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

Published online by Cambridge University Press:  01 July 2016

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]

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

Type
General Applied Probability
Copyright
Copyright © Applied Probability Trust 2004 

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