--- title: Cori–Vauquelin–Schaeffer Bijection url: https://www.emergentmind.com/topics/cori-vauquelin-schaeffer-bijection type: topic --- # Cori–Vauquelin–Schaeffer Bijection 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 $i$, define the "successor" $j > i$ as the first future corner where the label has decreased by 1. If no such $j$ exists, assign an extra point at infinity $\star$. - Draw non-crossing arcs from each corner to its successor (or to $\star$), 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: $d(v,\star) = \ell(v) - \ell(\star)$ with the normalization $\ell(\star) = \min \ell - 1$ [2502.13074, 2405.05677, 1201.1052]. ## 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 $\ell(u) - \ell(v) = \lim_{z \to \infty} (d(u,z) - d(v,z))$, thus encoding infinite-volume geodesic structure [1201.1052]. 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 $d(a,b) \leq \ell(a) + \ell(b) - 2 \min_{v\in[[a,b]]}\ell(v)$ 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 ($\alpha$-stable, $\alpha \in (1,2)$), 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 $\alpha$-stable spheres for heavy-tailed regimes [2502.13074, 2405.05677]. ## 4. The Continuum CVS Bijection and the Brownian Sphere The continuum version of the CVS bijection maps a pair $(f,g)$ of continuous functions—specifically, coding a real tree $T_f$ (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: \[ d_h(s,t) = \inf \left\{\sum_{i=1}^k d_g(s_i, t_i): k\geq 1, d_f(t_j,s_{j+1})=0\ \forall j