Electronic Communications in Probability

Growing random 3-connected maps or Comment s'enfuir de l'Hexagone

Louigi Addario-Berry

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We use a growth procedure for binary trees due to Luczak and Winkler, a bijection between binary trees and irreducible quadrangulations of the hexagon due to Fusy, Poulalhon and Schaeffer, and the classical angular mapping between quadrangulations and maps, to define a growth procedure for maps. The growth procedure is local, in that every map is obtained from its predecessor by an operation that only modifies vertices lying on a common face with some fixed vertex. As n tends to infinity, the probability that the n'th map in the sequence is 3-connected tends to 2^8/3^6. The sequence of maps has an almost sure limit G, and we show that G is the distributional local limit of large, uniformly random 3-connected graphs.

Article information

Electron. Commun. Probab., Volume 19 (2014), paper no. 54, 12 pp.

Accepted: 12 August 2014
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]

Random maps random trees random planar graphs growth procedures

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Addario-Berry, Louigi. Growing random 3-connected maps or Comment s'enfuir de l'Hexagone. Electron. Commun. Probab. 19 (2014), paper no. 54, 12 pp. doi:10.1214/ECP.v19-3314. https://projecteuclid.org/euclid.ecp/1465316756

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