Geodesic
On the spectral dimension of random trees
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities (2006).

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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 so-called uniform random tree. We show that generic random trees have spectral dimension $d_s=4/3$. 
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     author = {Durhuus, Bergfinnur and Jonsson, Thordur and Wheater, John},
     title = {On the spectral dimension of random trees},
     journal = {Discrete mathematics & theoretical computer science},
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     volume = {DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities},
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Durhuus, Bergfinnur; Jonsson, Thordur; Wheater, John. On the spectral dimension of random trees. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities (2006). doi : 10.46298/dmtcs.3507. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3507/

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