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Random walks on decorated Galton-Watson trees

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Random trees are the building blocks for a range of probabilistic structures, including percolation clusters on the lattice and a range of statistical physics models on random planar maps. In this talk we consider a random walk on a critical “decorated” Galton-Watson tree, by which we mean that we first sample a critical Galton-Watson tree T, replace each vertex of degree n with an independent copy of a graph G(n), and then glue these inserted graphs along the tree structure of T. We will determine the random walk exponents for this decorated tree model in terms of the exponents for the underlying tree and the exponents for the inserted graphs. We will see that the model undergoes several phase transitions depending on how these exponents balance out.

This talk is part of the Probability series.

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