Discrete Mathematics & Theoretical Computer Science (Jan 2006)

On the spectral dimension of random trees

  • Bergfinnur Durhuus,
  • Thordur Jonsson,
  • John Wheater

DOI
https://doi.org/10.46298/dmtcs.3507
Journal volume & issue
Vol. DMTCS Proceedings vol. AG,..., no. Proceedings

Abstract

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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 $d_s=4/3$.

Keywords