Cori–Vauquelin–Schaeffer Bijection

• The Cori–Vauquelin–Schaeffer bijection is a combinatorial construction that associates rooted, well-labeled plane trees with planar quadrangulations using a contour exploration algorithm.
• It extends to infinite trees and constructs objects like the Uniform Infinite Planar Quadrangulation, with labels encoding key distance properties between vertices.
• In scaling limits, the bijection bridges discrete map enumeration with continuum random geometry, revealing universal structures such as the Brownian sphere and α-stable spheres.

The Cori–Vauquelin–Schaeffer bijection (CVS bijection) is a fundamental combinatorial construction establishing a correspondence between well-labeled plane trees and planar quadrangulations, with extensive ramifications in the theory of random planar maps and their scaling limits. It serves as a pivotal tool for understanding geometric and probabilistic properties of random surfaces and, in modern refinements, extends to infinite and continuum settings, yielding scaling limits that include the Brownian sphere and its stable analogues.

1. Origin and Classical Statement of the CVS Bijection

The CVS bijection operates between the class of rooted, well-labeled plane trees with $n$ edges and the class of rooted, pointed planar quadrangulations with $n$ faces. A rooted plane tree $\tau$ of $n$ edges is endowed with a labeling $\ell:V(\tau)\to\mathbb{Z}$ satisfying:

• $\ell(\text{root}) = 0$,
• For each edge $(u,v)$, $|\ell(u) - \ell(v)| \le 1$.

The bijection proceeds via a detailed algorithmic process:

• Perform a contour exploration of $\tau$ to order the corners $(v_0,v_1,\ldots,v_{2n})$.
• For each corner $n$0, define the "successor" $n$1 as the first future corner where the label has decreased by 1. If no such $n$2 exists, assign an extra point at infinity $n$3.
• Draw non-crossing arcs from each corner to its successor (or to $n$4), ensuring arcs neither cross each other nor the tree.
• The union of tree-edges and these arcs forms a planar map of degree 4 faces, i.e., a quadrangulation.

The root of the quadrangulation is determined by an independent orientation choice. This construction yields a bijection between the set of well-labeled trees (with orientation bit) and rooted, pointed quadrangulations. The label at each vertex encodes distance to the distinguished point: $n$5 with the normalization $n$6 (Angel et al., 18 Feb 2025, Archer et al., 2024, Curien et al., 2012).

2. Extensions and Infinite Volume Limits

The CVS bijection extends naturally to infinite trees. An infinite spine is allowed, and the positivity constraint on labels is released, broadening applicability considerably. When applied to critical geometric Galton–Watson trees with appropriate label increments, this extended bijection produces the Uniform Infinite Planar Quadrangulation (UIPQ). The labels in the infinite map retain meaning as horofunctions, with $n$7, thus encoding infinite-volume geodesic structure (Curien et al., 2012).

The construction also admits an explicit combinatorial inverse: Given a rooted quadrangulation with a label-preserving assignment, the underlying labeled tree is reconstructed by adding diagonals in each face according to local rules dictated by cyclic label sequences.

3. Metric Structure and Scaling Limits

Analyzing the metric induced by labels yields tight probabilistic connections between the contour/label processes on trees and geometry in the associated quadrangulation. Notably, the label differences between adjacent tree vertices translate directly into graph distance inequalities within the quadrangulation; bounds such as the cactus bound $n$8 are central to establishing metric tightness needed for scaling limits.

In the scaling regime, when random trees are chosen according to offspring distributions with heavy tails ($n$9-stable, $\tau$0), the CVS bijection transports scaling limits from trees to planar maps. The resulting metric spaces are universal limits: the Brownian sphere for finite-variance trees and $\tau$1-stable spheres for heavy-tailed regimes (Angel et al., 18 Feb 2025, Archer et al., 2024).

4. The Continuum CVS Bijection and the Brownian Sphere

The continuum version of the CVS bijection maps a pair $\tau$2 of continuous functions—specifically, coding a real tree $\tau$3 (via a Brownian excursion or stable limit) and a labeling process (the Brownian snake)—to a random metric-measure space known as the Brownian sphere. The construction employs the "mating-of-trees" pseudometric: $\tau$4 where $\tau$5 is the contour metric and $\tau$6 the label-induced metric.

Under this mapping, $\tau$7 is recovered from the marked metric-measure space (the Brownian sphere) together with an orientation variable. Key geometric sets—cut-locus and geodesic skeleton—correspond to the skeletons of the respective trees and serve as loci for reconstructing the underlying coding functions. This mapping is a measurable isomorphism modulo orientation, satisfying invertibility properties almost surely with respect to the Brownian-sphere law (Angel et al., 18 Feb 2025).

5. The Role of CVS in Random Maps and Universality

The CVS bijection has enabled a series of deep results on the universal behavior of large random planar maps. Primary consequences include:

• Identification of the Brownian sphere as the universal scaling limit for uniformly random quadrangulations.
• Description of scaling exponents (Hausdorff dimension, volume fluctuation exponents) for planar maps via those for underlying trees, i.e., Hausdorff dimension $\tau$8 for $\tau$9-stable limits (Archer et al., 2024).
• Explicit transfer of probabilistic limit theorems: the convergence of contour and label processes for BGW trees to the stable snake processes is mirrored in the geometry of the limiting random maps.

The extension to master bijections by Bernardi–Fusy illustrates that the CVS construction generalizes to $n$0-angulations and clarifies its uniformity and systematization (see (Bernardi et al., 2010) for the detailed combinatorial perspective).

6. Inversion, Orientation, and Technical Lemmas

A fundamental advance is the explicit inversion of the continuum CVS bijection. The process reconstructs the encoding functions $n$1 by geometric analysis of the metric-measure (Brownian) sphere, using properties of cut-loci, branching structure, and the quadratic variation of labelings along geodesic branches. The orientation ambiguity is resolved via a measurable selection from the sphere’s structure, yielding true invertibility (Angel et al., 18 Feb 2025).

Measurability, uniqueness, and tightness of the correspondence are established via technical lemmas that link intrinsic geometry of the map to the analytic properties of the tree and label processes.

7. Enumerative and Combinatorial Foundations

The enumerative power of the CVS bijection is manifest in the closed-form generating function expressions for planar quadrangulations, e.g., $n$2 for the number of rooted simple quadrangulations with $n$3 inner faces (Bernardi et al., 2010). The construction is compatible with canonical orientation characterizations and bicolored tree encodings, confirming its centrality in map enumeration and bijective combinatorics.

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The Cori–Vauquelin–Schaeffer bijection and its continuum variants organize the profound connections between labeled trees, random geometry, and the universality of random surfaces, enabling a unified approach to the study of random planar maps and their scaling limits (Angel et al., 18 Feb 2025, Archer et al., 2024, Curien et al., 2012, Bernardi et al., 2010).