Trees, Animals, and Percolation on Hyperbolic Lattices

Neal Madras (York University)
C. Chris Wu (Penn State University)

Abstract


We study lattice trees, lattice animals, and percolation on non-Euclidean lattices that correspond to regular tessellations of two- and three-dimensional hyperbolic space. We prove that critical exponents of these models take on their mean field values. Our methods are mainly combinatorial and geometric.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 2019-2040

Publication Date: December 3, 2010

DOI: 10.1214/EJP.v15-837

References

  1. Benjamini, Itai; Lyons, Russell; Peres, Yuval; Schramm, Oded. Critical percolation on any nonamenable group has no infinite clusters. Ann. Probab. 27 (1999), no. 3, 1347--1356. MR1733151 (2000k:60197)
  2. Benjamini, I. and Schramm, O. (1996). Percolation beyond ${\bf Z}^d$, many questions and a few answers. Electronic Commun. Probab. 1, Paper no. 8, 71--82.
  3. Benjamini, Itai; Schramm, Oded. Percolation in the hyperbolic plane. J. Amer. Math. Soc. 14 (2001), no. 2, 487--507 (electronic). MR1815220 (2002h:82049)
  4. Borgs, Christian; Chayes, Jennifer; van der Hofstad, Remco; Slade, Gordon. Mean-field lattice trees.On combinatorics and statistical mechanics. Ann. Comb. 3 (1999), no. 2-4, 205--221. MR1772346 (2001i:82036)
  5. Bourdon, M. (1993). Actions quasi-convexes d'un groupe hyperbolique, flot geodesique. Thesis, Orsay.
  6. Phénomènes critiques, systèmes aléatoires, théories de jauge. Part I, II. [Critical phenomena, random systems, gauge theories. Part I, II] Papers from the 43rd session of the Les Houches summer school of theoretical physics held at Les Houches, August 1–September 7, 1984.Edited by Konrad Osterwalder and Raymond Stora.North-Holland Publishing Co., Amsterdam, 1986. Vol. I: xxxviii+537 pp.; Vol. II: pp. i--xxxviii and 539--1201. ISBN: 0-444-86980-8 MR0880522 (87m:81006)
  7. Bowen, Rufus; Series, Caroline. Markov maps associated with Fuchsian groups. Inst. Hautes Études Sci. Publ. Math. No. 50 (1979), 153--170. MR0556585 (81b:58026)
  8. Brydges, David C.; Imbrie, John Z. Branched polymers and dimensional reduction. Ann. of Math. (2) 158 (2003), no. 3, 1019--1039. MR2031859 (2005c:82051)
  9. Coxeter, H. S. M. Regular honeycombs in hyperbolic space. Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. III, pp. 155--169. Erven P. Noordhoff N.V., Groningen; North-Holland Publishing Co., Amsterdam, 1956. MR0087114 (19,304c)
  10. Coxeter, H. S. M. Regular polytopes.Second edition The Macmillan Co., New York; Collier-Macmillan Ltd., London 1963 xx+321 pp. (8 plates). MR0151873 (27 #1856)
  11. Grimmett, Geoffrey. Percolation.Second edition.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 321. Springer-Verlag, Berlin, 1999. xiv+444 pp. ISBN: 3-540-64902-6 MR1707339 (2001a:60114)
  12. Grimmett, G. R.; Newman, C. M. Percolation in $\infty+1$ dimensions. Disorder in physical systems, 167--190, Oxford Sci. Publ., Oxford Univ. Press, New York, 1990. MR1064560 (92a:60207)
  13. Grünbaum, Branko. Convex polytopes.Second edition.Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler.Graduate Texts in Mathematics, 221. Springer-Verlag, New York, 2003. xvi+468 pp. ISBN: 0-387-00424-6; 0-387-40409-0 MR1976856 (2004b:52001)
  14. Haagerup, U. and Munkholm, H.J. (1981). Simplices of maximal volume in hyperbolic $n$-space. Acta Math. 147, 1--11.
  15. Häggström, Olle; Jonasson, Johan. Uniqueness and non-uniqueness in percolation theory. Probab. Surv. 3 (2006), 289--344 (electronic). MR2280297 (2007m:60297)
  16. Iversen, Birger. Hyperbolic geometry.London Mathematical Society Student Texts, 25. Cambridge University Press, Cambridge, 1992. xiv+298 pp. ISBN: 0-521-43508-0; 0-521-43528-5 MR1205776 (94b:51023)
  17. van Rensburg, E. J. Janse. The statistical mechanics of interacting walks, polygons, animals and vesicles.Oxford Lecture Series in Mathematics and its Applications, 18. Oxford University Press, Oxford, 2000. x+379 pp. ISBN: 0-19-850561-2 MR1858028 (2003a:82032)
  18. Jonasson, Johan; Steif, Jeffrey E. Amenability and phase transition in the Ising model. J. Theoret. Probab. 12 (1999), no. 2, 549--559. MR1684757 (2000b:60238)
  19. Klarner, David A. Cell growth problems. Canad. J. Math. 19 1967 851--863. MR0214489 (35 #5339)
  20. Klein, D.J. (1981). Rigorous results for branched polymer models with excluded volume. J. Chem. Phys. 75, 5186--5189.
  21. Lalley, Steven P. Percolation on Fuchsian groups. Ann. Inst. H. Poincaré Probab. Statist. 34 (1998), no. 2, 151--177. MR1614583 (99g:60190)
  22. Lalley, S. P. Percolation clusters in hyperbolic tessellations. Geom. Funct. Anal. 11 (2001), no. 5, 971--1030. MR1873136 (2002i:60183)
  23. Lubensky, T.C. and Isaacson, J. (1979). Statistics of lattice animals and dilute branched polymers. Phys. Rev. A 20, 2130--2146.
  24. Madras, N. (1995). A rigorous bound on the critical exponent for the number of lattice trees, animals, and polygons. J. Statist. Phys. 78, 681--699.
  25. Madras, Neal; Wu, C. Chris. Self-avoiding walks on hyperbolic graphs. Combin. Probab. Comput. 14 (2005), no. 4, 523--548. MR2160417 (2006e:60061)
  26. Otter, Richard. The number of trees. Ann. of Math. (2) 49, (1948). 583--599. MR0025715 (10,53c)
  27. Pak, Igor; Smirnova-Nagnibeda, Tatiana. On non-uniqueness of percolation on nonamenable Cayley graphs. C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 6, 495--500. MR1756965 (2000m:60116)
  28. Parisi, Giorgio; Sourlas, Nicolas. Critical behavior of branched polymers and the Lee-Yang edge singularity. Phys. Rev. Lett. 46 (1981), no. 14, 871--874. MR0609853 (82d:82058)
  29. Polya, G. (1937). Kombinatorische Anzahlbestimmungen fur Gruppen, Graphen, und chemische Verbindungen. Acta Math. 68, 145--254.
  30. Pólya, G.; Read, R. C. Combinatorial enumeration of groups, graphs, and chemical compounds.Pólya's contribution translated from the German by Dorothee Aeppli.Springer-Verlag, New York, 1987. viii+148 pp. ISBN: 0-387-96413-4 MR0884155 (89f:05013)
  31. Ratcliffe, John G. Foundations of hyperbolic manifolds.Graduate Texts in Mathematics, 149. Springer-Verlag, New York, 1994. xii+747 pp. ISBN: 0-387-94348-X MR1299730 (95j:57011)
  32. Rietman, Ronald; Nienhuis, Bernard; Oitmaa, Jaan. The Ising model on hyperlattices. J. Phys. A 25 (1992), no. 24, 6577--6592. MR1210879 (94e:82026)
  33. Schonmann, Roberto H. Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs. Comm. Math. Phys. 219 (2001), no. 2, 271--322. MR1833805 (2002h:82036)
  34. Schonmann, Roberto H. Mean-field criticality for percolation on planar non-amenable graphs. Comm. Math. Phys. 225 (2002), no. 3, 453--463. MR1888869 (2003m:82039)
  35. Slade, G. The lace expansion and its applications.Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004.Edited and with a foreword by Jean Picard.Lecture Notes in Mathematics, 1879. Springer-Verlag, Berlin, 2006. xiv+228 pp. ISBN: 978-3-540-31189-8; 3-540-31189-0 MR2239599 (2007m:60301)
  36. Swierczak, Ewa; Guttmann, Anthony J. Self-avoiding walks and polygons on non-Euclidean lattices. J. Phys. A 29 (1996), no. 23, 7485--7500. MR1425834 (98a:82058)
  37. Tasaki, Hal; Hara, Takashi. Critical behavior in a system of branched polymers. Progr. Theoret. Phys. Suppl. No. 92 (1987), 14--25. MR0934664 (89f:82054)
  38. Tykesson, Johan. The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space. Electron. J. Probab. 12 (2007), no. 51, 1379--1401 (electronic). MR2354162 (2008k:60248)
  39. Vanderzande, C. (1998). Lattice Models of Polymers}. Cambridge Lecture Notes in Physics, v. 11. Cambridge University Press, Cambridge.
  40. Wilker, J. B.; Whittington, S. G. Extension of a theorem on supermultiplicative functions. J. Phys. A 12 (1979), no. 10, L245--L247. MR0545387 (81c:26019)
  41. Wu, C. Chris. Ising models on hyperbolic graphs. II. J. Statist. Phys. 100 (2000), no. 5-6, 893--904. MR1798548 (2002h:82027)
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.