) + 1) from the map, and then we identify edges pairwise along the boundary of D k . The edge between the two faces is removed, while the other faces on the boundary of the two faces are kept. Let f L (resp. f R ) be the face of M −∞,n which has e 4 (resp. e 2 ) on its boundary, but is not equal to t n ( α n (k)) or t n ( α n (k) + 1). If λ −1 (f L ) < λ −1 (f R ) (resp. λ −1 (f L ) > λ −1 (f R )) then we identify e 1 and e 4 (resp. e 1 and e 2 ), and e 2 and e 3 (resp. e 3 and e 4 ); notice that the two edges we identify will always have one common vertex and one not common vertex, and when we identify the edges we choose the orientation of the edges in the natural way, such that the common (resp. not common) vertices are identified. with the walk Z in place of Z.
(i) The map k → t(k) is a graph isomorphism modulo multiplicity (recall Definition 3.2) from H n to the dual of M n .
(ii) If we contract all the submaps D k of M n for k ∈ [−n, n] Z such that ∆Z k = c, then we get a map which is isomorphic to M n .
Proof. (i) This is immediate by Proposition 3.3, and since Z and M n (in the limit as n → ∞) are related by Mullin's bijection.
(ii) Consider two walks Z and Z with steps ∆Z and ∆Z , respectively, in {a, b, c, −a, −b}. Assume Z is obtained from Z by choosing a k 0 ∈ Z such that ∆Z k0 = c, and replacing this step c by two steps −a and −b. In other words, we have ∆Z k = ∆Z k for k < k 0 , ∆Z k0 = −a, ∆Z k0+1 = −b, and ∆Z k = ∆Z k−1 for k > k 0 + 1. Let M (resp. M ) denote the map we get when applying the sewing procedure described above to Z (resp. Z ), and let D be the submap of M of boundary length 4 corresponding to the faces added in steps k 0 and k 0 + 1 of the sewing procedure. Since we can proceed by induction, in order to conclude it is sufficient to show that when we contract D , we get the map M . This property is immediate by the sewing procedure applied to Z and Z , respectively.
From Lemma 3.5(ii) we see that there exists a natural map β n : V( M n ) → V(M n ) given by contracting all of the D k 's. See Figure 13. We define β n : V(M n ) → V( M n ) in an arbitrary way such that β n • β n is the identity map V(M n ) → V(M n ), and such that β n sends ∂M n to ∂ M n . Note that it is possible to find a map β n satisfying the latter property by the construction in the proof of Lemma 3.5(ii). LetF( M n ) denote the set of bounded faces of M n , i.e., F( M n ) \F( M n ) consist of the single unbounded face of M n . Let φ n : V( M n ) →F( M n ) map v ∈ V( M n ) to some arbitrarily chosen face of M n which has v on its boundary. Let ψ n :F( M n ) → V( M n ) map f ∈F( M n ) to some arbitrarily chosen vertex on its boundary. We also require that φ n , ψ n , β n are defined such that the maps φ n and ψ n defined just below satisfy the requirements on the root in (1.5) (note that by the definition of the contraction operation, it is possible to satisfy both this requirement and the second requirement on β n specified right above). By Lemma 3.5(i), we may identifẙ It follows from the definition of β n that (i) holds, and it follows from the definition of α and Z that (ii) and (iii), respectively, hold. Property (iv) follows from the argument for case 1 since the sewing procedure for M n and the definition of φ n are the same as for case 1.
Proof of Theorem 1.9 in case 2. Define the planar maps H and {H n } n∈N as in (2.2) and the discussion just after with Z the encoding walk for M . Throughout the proof, we fix A > 0 and n ∈ N and we couple Z with the correlated Brownian motion Z in such a way that the conclusion of Theorem 2.1 is satisfied for this choice of H.
To prove condition 1 from the theorem statement, we will argue that for sufficiently large C > 1, depending only on A, the following is true for each n ∈ N.
(i) For any v 1 ∼ v 2 in V(M n ) we can find a path P Mn v1,v2 from β n (v 1 ) to β n (v 2 ) in M n of length at most 3, and each edge of M n is on at most 10 of these paths. (Notice that when we bound the number of paths hitting each i ∈ V(G n ) below, it is sufficient to bound the number of paths hitting each edge, rather than each vertex, in this step.) (ii) With probability at least 1 − O n (n −A ), for any v 1 ∼ v 2 in V( M n ) we can find a path P Hn v1, v2 from φ n ( v 1 ) to φ n ( v 2 ) in H n of length at most C log n, and each i ∈ [−n − , n + ] Z is hit by at most C log n of the paths P Hn v1, v2 .
(iii) For any i 1 ∼ i 2 in H n we can find a path P Hn i1,i2 from α n (i 1 ) to α n (i 2 ) in H n of length at most 2, and each vertex i of H n is hit by at most 10 of the paths P Hn i1,i2 .
(iv) With probability at least 1 − O n (n −A ), for any i 1 ∼ i 2 in H n we can find a path P Gn i1,i2 from i 1 to i 2 in G n of length at most C(log n) 3 , and each i ∈ [−n, n] Z is hit by at most C(log n) 6 of the paths P Gn i1,i2 . Before we prove (i)-(iv), we will argue that they imply condition 1 upon replacing C by 100C 2 . Indeed, if the events in (i)-(iv) occur (which happens with probability at least 1 − O n (n −A )) and v 1 ∼ v 2 in V(M n ) we can construct the path P Gn v1,v2 from φ n (v 1 ) to φ n (v 2 ) in G by considering the path P Mn v1,v2 from (i), replacing each step ( v 1 , v 2 ) by the path P Hn v1, v2 from (ii), replacing each step of the resulting path by the appropriate path from (iii), then replacing each step of this last path by the appropriate path from (iv). It is immediate from (i)-(iv) that the length of the path will be at most 3 · C log n · 2 · C(log n) 3 = 6C 2 (log n) 4 , and that each i is hit by at most 10 · C log n · 10 · C(log n) 6 = 100C 2 (log n) 7 of the paths P Mn v1,v2 . The property (i) follows from Lemma 3.5(ii), while (iii) is immediate from the adjacency condition (2.2) for H n and H n . It follows from Theorem 2.1 that we can couple so that (iv) holds. Since the degree of the root of the UIPT has an exponential tail [AS03, Lemma 4.2], we know that for large enough C with probability at least 1 − O n (n −A ), The property (ii) follows from Lemma 3.5(i) and (3.5), since the definition of φ n implies that the face of M n associated with φ n ( v 1 ) (resp. φ n ( v 2 )) has v 1 (resp. v 2 ) on its boundary, so we can find a path of faces for M n from φ n ( v 1 ) to φ n ( v 2 ) of length at most C log n (c.f. the proof of condition 1 of Theorem 1.9 from Section 3.1.3 for a similar argument). The proof of condition 2 is very similar, and we therefore only give a brief sketch. First we prove variants of (i)-(iv) above for the inverse maps. For example, instead of (i) above, we show that for any v 1 ∼ v 2 in V( M n ) we can find a path P Mn v1, v2 from β n ( v 1 ) to β n ( v 2 ) in M n of length at most 3, and each edge of M n is on at most 10 of these paths. The new variants of (i)-(iv) are proved as before, and they imply condition 2 similarly as in the proof of condition 1.
To prove condition 3 we will only explain how to bound dist(φ n • ψ n (i), i; G n ), since the bound for dist(ψ n • φ n (v), v; M n ) is proven in a similar way. Assume the events in (i)-(iv) and (3.5) are satisfied, and choose an arbitrary i ∈ [−n, n] Z . Define By the definition of contraction and since v = β n (β n ( v)), we have dist( v, v ; M n ) ≤ 2. By the triangle inequality, The first term on the right side is bounded by C log n, since it follows from the definition of ψ n and φ n that the faces t( i) and t( φ n ( ψ n ( i))) have a common vertex, so the distance between the two faces in the dual of M n (equivalently, H n ) is at most C log n by (3.5). The second term on the right side is bounded by 2C log n, since (ii) and dist( v, v ; M n ) ≤ 2 imply that dist( φ n ( v), φ n ( v ); H n ) ≤ 2C log n. We conclude that dist( i, i ; H n ) ≤ 3C log n. Using this bound, (iii), the definition of α n , α n , and i = α n ( i ), we get further dist(i, i ; H n ) ≤ dist(i, α n ( i); H n ) + dist(α n ( i), i ; H n ) ≤ 1 + 2 · 3C log n.
Combining this with (iv) concludes the proof: dist(i, i ; G n ) ≤ (1 + 6C log n)(log n) 3 ≤ 7C(log n) 4 , where we assume that n is chosen sufficiently large in the last step.
Proof of Theorems 1.6, 1.7, and 1.8 in case 2. By [Ang03, Theorem 1.2], the metric ball B n (e 0 ; M ) of radius n centered at the root edge in the UIPT satisfies #V(B n (e 0 ; M )) = n 4+on(1) with probability tending to 1 as n → ∞, (3.6) which is stronger than the UIPT case of Theorem 1.6. (Observe that the weaker variant stated in Theorem 1.6 can also be proved by proceeding as in the UST case considered above.) Theorem 1.7 follows from Theorem 1.9 and [GHS19, Theorem 1.15]. Theorem 1.8 is immediate from (3.6) together with the UIPT versions of Theorem 1.9 and Lemma 1.11.

Bipolar-oriented planar maps and Schnyder woods
In this section we study the case of planar maps decorated by a bipolar orientation or Schnyder wood. We first do the necessary combinatorial arguments for finite bipolar-oriented maps in Section 3.3.1. Then we move to the infinite-volume setting in Section 3.3.2, and explain why the uniform infinite bipolar-oriented map, the local limit of finite bipolar-oriented maps, exists (the analogous arguments in the case of the other maps considered in this paper have already been carried out elsewhere). We then prove Theorem 1.9 in the case of the uniform infinite bipolar-oriented map (case 3) in Section 3.3.3. Finally in Section 3.3.4 we treat the case of bipolar-oriented maps with other face degree distributions (case 4) and show that Schnyder wood-decorated maps (case 5) can be treated as a special case of bipolar-oriented maps.

The mating-of-trees bijection for bipolar-oriented maps
An orientation on a graph is an assignment of a direction to each edge. A vertex is called a sink (resp. source) if there are no outgoing (resp. incoming) edges incident to the vertex. Sources and sinks are also called poles. A bipolar orientation of G with specified source and sink is an acyclic orientation with no source or sink except at the specified poles. A bipolar-oriented planar map is a planar map M , with a marked face f 0 and a bipolar orientation O on M whose source and sink are both on the boundary of f 0 . Bipolar-oriented planar maps have rich combinatorial structures and numerous applications in algorithms. For an overview of the graph theoretic perspective on bipolar orientations, we refer to [dFdMR95] and references therein. See also [FPS09,BBMF11] and the references therein for enumerative and bijective results. The plane R 2 is naturally associated with the notion of east, west, south and north. It is known [AK15] that any bipolar-orientated planar map can be embedded into R 2 in such a way that every edge is north oriented and the marked face is unbounded. See Figure 14 for an example. Given a bipolar-oriented map (M, f 0 , O), we always embed it this way so that O can be read off from the embedding and the source and sink can be identified as the south pole and north pole, respectively, of the map. We therefore denote the source and sink by S(M ) and N(M ), respectively. For any edge e ∈ E(M ) directed as in O, we also write S(e) and N(e), respectively, for the initial and terminal edge of e. We call a bipolar-oriented map consisting of a simple cycle (separating a single bounded face from the unbounded face) a bipolar-oriented face. The edges of a bipolar-oriented face can be divided into those which lie on the west and east boundary arcs connected the two poles, which we call west edges and east edges, respectively. We can think of any face inF(M ) as a bipolar-oriented face. We call the clockwise (resp. counterclockwise) arc on f 0 from S(M ) to N(M ) the west (resp. east) boundary of M . An edge e is on both the west and east boundaries of M if and only if removing e separates M into two connected components.
We will now review the mating-of-trees bijection for finite bipolar-oriented maps from [KMSW19]. For any v ∈ V(M ) \ {N(M )}, there is a unique edge incident to v which is oriented away from v and is the furthest west among all such edges. We call this edge the NW (northwest) edge of v. Similarly, there is a unique edge incident to v which is oriented toward v and is the furthest east among all such edges. We call this edge the SE (southeast) edge of v. The NW tree T NW of (M, f 0 , O) is the directed planar tree that can be drawn as follows: each NW edge is entirely in T NW ; for each other edge, a segment containing the head of  1. If λ(t − 1) is the SE edge of N(λ(t − 1)), let λ(t) be the NW edge of N(λ(t − 1)) and set λ(t) := S(λ(t)) ∈ V(M ).
See Figure 14 for an illustration of T NW , T SE , λ, λ. 3. If ∆Z k = (i, −j) for some i, j ≥ 0, to construct (M k , f k 0 , O k , e k ), we glue a bipolar-oriented face f k with i + 1 east edges and j + 1 west edges to M k−1 , so that N(f k ) = N(e k ), the west edges of f k are east edges of f k 0 , and the east edges of f k are not edges of M k−1 . The marked edge e k is updated to be the unique east edge of f k satisfying S(e k ) = S(f k ). By the definition of Z, λ and the sewing procedure, we get the following.  • when ∆Z t ∈ Z ≥0 × Z ≤0 . We write τ t−1 • as τ • for simplification. In the right panel, the straight line represents the east boundary of M t−1 . The green edge is λ(t − 1) and the purple edge is λ(t). The vertices to the south of N(λ(t − 1)) are λ(τ 1 ), λ(τ 2 ), . . . in order. At time t we add a face λ(t) with west boundary length = 7. We see that S( λ(t)) = λ(τ 7 ). Moreover, in the left panel for t, R fixed, the time s satisfies the inequality (3.9) if and only if s ∈ {τ i } 1≤i≤7 , which proves Assertion 1. Proof. See Figure 16 for an illustration of the proof. For all k ∈ [0, #E(M )] Z , let τ k 0 = k. For i ≥ 0 and τ k i ≥ 0, we set sup ∅ = −∞ by convention and inductively define τ k i+1 = sup{t < τ k i : ∆Z t = (−1, 1) and R t = R τ k i − 1}. We first prove the following claim by an induction on k: S(e k ) = λ(τ k 1 ) and the vertices on the east boundary of M k between S(e k ) and S(M k ) are λ(τ k 1 ), λ(τ k 2 ), · · · , λ(τ k R k +1 ) in order. (3.11) For k = 0, (3.11) holds by (3.8). For 1 ≤ k ≤ #E(M ), assume (3.11) holds for k − 1. If ∆Z k = (−1, 1), then τ k i = τ k−1 i−1 for i ≥ 1 and by the definition of the sewing procedure we add an edge at time k (so we do not cover up any boundary vertices), which shows that (3.11) holds for k. If ∆Z k ∈ Z ≥0 × Z ≤0 , then at time k we add a face λ(k) with west boundary length k := R k − R k−1 . Therefore τ k i = τ k−1 k +i−1 for i ≥ 1 and S(e k ) = λ(τ k−1 k ) = λ(τ k 1 ). Thus (3.11) holds for k. This proves (3.11). Note that s, t satisfies (3.9) if and only if s ∈ {τ t−1 i : 1 ≤ i ≤ t−1 }. Now Assertion 1 follows from (3.11). By rotating R 2 by 180 degrees, we reverse the direction of each edge on (M, f 0 , O) and swap the role of south and north as well as west and east. We can also reverse the path λ and the walk Z. Then the role of R is played by the time reversal of L and vice versa. Here we note that after the time reversal, (s − 1, s, t − 1, t) becomes (t, t − 1, s, s − 1). This time reversal trick gives Assertion 2 from Assertion 1.

3.3.2
The uniform infinite bipolar-oriented map In this section, we construct an infinite planar map which will be the local limit of the uniform bipolar-oriented maps as the size tends to ∞ (see Proposition 3.16 and Definition 3.18.). Consider a bi-infinite simple random walk Z = (L, R) with Z 0 = 0 and the step distribution ν supported on (3.7) such that As above, we write ∆Z t = Z t − Z t−1 for all t ∈ Z. For two integers m < n, we choose K ∈ N large enough so that Z([m, n] Z ) + (K, K) ⊂ (0, ∞) 2 . Let x = L m + R m + 2K and y = L n + R n + 2K so that the straight line segment between Z m + (K, K) (resp., Z n + (K, K)) and (x, 0) (resp., (0, y)) has slope −1. By the time reversal trick in the proof of Lemma 3.8 (see also Remark 3.9), we have the following.  Figure 18: We use the notations in the proof of Lemma 3.12. Left: An illustration of the times τ, ρ ± , T • , T • . The green step is ∆Z τ . Right: The corresponding picture for Nb 0 . The arrowed edge is e 0 . The green edge is λ(τ ). The faces of Nb 0 correspond to the blue and red steps in the left panel. From Assertion 1 in Lemma 3.8, the time τ is first time when S(λ(τ )) = S(e 0 ). Moreover, ρ + (resp. ρ − ) is the unique time t such that λ(t) is a face and S(e 0 ) is on the west (resp. east) boundary of λ(t) but is not a pole of λ(t). Finally, ) describes the set of faces where S(e 0 ) is the south (resp. north) pole.
{b m,n (i)} i>0 as the branch of the NW tree of M m,n from N(λ m,n ( )) to infinity, and {b m,n (i)} i<0 as the branch of the SE tree of M m,n from S(λ m,n ( )) to infinity. for each n ∈ N. For each t ∈ Z, let λ(t) = λ −n,n (t) for n > |t|. This defines a function λ : Z → E(M ). By definition λ is an injection. Lemma 3.14 below will show that λ is in fact a bijection. So far there is no probability involved. Now we prove a few probabilistic lemmas ensuring that almost surely M does not have certain pathological behaviors. We will use the following notation. For integers m < n, we call the union of the east (resp., west) boundary of M m,n,K for all large enough K's the west (resp. east) boundary of M m,n . Lemma 3.12. With probability 1, there exists n ∈ N such that S(e 0 ), i.e., the tail of e 0 , is not on the east or west boundary of M −n,n .
Let Nb 0 be the subgraph of M consisting of all faces of M with S(e 0 ) on their boundaries, and all vertices and edges of M that are on the boundaries of such faces. We claim that faces in Nb 0 all belong to F(M ), which will imply Lemma 3.12 with n = |ρ − | ∨ |ρ + |.
By Lemma 3.14 λ is a bijection between Z and E(M ). Moreover, the degree of each face of M is bounded. Therefore, given a face of M we can define its poles and east/west boundaries as in the finite map case. Now for each t ∈ Z, if λ(t − 1) is the SE edge of N(λ(t − 1)), set λ(t) := S(λ(t)). Otherwise, let λ(t) be the unique face where λ(t − 1) is a west edge. Then λ : Z → V(M ) ∪ F(M ) is a bijection. For each t ∈ Z, both Items 1 and 2 in Definition 3.6 hold in this infinite volume setting. The following fact is an immediate consequence of this extension of definition. We adapt the notion of Benjamini-Schramm convergence [BS01] to the setting of oriented rooted graphs. Given planar maps G and G with oriented root edges e and e , respectively, the Benjamini-Schramm distance [BS01] from (G, e) to (G , e ) is defined by where o and o are the tail of e and e respectively and B r (e; G) B r (e ; G ) means that the two metric balls are isomorphic as oriented rooted planar maps. Then d loc defines a metric on the space of finite rooted planar maps. Let M be the Cauchy completion of the space of finite oriented rooted planar maps under d loc . Elements in M which are not finite are infinite rooted oriented planar maps. The following proposition asserts that M is the Benjamini-Schramm limit of the uniform finite bipolar-oriented maps viewed from a typical edge. Proof. Let (ξ 1 , ξ 2 ) be independent of Z so that P[(ξ 1 , ξ 2 ) = (i, j)] = 2 −i−j−2 for all (i, j) ∈ Z ≥0 ×Z ≥0 . Let E n be the event that {(ξ 1 , 0) + Z t } 0≤t≤n−1 is a walk starting at (ξ 1 , 0), ending at (0, ξ 2 ) and staying in Z ≥0 × Z ≥0 during [0, n − 1]. Then conditioning on E n and applying the sewing procedure to {(ξ 1 , 0) + Z t } 0≤t≤n−1 , we obtain a sample of (M (n), O(n)) (see [KMSW19,Remark 2]). Let λ n be the edge-valued function associated with this sewing procedure and let U n be a uniform sample from [0, n − 1] Z . Then (M (n), λ(U n ), O(n)) is a sample of (M (n), e(n), O(n)). Now let Z n (•) be sampled from the conditional law of Z(• + U n ) − Z(U n ) given E n . Standard results for random walk in cones (see, e.g., [DW15b,Theorem 4] and [DW15a, Theorem 1]) yield that as long as P[E n ] > 0, it holds that P[E n ] decays sub-exponentially in n. Therefore, Cramer's large deviation theorem implies that for all N > 0 the walks Z n and Z can be coupled so that their restrictions to [−N, N ] agree with probability 1 − o n (1) (see [She16,Section 4.2] for a similar application of Cramer's theorem). Now Proposition 3.16 follows from Lemma 3.13.
For each ∈ Z, let M ,∞ be the submap of M obtained by taking the union n> M ,n , rooted at the edge λ( ). Then M ,∞ has a bi-infinite boundary, which coincides with the west boundary of M ,n for each integer n > . List these boundary edges as {b (i)} i∈Z such that b (0) = λ( ) and N(b (i)) = S(b (i + 1)) for each i ∈ Z. By Lemma 3.11, we have the following. Lemma 3.17. In the setting right above, almost surely, b (i + 1) is the NW edge of N(b (i)) for each i ≥ 0, and b (i − 1) is the SE edge of S(b (i)) for each i ≤ 0.
For each ∈ Z, we can also consider M −∞, = n< M n, , where λ( ) is assigned to be the root edge.
Then M −∞, and M ,∞ share the same boundary.   For the proof of Lemma 3.20, we need the following elementary estimate.
nonnegative random variables such that E[e bX1 ] < ∞ for some b > 0. Let ξ be a nonnegative random variable (not necessarily independent from the X i 's) with an exponential tail. Then there exists c > 0 only depending on the law of X 1 and ξ so that Proof. By a union bound, for δ > 0, By choosing δ small enough according to the law of X 1 , the lemma follows from Cramer's Large Deviation Theorem.
Proof of Lemma 3.20. The stationarity of {Nb i } i∈Z and { Nb i } i∈Z follows from the stationarity of Z. So it suffices to focus on i = 0 in the rest of the proof.
To deal with Nb 0 , we retain the notation used in the proof of Lemma 3.12. By the Markov property of Z, all of ξ − , ξ + and ζ have geometric distributions. Moreover, conditioning on ξ − = 0, we have that {∆Z Ti } ξ − <i≤0 are i.i.d and have exponential tails. The same also holds for {∆Z Ti } 1≤i≤ξ+ and {∆Z Ti } 1≤i≤ζ . (However, the laws of ∆Z T−1 and ∆Z T1 are not the same.) Now observations 1-5 in the proof of Lemma 3.12 combined with Lemma 3.21 and (3.12) imply that |Nb 0 | has an exponential tail. We obtain the same result for {Nb N(λ(i)) } i∈Z by reversing the time direction and apply Lemma 3.10.

Remark 3.22.
A bipolar orientation on a map can be associated with a bipolar orientation on its dual map via rotating the orientation of each edge by 90 degrees. Therefore the UIBOM and its dual map agree in law. Thus bounds on | Nb 0 | immediately follow from bounds on |Nb 0 |. However, our proof of Lemma 3.20 is more robust and works for the more general setting in Section 3.3.4.

Proof of Theorem 1.9 for the UIBOM
From the discussion in Section 3.3.2, Assertions 1 and 2 in Lemma 3.8 hold for the UIBOM for all s, t ∈ Z with s < t. We still refer to this fact as Lemma 3.8 for the UIBOM. Let Q := Q(M ) be the radial map associated with M , i.e., the quadrangulation obtained from M by connecting the center of each face of M with its adjacent vertices and removing all edges of M . We identify V(M ) ∪ F(M ) with V(Q) so that λ is a bijection between Z and V(Q). By Lemma 3.8 for the UIBOM, we have the following. This constructions gives a general bijection between bipolar orientations and 2-orientations. When M = Z 2 , the 2-orientation on Q is called a 6-vertex configuration. This special case is considered in [KMSW16], based on which a connection between SLE 12 and the 6-vertex model is found.
To use Theorem 2.1 and Remark 2.5 to prove Theorem 1.9 , we define a planar map H = H(Z) via the following discrete analogue of (1. One can give a more direct description of M n . Recall Definition 3.18. For n ∈ N, let {b n (i)} i∈Z and {b −n (i)} i∈Z be as in Lemma 3.17 with = n and = −n, respectively. Define τ n = inf{t ≥ n : λ(t) = b −n (i) for some i > 0} and v +,n = S(λ(τ n )); τ −n = sup{t ≤ −n : S(λ(t)) = S(b n (i)) for some i ≤ 0} and v −,n = S(λ(τ −n )). must be on the east boundary of M n . This gives sup{t < k : R t = R k − 1} < −n, which contradicts (1.21) for R.
Recall the notations above Lemma 3.20. The following fact follows from Lemma 3.12 and the definition of Q n . Proof. We leave the reader to check the following fact. Write F(Nb v ) as { λ(i k ) : 1 ≤ k ≤ #F(Nb v )} where i 1 < · · · < i #F (Nbv) . Then for any 1 ≤ k ≤ #F(Nb v ) − 1, if λ(i k ) and λ(i k+1 ) are both in F(M n ) but do not satisfy condition 3 in the definition of Q, then they are connected in Q n only using vertices and faces in Nb v . These facts together with the definition of Q n imply the statement of the lemma.
From here on the proof of Theorem 1.9 in this case is very similar to the case of the infinite spanning tree-decorated map (Section 3.1.3) but there are minor technical differences so we give the complete argument.
Let C 1 > 0 be the constant from Theorem 2.1 for A in Theorem 1.9 and for Z above. As in the spanning tree case, by the isomorphism (modulo multiplicity) λ between H and Q plus Remark 2.5, we see that with probability at least 1 − O n (n −A ), the following is true (A) For each i 1 , i 2 ∈ [−n, n] Z with λ(i 1 ) ∼ λ(i 2 ) in Q, there is a path P G i1,i2 from i 1 to i 2 in G n with | P G i1,i2 | ≤ C 1 (log n) 3 ; and each j ∈ [−n, n] Z is contained in at most C 1 (log n) 6 of the paths P G i1,i2 .
(B) For each i 1 , i 2 ∈ [−n, n] Z with i 1 ∼ i 2 in G, then there is a path P Q i1,i2 from λ(i 1 ) to λ(i 2 ) in Q n with | P Q i1,i2 | ≤ C 1 (log n) 3 ; and each j ∈ [−n, n] Z is contained in at most C 1 (log n) 6 of the paths P Q i1,i2 . By basic properties of Brownian motion, deg(i; G) has an exponential tail. See [GMS17, Lemma 2.2] for a proof. Combined with Lemma 3.20, a union bound implies that there exists C 2 = C 2 (A) such that with probability at least 1 − O n (n −A ), (C) max x∈V(Mn)∪F (Mn) |Nb x | ≤ C 2 log n and max i∈[−n,n] Z deg(i; G) ≤ C 2 log n.
From here on we work on the event that (A)-(C) hold.
Proof of condition 1. For v 1 , v 2 ∈ V(M n ) such that v 1 ∼ v 2 in M n , let e be the edge connecting them. If both of the two faces of M with e on their boundaries are not in M n , then one can check that v 1 , v 2 satisfy condition 2 in the definition of Q and v 1 , v 2 ∈ V(Q n ). Indeed, if {v 1 , v 2 } is not on a face, it must be of the form λ(t) for some t ∈ [−n, n) Z by the definition of M n . This means that it is the NW edge of its tail. If it is not the SE edge of its head, then the face on its right will be visited at t + 1. By the definition of φ n , we have that φ n (v i ) = λ −1 (v i ) for i = 1, 2. In this case we let P G v1,v2 = P G λ −1 (v1), λ −1 (v2) .
If there exists f ∈ F(M n ) with e on its boundary, then f ∈ F(Nb v1 ) ∩ F(Nb v2 ). Now Lemma 3.26 yields that λ(φ n (v i )) and f are connected by a path P i in Q n using only vertices and faces in Nb vi for i = 1, 2. Let P G v1,v2 be the concatenation of paths P G where {v, w} is an edge on P 1 or P 2 . Now Given f ∼ v in Q n , if {f, v} is an edge on P 1 or P 2 defined above for some v 1 , v 2 ∈ V(M ), we must have v 1 , v 2 ∈ V(Nb f ). Therefore for each j ∈ [−n, n] Z , the number of paths of the form P G v1,v2 hitting j is at most max f ∈F (Mn) |Nb f | · C 2 (log n) 6 (C) ≤ C 1 C 2 (log n) 7 . (3.17) Proof of condition 2. Given a path P on Q of length m, we write it as { λ(k i )} 1≤i≤m . In addition, if λ(k 1 ) ∈ F(M ) (resp. λ(k m ) ∈ F(M )), we add ψ(k 1 ) (resp. ψ(k m )) at the beginning (resp. end) of P . Denote this possibly augmented path by P . Now we define a path Π(P ) on M n in two steps: ≤ 2C 1 C 2 (log n) 7 . (3.18) For each v ∈ V(M n ), in order for v to be hit by some P M i1,i2 , there must exist a face in Nb v that is hit by some path P Q i1,i2 . By (B), the number of paths of the form P M i1,i2 hitting v is bounded by |Nb v | · C 1 (log n) 6 (C) ≤ C 1 C 2 (log n) 7 .

Bipolar orientations with other weights and Schnyder woods
The only properties of Z = (L, R) that we used in Sections 3.3.2 and 3.3.3 are as follows: {∆Z i } i∈Z are i.i.d with E[∆Z i ] = (0, 0) and the correlation of L and R is in (−1, 1) (3.20) The support of ∆Z 1 is contained in (3.7) and E[e b|∆Z1| ] < ∞ for some b > 0. (3.21) In particular, as long as Z satisfies (3.20)-(3.21), we can construct an infinite planar map associated with Z, analogous to the UIBOM, via the sewing procedure. This gives a large family of measures on infinite bipolar-oriented maps. By [KMSW19, Theorem 6, Remark 2] and the argument for Proposition 3.16, this family of infinite bipolar-oriented maps includes the Benjamini-Schramm limits of the uniform measures on size n bipolar-oriented k-angulations 8 for any k ≥ 3. More generally, one can consider Boltzmann-type distributions where each face has a certain weight. We note that the above family includes maps encoded by walks with any correlation in (−1, 0), which corresponds to κ > 8 and γ < √ 2. The exact same argument as in Section 3.3.3 shows that Theorem 1.9 still holds for this family of maps (with M n , φ n , and ψ n defined in exactly the same way and γ determined by corr(L 1 , R 1 ) = − cos(πγ 2 /4)). We remark that in Section 3.3.3 we sometimes rotate the plane 180 degree and switch Z t = (L t , R t ) to (R −t , L −t ). But we never need the fact that (L t , R t ) d = (R −t , L −t ). All we need is that the walk (R −t , L −t ) also satisfies (3.20)-(3.21), so we do not have to impose this kind of symmetry condition on ∆Z to ensure our generalization of Section 3.3.3. Thus, we have established Theorem 1.9 in case 4.