Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-24T07:00:22.576Z Has data issue: false hasContentIssue false
  • English
  • Français

A random hierarchical lattice: the series-parallel graph and its properties

Part of: Special processes Graph theory Equilibrium statistical mechanics

Published online by Cambridge University Press:  01 July 2016

B. M. Hambly  and
Jonathan Jordan
B. M. Hambly*
Affiliation:
University of Oxford
Jonathan Jordan*
Affiliation:
University of Sheffield
*
∗ Postal address: Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, UK.
∗∗ Postal address: Department of Probability and Statistics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a sequence of random graphs constructed by a hierarchical procedure. The construction replaces existing edges by pairs of edges in series or parallel with probability p. We investigate the effective resistance across the graphs, first-passage percolation on the graphs and the Cheeger constants of the graphs as the number of edges tends to infinity. In each case we find a phase transition at

Keywords

Hierarchical latticeeffective resistancephase transitionfirst-passage percolationCheeger constants

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 05C80: Random graphs 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 3 , September 2004 , pp. 824 - 838
Copyright
Copyright © Applied Probability Trust 2004 

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] Alon, N. and Naor, M. (1993). Coin-flipping games immune against linear sized coalitions. SIAM J. Comput. 22, 403–417.Google Scholar
[2] Barlow, M. T. (1998). Diffusions on fractals. In Lectures on Probability Theory and Statistics (Saint-Flour, July 1995; Lecture Notes Math. 1690), Springer, Berlin, pp. 1–121.Google Scholar
[3] Chung, F. R. K. (1997). Spectral Graph Theory (CBMS Reg. Conf. Ser. Math. 92). American Mathematical Society, Providence, RI.Google Scholar
[4] Cook, J. and Derrida, B. (1989). Polymers on disordered hierarchical lattices: a nonlinear combination of random variables. J. Statist. Phys. 57, 89–139.Google Scholar
[5] Derrida, B. (1986). Pure and random models of statistical mechanics on hierarchical lattices. In Critical Phenomena, Random Systems and Gauge Theories, eds Osterwalder, K. and Stora, R., North-Holland, Amsterdam, pp. 989–999.Google Scholar
[6] Essoh, C. D. and Bellisard, J. (1989). Resistance and fluctuation of a fractal network of random resistors: a non-linear law of large numbers. J. Phys. A 22, 4537–4548.CrossRefGoogle Scholar
[7] Griffiths, R. and Kaufman, M. (1982). Spin systems on hierarchical lattices. Phys. Rev. B 26, 5022–5032.Google Scholar
[8] Jordan, J. (2002). Almost sure convergence for iterated functions of independent random variables. Ann. Appl. Prob. 12, 985–1000.Google Scholar
[9] Jordan, J. (2003). Renormalisation of random hierarchical systems. , University of Oxford.Google Scholar
[10] Li, D. L. and Rogers, T. D. (1999). Asymptotic behavior for iterated functions of random variables. Ann. Appl. Prob. 9, 1175–1201.Google Scholar
[11] Moore, E. F. and Shannon, C. E. (1956). Reliable circuits using less reliable relays. I. J. Franklin Inst. 262, 191–208.CrossRefGoogle Scholar
[12] Moore, E. F. and Shannon, C. E. (1956). Reliable circuits using less reliable relays. II. J. Franklin Inst. 262, 281–297.Google Scholar
[13] Schenkel, A., Wehr, J. and Wittwer, P. (2000). Computer-assisted proofs for fixed point problems in Sobolev spaces. Math. Phys. Electron J. 6, No. 3.Google Scholar
[14] Shneiberg, I. Y. (1986). Hierarchical sequences of random variables. Theory Prob. Appl. 31, 137–141.Google Scholar
[15] Wehr, J. and Woo, J.-M. (2001). Central limit theorems for nonlinear hierarchical sequences of random variables. J. Statist. Phys. 104, 777–797.Google Scholar