The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees
By Guillaume Chapuy. 2008
55 citations recorded by the source corpus.
A unicellular map is a map which has only one face. We give a bijection between a dominant subset of rooted unicellular maps of given genus and a set of rooted plane trees with distinguished vertices. The bijection applies as well to the case of labelled unicellular maps, which are related to all rooted maps by Marcus and Schaeffer’s bijection. This gives an immediate derivation of the asymptotic number of unicellular maps of given genus, and a simple bijective proof of a formula of Lehman and Walsh on the number of triangulations with one vertex. From the labelled case, we deduce an expression of the asymptotic number of maps of genus g with n edges involving the ISE random measure, and an explicit characterization of the limiting profile and radius of random bipartite quadrangulations of genus g in terms of the ISE.
Subfields: Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/e897e8df6209b55513839e0cf1781d45f5ae1d56
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In Mathematical Physics
In Statistics and Probability
Citations in the corpus
- Scaling Limits for Random Quadrangulations of Positive Genus (2010, background): “In positive genus, Chapuy, Marcus, and Schaeffer [6] extended the bijective approaches known for the planar case, leading Chapuy [5] to establish the convergence of the rescaled profile of a uniform random bipartite quadrangulation of fixed genus.”