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Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree. (English) Zbl 1187.60083
It is shown that the Brownian motion on the continuum random tree is the scaling limit of the simple random walk on any family of discrete \(n\)-vertex ordered graph trees whose search-depth functions converge to the Brownian excursion as \(n\rightarrow \infty \). The author prove both a quenched version (for typical realisations of the trees) and an annealed version (averaged over all realisations of the trees) of this result. The assumptions made cover the case of simple random walk on the trees generated by the Galton-Watson branching process, conditioned on the total population size.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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