Record process on the Continuum Random Tree

Details Record process on the Continuum Random Tree Record process on the Continuum Random Tree

2011-07-19

arxiv.org/abs/1107.3657v1

Mathematics

Probability

Scientific paper

We consider the number of cuts $X_n^*$ needed to isolate the root of the sub-tree spanned by $n$ leaves uniformly chosen at random in Aldous's continuum random tree $\ct$. We prove the almost sure convergence of $X_n^*/\sqrt{2 n}$ to a Rayleigh random variable $Z$. We get from the a.s. convergence a representation of $Z$ as the integral on the leaves of $\ct$ of a record process indexed by the tree $\ct$. The proof relies on a Brownian Snake approach. This result was motivated by Janson's convergence in distribution of the renormalized number of cuts in a discrete random tree.

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