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Random walks on decorated Galton-Watson trees. (English. French summary) Zbl 07923683

Summary: In this article we study a simple random walk on a decorated Galton-Watson tree, obtained from a Galton-Watson tree by replacing each vertex of degree \(n\) with an independent copy of a graph \(G_n\) and gluing the inserted graphs along the tree structure. We assume that there exist constants \(d, R \geq 1\), \(v < \infty\) such that the diameter, effective resistance across and volume of \(G_n\) respectively grow like \(n^{\frac{1}{d}}\), \(n^{\frac{1}{R}}\), \(n^v\) as \(n\to\infty\). We also assume that the underlying Galton-Watson tree is critical with offspring tails \(\xi(x)\) decaying like \(cx^{-\alpha - 1}\) as \(x\to\infty\) for some constant \(c\) and some \(\alpha\in(1, 2)\). We establish the fractal dimension, spectral dimension, walk dimension and simple random walk displacement exponent for the resulting metric space as functions of \(\alpha\), \(d\), \(R\) and \(v\), along with bounds on the fluctuations of these quantities.

MSC:

60K37 Processes in random environments
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J35 Transition functions, generators and resolvents
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)

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