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Condensation in Nongeneric Trees

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Abstract

We study nongeneric planar trees and prove the existence of a Gibbs measure on infinite trees obtained as a weak limit of the finite volume measures. It is shown that in the infinite volume limit there arises exactly one vertex of infinite degree and the rest of the tree is distributed like a subcritical Galton-Watson tree with mean offspring probability m<1. We calculate the rate of divergence of the degree of the highest order vertex of finite trees in the thermodynamic limit and show it goes like (1−m)N where N is the size of the tree. These trees have infinite spectral dimension with probability one but the spectral dimension calculated from the ensemble average of the generating function for return probabilities is given by 2β−2 if the weight w n of a vertex of degree n is asymptotic to n −β.

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Authors and Affiliations

  1. The Science Institute, University of Iceland, Dunhaga 3, 107, Reykjavik, Iceland

    Thordur Jonsson & Sigurdur Örn Stefánsson

  2. NORDITA, Roslagstullsbacken 23, 106 91, Stockholm, Sweden

    Sigurdur Örn Stefánsson

Authors
  1. Thordur Jonsson
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  2. Sigurdur Örn Stefánsson
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Correspondence to Thordur Jonsson.

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Jonsson, T., Stefánsson, S.Ö. Condensation in Nongeneric Trees. J Stat Phys 142, 277–313 (2011). https://doi.org/10.1007/s10955-010-0104-8

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  • DOI: https://doi.org/10.1007/s10955-010-0104-8

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