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

Louigi Addario-Berry (McGill University)

Abstract


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.

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Pages: 1-12

Publication Date: August 12, 2014

DOI: 10.1214/ECP.v19-3314

References



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