A Uniform Sampling Procedure for Abstract Triangulations of Surfaces
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We present a procedure to sample uniformly from the set of combinatorial isomorphism types of balanced triangulations of surfaces - also known as graph-encoded surfaces. For a given number n, the sample is a weighted set of graph-encoded surfaces with 2n triangles. The sampling procedure relies on connections between graph-encoded surfaces and permutations, and basic properties of the symmetric group. We implement our method and present a number of experimental findings based on the analysis of 138 million runs of our sampling procedure, producing graph-encoded surfaces with up to 280 triangles. Namely, we determine that, for n fixed, the empirical mean genus g̅(n) of our sample is very close to g̅(n) = n-1/2 - (16.98n -110.61)^1/4. Moreover, we present experimental evidence that the associated genus distribution more and more concentrates on a vanishing portion of all possible genera as n tends to infinity. Finally, we observe from our data that the mean number of non-trivial symmetries of a uniformly chosen graph encoding of a surface decays to zero at a rate super-exponential in n.
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