Journal of Statistical Physics

, Volume 147, Issue 5, pp 942–962 | Cite as

Spectral Dimension of Trees with a Unique Infinite Spine



Using generating functions techniques we develop a relation between the Hausdorff and spectral dimension of trees with a unique infinite spine. Furthermore, it is shown that if the outgrowths along the spine are independent and identically distributed, then both the Hausdorff and spectral dimension can easily be determined from the probability generating function of the random variable describing the size of the outgrowths at a given vertex, provided that the probability of the height of the outgrowths exceeding n falls off as the inverse of n. We apply this new method to both critical non-generic trees and the attachment and grafting model, which is a special case of the vertex splitting model, resulting in a simplified proof for the values of the Hausdorff and spectral dimension for the former and novel results for the latter.


Random trees Randomly grown trees Spectral dimension Random walk Hausdorff dimension Simply generated trees Non-generic trees Galton-Watson process Vertex splitting model 



The authors would like to thank Bergfinnur Durhuus, Georgios Giasemidis, Thordur Jonsson and John Wheater for discussions as well as the anonymous referee for useful comments which improved the manuscript. S.Ö.S. acknowledges hospitality at the University of Iceland. S.Z. acknowledges financial support of STFC grant ST/G000492/1. Furthermore, he would like to thank NORDITA for kind hospitality and financial support for a visit during which this work was initiated.


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.NORDITAStockholmSweden
  2. 2.Rudolf Peierls Centre for Theoretical PhysicsUniversity of OxfordOxfordUK

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