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The spectral dimension of generic trees

Bergfinnur Durhuus·Thordur Jonsson·John F. Wheater
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Abstract

We define generic ensembles of infinite trees. These are limits as $N\to\infty$ 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 their spectral dimension 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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Bergfinnur Durhuus, Thordur Jonsson, John F. Wheater. 2006-07-11. The spectral dimension of generic trees. https://doi.org/10.1007/s10955-007-9348-3

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