The first cycles in an evolving graph
Discrete Mathematics
Almost all maps are asymmetric
Journal of Combinatorial Theory Series B
Marking in combinatorial constructions: generating functions and limiting distributions
Theoretical Computer Science - Special volume on mathematical analysis of algorithms (dedicated to D. E. Knuth)
Largest 4-connected components of 3-connected planar triangulations
Random Structures & Algorithms
Random sampling of large planar maps and convex polyhedra
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Random graphs
The Size of the Largest Components in Random Planar Maps
SIAM Journal on Discrete Mathematics
Combinatorial Enumeration
An Information-Theoretic Upper Bound of Planar Graphs Using Triangulation
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
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A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponential-quadratic type (e-x2), that is, Gaussian. We exhibit here a new class of "universal" phenomena that are of the exponential-cubic type (eix3), corresponding to nonstandard distributions that involve the Airy function. Such Airy phenomena are expected to be found in a number of applications, when confluences of critical points and singularities occur. About a dozen classes of planar maps are treated in this way, leading to the occurrence of a common Airy distribution that describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and fine optimization of random generation algorithms for multiply connected planar graphs.