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The Spectral Dimension of Generic Trees

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Abstract

We define generic ensembles of infinite trees. These are limits as N→∞ of ensembles of finite trees of fixed size N, defined in terms of a set of branching weights. Among these ensembles are those supported on trees with vertices of a uniformly bounded order. The associated probability measures are supported on trees with a single spine and Hausdorff dimension d h =2. Our main result is that the spectral dimension of the ensemble average is d s =4/3, and that the critical exponent of the mass, defined as the exponential decay rate of the two-point function along the spine, is 1/3.

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Authors and Affiliations

  1. Matematisk Institut, Universitetsparken 5, 2100, Copenhagen, Ø, Denmark

    Bergfinnur Durhuus

  2. University of Iceland, Dunhaga 3, 107, Reykjavik, Iceland

    Thordur Jonsson

  3. Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX13 NP, UK

    John F. Wheater

Authors
  1. Bergfinnur Durhuus
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  2. Thordur Jonsson
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  3. John F. Wheater
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Correspondence to Thordur Jonsson.

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Durhuus, B., Jonsson, T. & Wheater, J.F. The Spectral Dimension of Generic Trees. J Stat Phys 128, 1237–1260 (2007). https://doi.org/10.1007/s10955-007-9348-3

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  • DOI: https://doi.org/10.1007/s10955-007-9348-3

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