Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-13T05:59:53.686Z Has data issue: false hasContentIssue false
  • English
  • Français

AB percolation on bond-decorated graphs

Part of: Special processes Equilibrium statistical mechanics

Published online by Cambridge University Press:  14 July 2016

Martin J. B. Appel  and
John C. Wierman
Martin J. B. Appel*
Affiliation:
University of Iowa
John C. Wierman*
Affiliation:
The Johns Hopkins University
*
∗ Postal address: Department of Statistics and Actuarial Science, University of Iowa, Iowa City, IA 52242, USA.
∗∗ Postal address: Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, MD 21218, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

It is known [8] that a certain class of bond-decorated graphs exhibits multiple AB percolation phase transitions. Sufficient conditions are given under which the corresponding AB percolation critical probabilities may be identified as points of intersection of the graph of a certain polynomial with the boundary of the percolative region of an associated two-parameter bond-site percolation model on the underlying undecorated graph. The main result of the article is used to prove that the graphs in [8] exhibit multiple AB percolation critical probabilities. The possibility of identifying AB percolation critical exponents with corresponding limits for the bond-site model is discussed.

Keywords

PERCOLATIONBOND-SITE MODELCRITICAL PROBABILITIESCRITICAL EXPONENTS

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82B43: Percolation
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 1 , March 1993 , pp. 153 - 166
Copyright
Copyright © Applied Probability Trust 1993 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Appel, M. J. and Wierman, J. C. (1987) On the absence of infinite AB percolation clusters in bipartite graphs. J. Phys. A: Math. Gen. 20, 2527–2531.CrossRefGoogle Scholar
[2] Appel, M. J. B. (1990) AB Percolation. , The Johns Hopkins University.Google Scholar
[3] Appel, M. J. B. and Wierman, J. C. (1992) Two percolation critical exponents for Galton-Watson trees. Technical Report 199, University of Iowa.Google Scholar
[4] Essam, J. W. (1980) Percolation theory. Rep. Prog. Phys. 43, 833–912.Google Scholar
[5] Fortuin, C. M, Kasteleyn, P. W. and Ginibre, J. (1971) Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22, 89–103.Google Scholar
[6] Kesten, H. (1980) The critical probability of bond percolation on the square lattice equals 1/2. Commun. Math. Phys. 74, 41–59.CrossRefGoogle Scholar
[7] Kesten, H. (1982) Percolation Theory for Mathematicians. Birkhäuser, Basel.CrossRefGoogle Scholar
[8] Luczak, T. and Wierman, J. C. (1989) Counterexamples in AB percolation. J. Phys. A: Math. Gen. 22, 185–191.CrossRefGoogle Scholar
[9] Nakanishi, H. (1987) Critical behavior of AB percolation in two dimensions. J. Phys. A : Math. Gen. 20, 6075–6083.Google Scholar
[10] Newman, M. H. A. (1951) Elements of the Topology of Plane Sets of Points, 2nd edn. Cambridge University Press.Google Scholar
[11] Ord, G., Whittington, S. G. and Wilker, J. B. (1984) Critical probabilities in percolation on decorated graphs. J. Phys. A: Math. Gen. 17, 3195–3199.CrossRefGoogle Scholar
[12] Toth, B. (1985) A lower bound for the critical probability of the square lattice. Z. Wahrschlichkeitsth. 69, 19–22.Google Scholar
[13] Van Den Berg, J. and Kesten, H. (1985) Inequalities with applications to percolation and reliability. J. Appl. Prob. 22, 556–569.Google Scholar
[14] Wierman, J. C. (1981) Bond percolation on honeycomb and triangular lattices. Adv. Appl. Prob. 13, 293–313.Google Scholar
[15] Wierman, J. C. (1988) AB percolation: a brief survey. In Proc. Sem. Combinatorics and Graph Theory. Banach Center Publications.Google Scholar
[16] Wierman, J. C. (1988) On AB percolation on bipartite graphs. J . Phys. A: Math. Gen. 21, 1945–1949.CrossRefGoogle Scholar