Non-amenable Cayley graphs of high girth have $p_c < p_u$ and mean-field exponents

Asaf Nachmias (University of British Columbia)
Yuval Peres (Microsoft Research)

Abstract


In this note we show that percolation on non-amenable Cayley graphs of high girth has a phase of non-uniqueness, i.e., $p_c< p_u$. Furthermore, we show that percolation and self-avoiding walk on such graphs have mean-field critical exponents. In particular, the self-avoiding walk has positive speed.

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Pages: 1-8

Publication Date: December 3, 2012

DOI: 10.1214/ECP.v17-2139

References

  • Aizenman, Michael; Barsky, David J. Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108 (1987), no. 3, 489--526. MR0874906
  • Aizenman, Michael; Newman, Charles M. Tree graph inequalities and critical behavior in percolation models. J. Statist. Phys. 36 (1984), no. 1-2, 107--143. MR0762034
  • Akhmedov, Azer. The girth of groups satisfying Tits alternative. J. Algebra 287 (2005), no. 2, 275--282. MR2134144
  • Barsky, D. J.; Aizenman, M. Percolation critical exponents under the triangle condition. Ann. Probab. 19 (1991), no. 4, 1520--1536. MR1127713
  • Bauerschmidt R., Duminil-Copin H., Goodman J. and G. Slade (2010), Lectures on self-avoiding walks, preprint.
  • Benjamini, Itai; Nachmias, Asaf; Peres, Yuval. Is the critical percolation probability local? Probab. Theory Related Fields 149 (2011), no. 1-2, 261--269. MR2773031
  • Benjamini, Itai; Schramm, Oded. Percolation beyond $\bold Z^ d$, many questions and a few answers. Electron. Comm. Probab. 1 (1996), no. 8, 71--82 (electronic). MR1423907
  • Benjamini, Itai; Schramm, Oded. Percolation in the hyperbolic plane. J. Amer. Math. Soc. 14 (2001), no. 2, 487--507 (electronic). MR1815220
  • Bollobás, Béla; Riordan, Oliver. Percolation. Cambridge University Press, New York, 2006. x+323 pp. ISBN: 978-0-521-87232-4; 0-521-87232-4 MR2283880
  • Brydges, David; Spencer, Thomas. Self-avoiding walk in $5$ or more dimensions. Comm. Math. Phys. 97 (1985), no. 1-2, 125--148. MR0782962
  • Burton, R. M.; Keane, M. Density and uniqueness in percolation. Comm. Math. Phys. 121 (1989), no. 3, 501--505. MR0990777
  • Duminil-Copin H. and Hammond A. (2012), Self-avoiding walk is sub-ballistic, preprint, ARXIV1205.0401.
  • 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
  • Hara, Takashi; Slade, Gordon. Mean-field critical behaviour for percolation in high dimensions. Comm. Math. Phys. 128 (1990), no. 2, 333--391. MR1043524
  • Hara, Takashi; Slade, Gordon. Self-avoiding walk in five or more dimensions. I. The critical behaviour. Comm. Math. Phys. 147 (1992), no. 1, 101--136. MR1171762
  • Häggström, Olle; Jonasson, Johan. Uniqueness and non-uniqueness in percolation theory. Probab. Surv. 3 (2006), 289--344. MR2280297
  • R. Lyons with Y. Peres, Probability on Trees and Networks, In preparation, link
  • Madras N. (2012) A Lower Bound for the End-to-End Distance of Self-Avoiding Walk, preprint, link
  • Madras, Neal; Slade, Gordon. The self-avoiding walk. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1993. xiv+425 pp. ISBN: 0-8176-3589-0 MR1197356
  • Madras, Neal; Wu, C. Chris. Self-avoiding walks on hyperbolic graphs. Combin. Probab. Comput. 14 (2005), no. 4, 523--548. MR2160417
  • Olʹshanskiĭ, A. Yu.; Sapir, M. V. On $F_ k$-like groups. (Russian) Algebra Logika 48 (2009), no. 2, 245--257, 284, 286--287; translation in Algebra Logic 48 (2009), no. 2, 140--146 MR2573020
  • 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
  • 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
  • Schonmann, Roberto H. Mean-field criticality for percolation on planar non-amenable graphs. Comm. Math. Phys. 225 (2002), no. 3, 453--463. MR1888869
  • 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
  • Woess, Wolfgang. Random walks on infinite graphs and groups. Cambridge Tracts in Mathematics, 138. Cambridge University Press, Cambridge, 2000. xii+334 pp. ISBN: 0-521-55292-3 MR1743100
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