Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 30184
The Sizes of Large Hierarchical Long-Range Percolation Clusters

Authors: Yilun Shang

Abstract:

We study a long-range percolation model in the hierarchical lattice ΩN of order N where probability of connection between two nodes separated by distance k is of the form min{αβ−k, 1}, α ≥ 0 and β > 0. The parameter α is the percolation parameter, while β describes the long-range nature of the model. The ΩN is an example of so called ultrametric space, which has remarkable qualitative difference between Euclidean-type lattices. In this paper, we characterize the sizes of large clusters for this model along the line of some prior work. The proof involves a stationary embedding of ΩN into Z. The phase diagram of this long-range percolation is well understood.

Keywords: percolation, component, hierarchical lattice, phase transition.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1058931

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 843

References:


[1] M. Aizenman and C. M. Newman, Discontinuity of the percolation density in one-dimensional 1/|x−y|2 percolation models. Comm. Math. Phys., 107(1986) 611-647.
[2] S. R. Athreya and J. M. Swart, Survival of contact processes on the hierarchical group. Probab. Theory Relat. Fields, 147(2010) 529-563.
[3] A. L. Barab'asi and E. Ravaz, Hierarchical organization in complex networks. Phys. Rev. E, 67(2003) 026112.
[4] I. Benjamini and O. Schramm, Percolation beyond Zd, many questions and a few answers. Elect. Commun. Probab., 1(1996) 71-82.
[5] N. Berger, Transience, recurrence and critical behavior for long-range percolation. Comm. Math. Phys., 226(2002) 531-558.
[6] M. Biskup, On the scaling of chemical distance in long-range percoaltion models. Ann. Probab., 32(2004) 2938-2977.
[7] B. Bollob'as, S. Janson and O. Riordan, The phase transition in inhomogeneous random graphs. Random Struct. Algorithms, 31(2007) 3-122.
[8] D. Coppersmith, D. Gamarnik and M. Sviridenko, The diameter of a longrange percolation graph. Random Struct. Algorithms, 21(2002) 1-13.
[9] D. A. Dawson and L. G. Gorostiza, Percolation in an ultrametric space. arXiv:1006.4400v2, 2011.
[10] D. A. Dawson and L. G. Gorostiza, Percolation in a hierarchical random graph. Comm. Stochastic Analysis, 1(2007) 29-47.
[11] G. Grimmett, Percolation. New York: Springer, 1999.
[12] V. Koval, R. Meester and P. Trapman, Long-range percolation on the hierarchical lattice. arXiv:1004.1251v1, 2010.
[13] R. Rammal, G. Toulouse and M. A. Virasoro, Ultrametricity for physicists. Rev. Mod. Phys., 58(1986) 765-788.
[14] P. Schneider, Nonarchimedean Functional Analysis. New York: Springer, 2002.
[15] L. S. Schulman, Long range percolation in one dimension. J. Phys. A: Math. Gen., 16(1983) L639-L641.
[16] Y. Shang, Percolation in a hierarchical lattice. Submitted to Ann. Probab.
[17] Y. Shang, Uniqueness of the infinite component for percolation on a hierarchical lattice. Submitted to Random Struct. Algorithms.
[18] Y. Shang, The giant component in a random subgraph of a weak expander. Int. J. Math. Comput. Sci., 7(2011) 95-99.
[19] Y. Shang, Leader-following consensus problems with a time-varying leader under measurement noises. arXiv:0909.4349, to appear in Adv. Dyn. Syst. Appl.
[20] J. Shen, Cucker-Smale flocking under hierarchical leadership. SIAM J. Appl. Math., 68(2007) 694-719.
[21] P. Trapman, The growth of the infinite long-range percolation cluster. Ann. Probab., 38(2010) 1583-1608.