×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bessis, Adv Appl Math 1 pp 109– (1980)
[2] , , On recursively unsolvable problems in topology and their classification, in Contributions to Mathematical Logic, North-Holland, Amsterdam, 1968, pp. 37–74.
[3] Brezin, Commun Math Phys 59 pp 35– (1978)
[4] Brezin, Phys Lett B 236 pp 144– (1990)
[5] Callan, J Integer Seg 8 (2004)
[6] Carlip, Classic Quantum Grav 15 pp 2629– (1998)
[7] Carlip, Phys Rev Lett 79 pp 4071– (1997)
[8] Di Francesco, Phys Rep 254 pp 1– (1995)
[9] Douglas, Nucl Phys B 335 pp 635– (1990)
[10] Ercolani, Int Math Res Not 2003 pp 755– (2003)
[11] Gessel, J Symbol Comp 14 pp 179– (1992)
[12] Gessel, J Integer Seg 8 (2004)
[13] Goulden, Trans Am Math Soc 353 pp 4405– (2001)
[14] Gross, Nucl Phys B 340 pp 333– (1990)
[15] Gross, Phys Rev Lett 64 pp 127– (1990)
[16] , Topological Graph Theory, John Wiley & Sons, New York, 1987.
[17] Harer, Invent Math 85 pp 457– (1986)
[18] Hartle, Class Quantum Grav 2 pp 707– (1985)
[19] Hawking, Nucl Phys B 144 pp 349– (1978)
[20] The path-integral approach to quantum gravity, in (Editors), General Relativity: An Einstein Centenary Survey, Cambridge University Press, New York, 1979, pp. 759–785.
[21] Heffter, Math Ann 38 pp 477– (1891)
[22] ’t Hooft, Nucl Phys B 72 pp 461– (1974)
[23] ’t Hooft, Nucl Phys B 75 pp 461– (1974)
[24] Itzykson, Commun Math Phys 134 pp 197– (1990)
[25] Jackson, Trans Am Math Soc 299 pp 785– (1987)
[26] Kontsevich, Commun Math Phys 147 pp 1– (1992)
[27] Special Functions and Their Applications, Prentice–Hall, Englewood Cliffs, NJ, 1965.
[28] The problem of homeomorphy, (Russian) Proc Int Congr Math, Cambridge University Press, London, 1960, pp. 300–306.
[29] Random Matrices, Academic Press, San Diego, 1991. · Zbl 0780.60014
[30] Oakley, Am Math Month 64 pp 143– (1957)
[31] Penner, J Diff Geom 27 pp 35– (1988)
[32] Regge, Nuovo Cim 19 pp 558– (1961)
[33] The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Springer-Verlag, New York, 2001. · Zbl 0964.05070
[34] Schleich, Nucl Phys B 402 pp 469– (1993)
[35] Geometrodynamics and the issue of the final state, in (Editors), Relativité, Groupes et Topologie, Gordon and Breach, New York, 1964, pp. 315–520.
[36] Zagier, Nieuw Arch Wisk 13 pp 489– (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.