Skip to main content
SpringerLink
Log in

Combinatorics and Probability of Maps

  • Chapter
Asymptotic Combinatorics with Application to Mathematical Physics

Part of the book series: NATO Science Series ((NAII,volume 77))

  • 298 Accesses

Abstract

We give an introductory pedagogical review of rigorous mathematical results concerning combinatorial enumeration and probability distributions for maps on compact orientable surfaces, with an emphasize on applications to two-dimensional quantum gravity.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Fernandez, R., Frolich, J. and Sokal A. (1992) Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory, Springer-Verlag.

    MATH  Google Scholar 

  2. Gibbs, Ph. (1995) The Small Scale Structure of Space-Time: A Bibliographical Review, Preprint, hep-th/9506171.

    Google Scholar 

  3. Wallace, D. (2000) The Quantization of Gravity-an introduction, Preprint, gr-qc/0004005.

    Google Scholar 

  4. Rovelli C. (1998) Strings, loops and others: a critical survey of the present approaches to quantum gravity, Preprint, gr-qc/9803024.

    Google Scholar 

  5. Schnepps, L., Ed. (1994) The Grothendieck theory of dessins d’enfants, London Math. Soc. Lect. Note Series, v. 200, Cambridge Univ. Press.

    Google Scholar 

  6. Schnepps, L. and Lochak, P., Eds. (1997) Geometric Galois Actions, London Math. Soc. Lect. Note Series, v. 242, 243, Cambridge Univ. Press.

    Google Scholar 

  7. Edmonds, J. (1960) A combinatorial representation for polyhedral surfaces, Notices Amer. Math. Soc., 7, 646.

    Google Scholar 

  8. Kontsevich, M. (1992) Intersection Theory on the Moduli Spaces of Curves and the Matrix Airy Function, Com. Math. Phys., 147, no. 1, 1–23.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Liskovets, V. A. (1996) Some asymptotical estimates for planar maps. Combinatorics, Probability and Computing, 5, 131–138.

    Article  MathSciNet  MATH  Google Scholar 

  10. Liu Yanpei, (1999) Enumerative theory of Maps, Kluwer.

    MATH  Google Scholar 

  11. Goulden, I. and Jackson D. (1983) Combinatorial Enumeration, Wiley.

    MATH  Google Scholar 

  12. Tutte, W. (1973) The Enumerative Theory of Planar Maps, in J. Srivastava et al. (Eds.), A Survey of Combinatorial Theory, North Holland.

    Google Scholar 

  13. Malyshev, V. (1999) Probability around the quantum gravity, Russian Math. Reviews, 54, no. 4, 3–46.

    MathSciNet  Google Scholar 

  14. Bender, E., Canfield, E. and Richmond, L. (1993) The asymptotic number of rooted maps on a surface. II. Enumeration by vertices and faces. J. Comb. Theory A, 63, 318–329.

    Article  MathSciNet  Google Scholar 

  15. Gao Zhi-Cheng (1992) The asymptotic number of rooted 2-connected triangular maps on a surface, J. Comb. Theory B, 54, no. 1, 102–112.

    ADS  Google Scholar 

  16. Frohlich, J. (1990) Regge calculus and discretized gravitational functional integral, Preprint, Zurich.

    Google Scholar 

  17. Ambjorn, J., Carfora, M. and Marzuoli, A. (1997) The geometry of dynamical triangulations, 1997; http://xxx.lanl.gov/hep-th/9612069.

    Google Scholar 

  18. Ambjorn, J., Durhuus, B. and Jonsson, T. (1997) Quantum geometry, Cambridge.

    Book  Google Scholar 

  19. Walsh, T. and Lehman, A. (1972) Counting Rooted Maps by Genus, parts 1, 2, 3, J. Comb. Theory B, 13, 192–218; (1972) 13, 122–141; (1975) 18, 222–259.

    MathSciNet  Google Scholar 

  20. Malyshev, V. and Minlos, R.. (1989) Gibbs Random Fields, Kluwer.

    Google Scholar 

  21. Brezin, E., Itzykson, C., Parisi, G. and Zuber, J. (1978) Planar Diagrams, Comm. Math. Phys., 59, 35–47.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Kazakov, V., Staudacher, M. and Wynter, Th. (1996) Exact solution of discrete two-dimensional R2 gravity, Nuclear Physics B 471, 309–333.

    MathSciNet  ADS  Google Scholar 

  23. Kazakov, V. A. (2000) Solvable Matrix Models. Talk delivered at Workshop “Matrix Models and Painleve equations”, 1999, Berkeley; hep-th/0003064.

    Google Scholar 

  24. Zvonkin, A. (1997) Matrix Integrals and Map Enumeration: An Accessible Introduction, Mathl. Comput. Modelling, 26, no. 8–10, 281–304.

    Article  MathSciNet  MATH  Google Scholar 

  25. Pastur, L. A. (1996) Spectral and Probabilistic Aspects of Random Matrix Models, in A. Boutet de Monvel and V.A. Marchenko (Eds.), Algebraic and Geometric methods in Mathematical Physics, Kluwer, 205–242.

    Google Scholar 

  26. Harer, J. and Zagier, D. (1986) The Euler characteristic of the moduli space of curves, Invent. Math., 85, 457–485.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Itzykson, C. and Zuber, J.-B. (1990) Comm. Math. Phys., 134, 197–207.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Okounkov, A. (2001) Gromov-Witten theory, Hurwitz numbers, and Matrix Models. I; math.AG/0101147.

    Google Scholar 

  29. Witten, E. (1991) Two dimensional gravity and intersection theory on moduli space, Surveys in Diff. Geometry, 1, 243–310.

    MathSciNet  Google Scholar 

  30. Dobrushin, R. (1968) The description of random field by means of conditional probabilities and conditions of its regularity, Theory of Probability and Appl., 13, 197–224.

    Article  Google Scholar 

  31. Lanford, O. and Ruelle, D. (1969) Observables at infinity and states with short range correlations in statistical mechanics. Comm. Math. Phys., 13, 174–215.

    Article  MathSciNet  ADS  Google Scholar 

  32. Baez, J. (1996) Spin network states in gauge theory, Adv. Math., 117, 253–272.

    Article  MathSciNet  MATH  Google Scholar 

  33. Penrose, R. (1971) Angular momentum: an approach to combinatorial space-time, in T. Bastin (Ed.), Quantum Theory and Beoynd, Cambridge Univ. Press.

    Google Scholar 

  34. Malyshev, V. (2001) Gibbs and Quantum Discrete Spaces, Russian Math. Reviews, 56, no. 5., 117–172.

    MathSciNet  Google Scholar 

  35. Atiyah, M. (1991) The geometry and physics of knots Cambridge Univ. Press, Cambridge.

    Google Scholar 

  36. Dubrovin, B., Frenkel, E. and Previato, E. (1993) Integrable Systems and Quantum Groups, Springer.

    Google Scholar 

  37. Ambjorn, J., Jurkiewich, J. and Loll, R. (2000) Lorentzian and Euclidean quantum gravity- analytical and numerical results, http://xxx.lanl.gov/hep-th/0001124.

    Google Scholar 

  38. Ambjorn, J. and Loll, R. (1998) Non-perturbative Lorentzian Quantum Gravity, Causality and Topology Change, Nucl. Phys. B, 536, 407.

    Article  MathSciNet  ADS  Google Scholar 

  39. Di Francesco, P., Guitter, E. and Kristjansen, C. (2000) Integrable 2D Lorentzian Gravity and Random Walks. Nucl. Phys. B, 567, 515–553.

    Article  ADS  Google Scholar 

  40. Malyshev, V., Yambartsev, A. and Zamyatin, A. (2001) Two-dimensional Lorentzian models, Moscow Math. Journal, 1, no. 3, 439–456.

    MathSciNet  MATH  Google Scholar 

  41. Malyshev, V. (in press) Dynamical triangulation models with matter fields: high temperature region, Comm. Math. Phys.

    Google Scholar 

  42. Glimm, J. and Jaffe, A. (1981) Quantum Physics.

    Book  MATH  Google Scholar 

  43. Rivasseau, V. (1991) Isosystolic inequalities and the topological expansion for random surfaces and matrix models. Comm. Math. Phys., 139, 183–200.

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. I. N. R. I. A., France

    V. A. Malyshev

Authors
  1. V. A. Malyshev
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. I.N.R.I.A., Le Chesnay, France

    Vadim Malyshev

  2. Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg, Russia

    Anatoly Vershik

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Malyshev, V.A. (2002). Combinatorics and Probability of Maps. In: Malyshev, V., Vershik, A. (eds) Asymptotic Combinatorics with Application to Mathematical Physics. NATO Science Series, vol 77. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0575-3_4

Download citation

  • DOI: https://doi.org/10.1007/978-94-010-0575-3_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-4020-0793-4

  • Online ISBN: 978-94-010-0575-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation