Miermont Bijection for Quadrangulations

• The Miermont bijection establishes a canonical correspondence between well-labeled unicycles and delayed quadrangulations, forming a bridge between discrete maps and metric geometry.
• It employs a systematic face-by-face construction with delay parameters that encode geodesic distances, enabling precise enumeration via generating functions.
• The bijection underpins scaling limits by linking discrete quadrangulations to continuum objects like the Brownian sphere, applicable to both orientable and non-orientable surfaces.

The Miermont bijection provides a canonical and combinatorially rich correspondence between classes of labeled maps (typically unicyclic or unicellular maps with additional metric structure) and multipointed rooted quadrangulations with admissible “delays.” It plays a central role in the combinatorial encoding of distances and the construction of scaling limits of random two-dimensional surfaces, notably in the explicit construction of the Brownian sphere and its variants. The bijection generalizes the Marcus–Schaeffer bijection to multiple points and applies uniformly to maps embedded in arbitrary compact surfaces, both orientable and non-orientable, and underpins the connection between discrete and continuum random geometry.

1. Definition and Combinatorial Framework

The classical setting of the Miermont bijection involves rooted planar quadrangulations with marked vertices and certain delay parameters. A rooted quadrangulation is a planar map where every face has degree 4 and a distinguished oriented edge (the root edge). For $n$ faces, the set is $\mathcal Q_n$. A bi-pointed quadrangulation is a triple $(q,v_1,v_2)$, $q\in \mathcal Q_n$, with two distinct labeled vertices. Introducing a delay $d\in\mathbb{Z}$, $d\equiv d_q(v_1,v_2)$ mod 2, $|d|< d_q(v_1,v_2)$, one constructs a delayed quadrangulation $(q,v_1,v_2,d)$, forming the set $\mathcal Q_n^{(d)}$.

On the labeled map side, the bijection associates these objects to well-labeled unicycles: rooted planar maps with two faces and an integer vertex labeling $\ell:V(u)\to\mathbb{Z}$ subject to $|\ell(x) - \ell(y)|\leq 1$ for adjacent $x,y$, with the root vertex labeled $0$. Notationally, $\mathcal U_n$ denotes well-labeled unicycles with $n$ edges, and $\widetilde{\mathcal U}_n$ includes additional structure for a distinguished oriented edge on the unique cycle.

The bijection (also termed the CVS-bijection in the bi-pointed case $k=2$) is two-to-one; fixing the orientation of the root edge in the quadrangulation gives a canonical correspondence.

2. Construction of the Miermont Bijection

The construction of the bijection proceeds face-by-face and leverages the labeling structure:

• For a well-labeled unicycle $(u, \ell)\in \mathcal U_n$, consider traversal of the corners of the external face.
• For each corner $c_i$, compute $\sigma(i) = \inf\{j>i: \ell(c_j) < \ell(c_i)\}$ with cyclic convention and $\inf \emptyset = +\infty$.
• Introduce a new external vertex $v_1$ with label $\ell(v_1) = \min_{v\in V(u)} \ell(v) - 1$. Draw arcs from each $c_i$ to $c_{\sigma(i)}$ if defined, otherwise to $v_1$, so as to avoid crossings.
• Symmetric construction for the internal face yields vertex $v_2$, same label law. After adding all arcs in both faces and erasing the unicycle edges, one obtains a quadrangulation with two marked vertices and root determined by the original root corner.
• The delay is $d = \ell(v_2) - \ell(v_1)$.

The inverse map reconstructs the labeled unicycle uniquely (up to root orientation) from the quadrangulation and delay data.

In the more general setting, the bijection extends to quadrangulations on arbitrary compact surfaces $\mathbb{S}$ with $k\geq1$ sources, each with a delay. The general criteria for delays involve constraints:

• $\min_i d(w_i) = 0$,
• $|d(w_i) - d(w_j)| < d_q(w_i, w_j)$ for $i\neq j$,
• $d(w_i) - d(w_j) + d_q(w_i, w_j) \equiv 0\pmod{2}$.

Vertex labeling $$\ell^d(v) = \min_{1\leq i\leq k} [d_q(v, w_i) + d(w_i)]$$ propagates distance information; across each edge, $|\ell^d(u)-\ell^d(v)|=1$. The bijection creates a red-labeled map (edges within faces connect increasing label corners), and a dual exploration graph (blue) that guides the construction and inverts the mapping.

3. Metric Encoding and Generating Functions

A key feature of the Miermont bijection is the encoding of metric information:

• The delay $d$ between the two distinguished vertices relates to the graph distance via $$d_q(v_1, v_2) = |\ell(v_2) - \ell(v_1)| + 1 = |d|+1$$.
• For enumeration, the bivariate generating function

$$F_n(x, y) = \sum_{q\in \mathcal Q_n} \sum_{\,d \in\,\mathrm{Adm}(q)} x^{\#V(q)} y^d$$

is naturally expressed using the bijection—$F_n(x, y)$ appears as the $z^n$ coefficient in an explicit algebraic series.

To paper bias by two-point distance, an exponential factor is inserted, leading to

$$Z_n(\lambda) = \sum_{(q,v_1,v_2,d)\in\mathcal Q_n^{(d)}} e^{-\lambda(|d| + 1)},$$

and this, along with its multivariate extensions, can be expressed using generating functions for well-labeled unicycles with additional weights $e^{-\lambda|\ell(v_2)-\ell(v_1)|}$.

On arbitrary surfaces, the enumeration extends to genus $h$ with generating series $Q_h(z)$ for maps of type $h$, known to be algebraic. Asymptotically,

$q_h(n) \sim p_h n^{\frac{5(h-1)}{2}} 12^n,$

which is uniform in orientable versus non-orientable cases, and the exponent $5(h-1)/2$ arises from the labeled mobile decomposition.

4. Scaling Limits and the Continuum Random Unicycle

In the scaling regime, the bijection governs convergence of planar quadrangulations to continuum objects:

• Graph distances and labels scale as $n^{1/4}$; i.e., if $Q_n^{(d)}$ is a uniform delayed quadrangulation, the rescaled metric measure space converges in the Gromov–Hausdorff–Prokhorov sense:

$$\left(V(Q_n), n^{-1/4}d_n, \mu_n\right) \Longrightarrow (\mathcal S, D, \mu),$$

with $(\mathcal S, D, \mu)$ the biased Brownian sphere.

• The limiting encoding in the continuum is via a continuum random labeled unicycle (CRLU): a pair $(\mathfrak e, W)$ where $\mathfrak e$ is a Brownian excursion (random duration), and $W$ is a Brownian snake head process driven by $\mathfrak e$, with an extra bias proportional to $$(\mathfrak e_{\min} + 1)^3(\mathfrak e_{\max} + 1)^3$$. The structure is a tree-of-trees with a cycle-spine.
• The continuous Miermont bijection (Theorem 3.8) collapses the CRLU along the pseudo-distance

$$D'(s, t) = \inf\sum_{i=1}^k \left(\ell(u_{i-1}) + \ell(u_i) - 2 \min_{v\in [u_{i-1},u_i]} \ell(v)\right),$$

yielding a random compact metric space isometric to the biased Brownian sphere.

• Universal constants such as $1/(2\,\Gamma(1/4)\,\sqrt{\pi})$ are determined via enumeration and scaling limits.

5. Conditionings, Voronoï Cells, and Applications

The continuum bijection has powerful applications for the construction of conditional laws and geometric features:

• In $(\mathcal S, D, x_1, x_2)$, define for any real $\delta \in [-D(x_1,x_2), D(x_1,x_2)]$ the $\delta$–delayed Voronoï cells:

$$\mathcal V_1^\delta = \{z\in\mathcal S : D(z,x_1) \leq D(z,x_2) + \delta\},\quad \mathcal V_2^\delta = \{z\in\mathcal S : D(z,x_1) \geq D(z,x_2) + \delta\}.$$

These capture the metric decomposition of the sphere into cells associated with the two points.

• Conditioning the biased Brownian sphere on a fixed two-point distance $\{D(x_1, x_2) = b\}$ (Theorem 4.10) yields a quotient of a CRLU with two faces of areas $s$ and $b-s$, glued along geodesic boundaries. The law of $(s, b-s)$ is explicitly given by

$$\frac{(s(b-s))^{-3/2}}{\int_0^b (t(b-t))^{-3/2}dt}, \qquad s \in (0, b).$$

This construction uses concatenated first-passage Brownian bridges and continuum snakes.

• The bigeodesic Brownian plane (Theorem 5.1) is obtained as the local limit by gluing Poisson clouds of snake excursions above two independent 3-dimensional Bessel processes, corresponding to the local maximal--minimal geodesic structure that arises around typical points in the sphere.

6. Extension to Arbitrary Surfaces

The Miermont bijection extends to all compact surfaces $\mathbb{S}$ (of orientable or non-orientable type $h$) with $k$ sources and delays as specified. The construction involves:

• Labeling vertices by $\ell^d$ derived from source distances and delays.
• Building a dual-exploration graph (DEG) that recursively crosses all edges and organizes blue vertices and edges according to label increments.
• Insertion of red map edges within each face, guided by the cyclic or local orientation.
• The correspondence is $2:1$ because the root orientation induces a sign, but can be made bijective by additional marking.

Metric consequences are uniform: distances (e.g., the radius of a random quadrangulation or typical two-point distances) scale as $n^{1/4}$, and profiles converge to occupation measures of generalized Brownian snakes on $\mathbb{S}$.

7. Key Theorems, Proof Ideas, and Implications

The principal theorems include:

• Discrete CVS-bijection: A $2$–to–$1$ correspondence between well-labeled unicycles and delayed quadrangulations of equal size; noncrossing successor arcs define quadrangular faces with two marked vertices.
• Unicycle ↔ Quadrilateral with Geodesic Sides: Transition from unicycles to "vertebrates" (quadrilaterals with geodesic boundary), essential for generalizing to higher genus.
• Continuum Limit Theorem: Scaling limits of quadrilaterals with geodesic boundary converge (in the 4-marked GHP topology) to continuous objects coded by Brownian bridges and a snake.
• Continuum Miermont Bijection: The metric scaling/bias by the two-point distance persists, and the limiting quotient space matches the Brownian sphere law.
• Conditioning at Fixed Distance: The sphere conditioned on two points at prescribed distance arises from gluing CRLU faces with explicit area and geodesic length distributions.
• Bigeodesic Brownian Plane: The local geometry about generic geodesic points is described by Bessel process-based constructions with Poisson snakes.

Proof approaches include generating function enumeration, integrability and total mass control under delay-bias, joint scaling analysis for cycle-length and area, passing to limits in the Gromov–Hausdorff–Prokhorov framework, and the structural properties of the Brownian snake and first-passage bridges.

A plausible implication is that the bijective and measure-biasing techniques developed for quadrangulations transfer robustly to surfaces of arbitrary topology, facilitating a unified framework for random geometry and providing fine control on scaling limits and conditional structures. Convergence in all these cases relies on metric and measure-theoretic stability under the scaling dictated by the combinatorics and the associated stochastic processes.

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References:

• (Mourichoux, 11 Nov 2025): Detailed construction of the Brownian sphere via the Miermont bijection and continuum random unicycle, including proofs of all key theorems and applications to Voronoï cells and the bigeodesic plane.
• (Chapuy et al., 2015): Generalization of the Miermont bijection to quadrangulations on arbitrary compact surfaces, the combinatorial and enumerative consequences, and scaling limit results.