Bergfinnur Durhuus ; Thordur Jonsson ; John Wheater

On the spectral dimension of random trees
dmtcs:3507 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2006,
DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities

https://doi.org/10.46298/dmtcs.3507
On the spectral dimension of random treesArticle
Authors: Bergfinnur Durhuus ^{1}; Thordur Jonsson ^{2}; John Wheater ^{3,}^{4}
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Bergfinnur Durhuus;Thordur Jonsson;John Wheater
1 Department of Mathematical Sciences [Copenhagen]
2 Science Institute
3 Rudolf Peierls Center for Theoretical Physics
4 Rudolf Peierls Centre for Theoretical Physics
We determine the spectral dimensions of a variety of ensembles of infinite trees. Common to the ensembles considered is that sample trees have a distinguished infinite spine at whose vertices branches can be attached according to some probability distribution. In particular, we consider a family of ensembles of $\textit{combs}$, whose branches are linear chains, with spectral dimensions varying continuously between $1$ and $3/2$. We also introduce a class of ensembles of infinite trees, called $\textit{generic random trees}$, which are obtained as limits of ensembles of finite trees conditioned to have fixed size $N$, as $N \to \infty$. Among these ensembles is the socalled uniform random tree. We show that generic random trees have spectral dimension $d_s=4/3$.