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

Part of:Graph theorySpecial processesEquilibrium 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: jonathan.jordan@sheffield.ac.uk
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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 graphs82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 3 , September 2004, pp. 824 - 838
Copyright
Copyright © Applied Probability Trust 2004 

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