## Topological characteristics of random triangulated surfaces.(English)Zbl 1145.52009

The paper considers topological characteristics of orientable surfaces generated by gluing $$n$$ triangles together. The results are expressed in terms of a parameter $$h=n/2+ \chi$$, where $$\chi$$ is the Euler characteristic of the surface (number of vertices minus number of edges plus number of faces). A class of similar models is considered all giving the same probability distributions for various parameters of interest. Simulations and results suggest that $\text{Ex} [h] = \text{log}(3n) + \gamma + o(1) , \quad \text{Var} [h] = \log(3n) + \gamma - \pi^2/6 + o(1)$ hold for similar models, where $$\gamma=0.5772 \ldots$$ is Euler’s constant. It is proven that $\text{Ex} [h] = \log n + O(1),\quad \text{Var} [h] = O(\text{log} \, n).$ It is proven also that the probability of connectedness of a random surface for the map model with $$n$$ triangles is $\Pr [c=1] = 1- \frac{5}{18n} + O \left( \frac{1}{n^2} \right),$ where $$c$$ is the number of connected components. Results for a number of topological invariants and combinatorial characteristics of the random surfaces are derived.
Reviewer: J. Kaupužs (Riga)

### MSC:

 52B70 Polyhedral manifolds 57Q15 Triangulating manifolds 60B05 Probability measures on topological spaces 60D05 Geometric probability and stochastic geometry

### Keywords:

random surfaces; Euler characteristic
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### References:

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