Sharp Concentration of the Number of Submaps in Random Planar Triangulations

We show that the maximum vertex degree in a random 3-connected planar triangulation is concentrated in an interval of almost constant width. This is a slightly weaker type of result than our earlier determination of the limiting distribution of the maximum vertex degree in random planar maps and in random triangulations of a (convex) polygon. We also derive sharp concentration results on the number of vertices of given degree in random planar maps of all three types. Some sharp concentration results about general submaps in 3-connected triangulations are also given.

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Correspondence to Zhicheng Gao*.

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* Research supported by NSERC and Australian Research Council

† Research supported by the Australian Research Council

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Gao*, Z., Wormald†, N.C. Sharp Concentration of the Number of Submaps in Random Planar Triangulations. Combinatorica 23, 467–486 (2003). https://doi.org/10.1007/s00493-003-0028-x

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AMS Subject Classification (2000):

  • 05C30
  • 05A16
  • 05C80