Long-range percolation on the hierarchical lattice

Vyacheslav Koval (Utrecht University)
Ronald Meester (VU University Amsterdam)
Pieter Trapman (Stockholm University)


We study long-range percolation on the hierarchical lattice of order $N$, where any edge of length $k$ is present with probability $p_k=1-\exp(-\beta^{-k} \alpha)$, independently of all other edges. For fixed $\beta$, we show that $\alpha_c(\beta)$ (the infimum of those $\alpha$ for which an infinite cluster exists a.s.) is non-trivial if and only if $N < \beta < N^2$. Furthermore, we show uniqueness of the infinite component and continuity of the percolation probability and of $\alpha_c(\beta)$ as a function of $\beta$. This means that the phase diagram of this model is well understood.

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

Pages: 1-21

Publication Date: July 23, 2012

DOI: 10.1214/EJP.v17-1977


  • Aizenman, Michael; Barsky, David J. Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108 (1987), no. 3, 489--526. MR0874906
  • Aizenman, M.; Newman, C. M. Discontinuity of the percolation density in one-dimensional $1/\vert x- y\vert ^ 2$ percolation models. Comm. Math. Phys. 107 (1986), no. 4, 611--647. MR0868738
  • Athreya, Siva R.; Swart, Jan M. Survival of contact processes on the hierarchical group. Probab. Theory Related Fields 147 (2010), no. 3-4, 529--563. MR2639714
  • Berger, Noam. Transience, recurrence and critical behavior for long-range percolation. Comm. Math. Phys. 226 (2002), no. 3, 531--558. MR1896880
  • Biskup, Marek. On the scaling of the chemical distance in long-range percolation models. Ann. Probab. 32 (2004), no. 4, 2938--2977. MR2094435
  • Biskup, Marek. Graph diameter in long-range percolation. Random Structures Algorithms 39 (2011), no. 2, 210--227. MR2850269
  • Bollobás, Béla; Janson, Svante; Riordan, Oliver. The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31 (2007), no. 1, 3--122. MR2337396
  • Chayes, Lincoln; Schonmann, Roberto H. Mixed percolation as a bridge between site and bond percolation. Ann. Appl. Probab. 10 (2000), no. 4, 1182--1196. MR1810870
  • Cox, J. T.; Durrett, Richard. Limit theorems for the spread of epidemics and forest fires. Stochastic Process. Appl. 30 (1988), no. 2, 171--191. MR0978353
  • D.A. Dawson and L.G. Gorostiza, phPercolation in an ultrametric space, ArXiv e-prints.
  • Dawson, D. A.; Gorostiza, L. G. Percolation in a hierarchical random graph. Commun. Stoch. Anal. 1 (2007), no. 1, 29--47. MR2404371
  • Dekking, F. M.; Meester, R. W. J. On the structure of Mandelbrot's percolation process and other random Cantor sets. J. Statist. Phys. 58 (1990), no. 5-6, 1109--1126. MR1049059
  • Diekmann, Odo; Heesterbeek, J. A. P. Mathematical epidemiology of infectious diseases. Model building, analysis and interpretation. Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2000. xvi+303 pp. ISBN: 0-471-49241-8 MR1882991
  • Friedman, Nathaniel A. Introduction to ergodic theory. Van Nostrand Reinhold Mathematical Studies, No. 29. Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1970. v+143 pp. MR0435350
  • Friedman, Nathaniel A. Replication and stacking in ergodic theory. Amer. Math. Monthly 99 (1992), no. 1, 31--41. MR1140275
  • Gandolfi, A.; Keane, M. S.; Newman, C. M. Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses. Probab. Theory Related Fields 92 (1992), no. 4, 511--527. MR1169017
  • 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
  • Grimmett, Geoffrey. The random-cluster model. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 333. Springer-Verlag, Berlin, 2006. xiv+377 pp. ISBN: 978-3-540-32890-2; 3-540-32890-4 MR2243761
  • Hara, Takashi; Hattori, Tetsuya; Watanabe, Hiroshi. Triviality of hierarchical Ising model in four dimensions. Comm. Math. Phys. 220 (2001), no. 1, 13--40. MR1882398
  • Meester, Ronald; Trapman, Pieter. Bounding basic characteristics of spatial epidemics with a new percolation model. Adv. in Appl. Probab. 43 (2011), no. 2, 335--347. MR2848379
  • Newman, C. M.; Schulman, L. S. One-dimensional $1/\vert j-i\vert ^ s$ percolation models: the existence of a transition for $s\leq 2$. Comm. Math. Phys. 104 (1986), no. 4, 547--571. MR0841669
  • Zhang, Z. Q.; Pu, F. C.; Li, B. Z. Long-range percolation in one dimension. J. Phys. A 16 (1983), no. 3, L85--L89. MR0701466
  • Trapman, Pieter. The growth of the infinite long-range percolation cluster. Ann. Probab. 38 (2010), no. 4, 1583--1608. MR2663638
  • Van Rooij, A.C.M., Non-Archimedean functional analysis, vol. 51 of Monographs and Textbooks in Pure and Applied Math, 1978.

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