Skip to main content
SpringerLink
Log in

Spanning Forests on Random Planar Lattices

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

The generating function for spanning forests on a lattice is related to the q-state Potts model in a certain q→0 limit, and extends the analogous notion for spanning trees, or dense self-avoiding branched polymers. Recent works have found a combinatorial perturbative equivalence also with the (quadratic action) O(n) model in the limit n→−1, the expansion parameter t counting the number of components of the forest.

We give a random-matrix formulation of this model on the ensemble of degree-k random planar lattices. For k=3, a correspondence is found with the Kostov solution of the loop-gas problem, which arise as a reformulation of the (logarithmic action) O(n) model, at n=−2.

Then, we show how to perform an expansion around the t=0 theory. In the thermodynamic limit, at any order in t we have a finite sum of finite-dimensional Cauchy integrals. The leading contribution comes from a peculiar class of terms, for which a resummation can be performed exactly.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Balian, R., Toulouse, G.: Phys. Rev. Lett. 30, 544 (1973)

    Article  ADS  Google Scholar 

  2. Fisher, M.: Phys. Rev. Lett. 30, 679 (1973)

    Article  ADS  MathSciNet  Google Scholar 

  3. Abe, R.: Prog. Theor. Phys. 48, 1414 (1972)

    Article  ADS  Google Scholar 

  4. Duplantier, B., David, F.: J. Stat. Phys. 51, 327 (1988)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Parisi, G., Sourlas, N.: J. Phys. Lett. 41, L403 (1980)

    Article  Google Scholar 

  6. De Gennes, P.G.: Phys. Lett. 38A, 339 (1972)

    ADS  Google Scholar 

  7. Des Cloiseaux, J.: J. Phys. 36, 281 (1975)

    Google Scholar 

  8. McKane, A.J.: Phys. Lett. 76A, 22 (1980)

    ADS  MathSciNet  Google Scholar 

  9. Aragão De Carvalho, C., Caracciolo, S., Fröhlich, J.: Nucl. Phys. B 215, 209 (1983)

    Article  ADS  Google Scholar 

  10. Caracciolo, S., Jacobsen, J.L., Saleur, H., Sokal, A.D., Sportiello, A.: Phys. Rev. Lett. 93, 080601 (2004). arXiv:cond-mat/0403271

    Article  ADS  MathSciNet  Google Scholar 

  11. Caracciolo, S., Sokal, A.D., Sportiello, A.: J. Phys. A: Math. Theor. 40, 13799 (2007). arXiv:0706.1509

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. Sokal, A.D.: In: Webb, B.S. (ed.) Surveys in Combinatorics, 2005, pp. 173–226. Cambridge University Press, Cambridge (2005). arXiv:math/0503607

    Google Scholar 

  13. Nienhuis, B.: Phys. Rev. Lett. 49, 1062 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  14. Temperley, H.N.V., Lieb, E.H.: Proc. R. Soc. Lond. A, Math. Phys. Sci. 322, 251 (1971)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. Baxter, R.J., Kelland, S.B., Wu, F.Y.: J. Phys. A: Math. Gen. 9, 397 (1976)

    Article  ADS  Google Scholar 

  16. van Beijeren, H.: Phys. Rev. Lett. 38, 993 (1977)

    Article  ADS  Google Scholar 

  17. Nienhuis, B.: In: Domb, C., Lebowitz, J.L. (eds.) Phase Transitions and Critical Phenomena. Academic Press, New York (1987)

    Google Scholar 

  18. Caracciolo, S., Pelissetto, A.: Nucl. Phys. B 420, 141 (1994). arXiv:hep-lat/9401015

    Article  ADS  Google Scholar 

  19. Caracciolo, S., Pelissetto, A.: Nucl. Phys. B 455, 619 (1995). arXiv:hep-lat/9510015

    Article  ADS  Google Scholar 

  20. Shin, D.-S.: Nucl. Phys. B 546, 669 (1999). arXiv:hep-lat/9810025

    Article  ADS  Google Scholar 

  21. Allés, B., Caracciolo, S., Pelissetto, A., Pepe, M.: Nucl. Phys. B 562, 581 (1999). arXiv:hep-lat/9906014

    Article  ADS  Google Scholar 

  22. Caracciolo, S., De Grandi, C., Sportiello, A.: Nucl. Phys. B 787, 260 (2007). arXiv:0705.3891

    Article  MATH  ADS  Google Scholar 

  23. ’t Hooft, G.: Nucl. Phys. B 72, 461 (1974)

    Article  ADS  MathSciNet  Google Scholar 

  24. Brézin, E., Itzykson, C., Parisi, G., Zuber, J.B.: Commun. Math. Phys. 59, 35 (1978)

    Article  MATH  ADS  Google Scholar 

  25. Bessis, D.: Commun. Math. Phys. 69, 147 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  26. Itzykson, C., Zuber, J.B.: J. Math. Phys. 21, 411 (1980)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  27. Polyakov, A.M.: Mod. Phys. Lett. A 2, 893 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  28. Knizhnik, V.G., Polyakov, A.M., Zamolodchikov, A.B.: Mod. Phys. Lett. 3A, 819 (1988)

    ADS  MathSciNet  Google Scholar 

  29. David, F.: C.R. Acad. Sci. Paris 307(II), 1051 (1988)

    Google Scholar 

  30. David, F.: Mod. Phys. Lett. A 3, 1651 (1988)

    Article  ADS  Google Scholar 

  31. Distler, J., Kawai, H.: Nucl. Phys. B 321, 509 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  32. Distler, J., Hlousek, Z., Kawai, H.: Int. J. Mod. Phys. A 5, 1093 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  33. Duplantier, B.: Phys. Rev. Lett. 84, 1363 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  34. Duplantier, B.: Conformal random geometry. In: Bovier, A., Dunlop, F., denHollander, F. (eds.) Les Houches Summer School, Session LXXXIII, 2005, Mathematical Statistical Physics, pp. 101–217. Elsevier B.V., Amsterdam (2006). arXiv:math-ph/0608053v1

    Chapter  Google Scholar 

  35. Duplantier, B., Kostov, I.K.: Phys. Rev. Lett. 61, 1433 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  36. Duplantier, B., Kostov, I.K.: Nucl. Phys. B 340, 491 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  37. Duplantier, B., Sheffield, S.: Liouville quantum gravity and KPZ. arXiv:0808.1560 (2008)

  38. David, F., Bauer, M.: Another derivation of the geometrical KPZ relations. arXiv:0810.2858 (2008)

  39. Di Francesco, P.: Lectures given at the summer school “Applications of random matrices in physics”. Les Houches, June 2004. arXiv:math-ph/0406013 (2004)

  40. Kazakov, V.A., Kostov, I.K., Migdal, A.A.: Phys. Lett. B 157, 295 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  41. Boulatov, D., Kazakov, V.A., Kostov, I.K., Migdal, A.A.: Phys. Lett. B 174, 87 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  42. Boulatov, D., Kazakov, V.A., Kostov, I.K., Migdal, A.A.: Nucl. Phys. B 275(FS17), 641 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  43. Kazakov, V.A.: Phys. Lett. A 119, 140 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  44. Kazakov, V.A.: Nucl. Phys. B 4, 93 (1988) (Proc. Supp.)

    Article  MATH  MathSciNet  Google Scholar 

  45. Kostov, I.: Mod. Phys. Lett. A 4, 217 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  46. Gaudin, M., Kostov, I.: Phys. Lett. B 220, 200 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  47. Eynard, B., Zinn-Justin, J.: Nucl. Phys. B 386, 558 (1992). arXiv:hep-th/9204082

    Article  ADS  MathSciNet  Google Scholar 

  48. Eynard, B., Kristjansen, C.: Nucl. Phys. B 455, 577 (1995). arXiv:hep-th/9506193

    Article  MATH  ADS  MathSciNet  Google Scholar 

  49. Eynard, B., Kristjansen, C.: Nucl. Phys. B 466, 463 (1996). arXiv:hep-th/9512052

    Article  MATH  ADS  MathSciNet  Google Scholar 

  50. Durhuus, B., Kristjansen, C.: Nucl. Phys. B 483, 535 (1997). arXiv:hep-th/9609008

    Article  MATH  ADS  MathSciNet  Google Scholar 

  51. Eynard, B., Guitter, E., Kristjansen, C.: Nucl. Phys. B 528, 523 (1998). arXiv:cond-mat/9801281

    Article  MATH  ADS  MathSciNet  Google Scholar 

  52. Guitter, E., Kristjansen, C., Nielsen, J.L.: Nucl. Phys. B 546, 731 (1999). arXiv:cond-mat/9811289

    Article  MATH  ADS  MathSciNet  Google Scholar 

  53. Kostov, I.K., Staudacher, M.: Nucl. Phys. B 384, 459 (1992). arXiv:hep-th/9203030

    Article  ADS  MathSciNet  Google Scholar 

  54. Caracciolo, S., Sportiello, A.: J. Stat. Mech. 0502, P002 (2005). arXiv:cond-mat/0412327

    Google Scholar 

  55. Kostov, I.: Mod. Phys. Lett. A 4, 217 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  56. Gaudin, M., Kostov, I.: Phys. Lett. B 220, 200 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  57. Eynard, B., Bonnet, G.: Phys. Lett. B 463, 273 (1999). arXiv:hep-th/9906130

    Article  MATH  ADS  MathSciNet  Google Scholar 

  58. Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman & Hall/CRC Press, London, Boca Raton (2003)

    Google Scholar 

  59. Deng, Y., Garoni, T.M., Sokal, A.D.: Phys. Rev. Lett. 98, 030602 (2007). arXiv:cond-mat/0610193

    Article  ADS  Google Scholar 

  60. Bedini, A., Caracciolo, S., Sportiello, A.: J. Phys. A: Math. Theor. 41, 205003 (2008). arXiv:0802.1506

    Article  MathSciNet  Google Scholar 

  61. Slater, L.J.: Generalized Hypergeometric Functions. Cambridge University Press, Cambridge (1966)

    MATH  Google Scholar 

  62. Sloane, N.J.A.: On-Line Encyclopedia of Integer Sequences. http://www.research.att.com/~njas/ (2009). C n ≡A 1,n : A000108; A 2,n : A001764; A 3,n ⋅⋅⋅ A 7,n : A002293 ⋅⋅⋅ A002297

Download references

Author information

Authors and Affiliations

  1. Dip. di Fisica dell’Università degli Studi di Milano and INFN, via Celoria 16, 20133, Milan, Italy

    Sergio Caracciolo & Andrea Sportiello

Authors
  1. Sergio Caracciolo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrea Sportiello
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergio Caracciolo.

Additional information

Dedicated to Édouard Brézin and Giorgio Parisi pioneers also in this subject, on the occasion of their special birthday.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Caracciolo, S., Sportiello, A. Spanning Forests on Random Planar Lattices. J Stat Phys 135, 1063–1104 (2009). https://doi.org/10.1007/s10955-009-9733-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-009-9733-1

Keywords

Search

Navigation