Abstract
We state some unsolved problems and describe relevant examples concerning random walks on trees. Most of the problems involve the behavior of random walks with drift: e.g., is the speed on Galton-Watson trees monotonic in the drift parameter? These random walks have been used in Monte-Carlo algorithms for sampling from the vertices of a tree; in general, their behavior reflects the size and regularity of the underlying tree. Random walks are related to conductance. The distribution function for the conductance of Galton-Watson trees satisfies an interesting functional equation; is this distribution function absolutely continuous?
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Partially supported by an Alfred P. Sloan Foundation Research Fellowship (Pemantle), by NSF Grants DMS-9306954 (Lyons), DMS-9300191 (Pemantle), and DMS-9404391 (Peres), and by a Presidential Faculty Fellowship (Pemantle). The authors are grateful to the IMA at the University of Minnesota for its hospitality.
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Lyons, R., Pemantle, R., Peres, Y. (1997). Unsolved Problems Concerning Random Walks on Trees. In: Athreya, K.B., Jagers, P. (eds) Classical and Modern Branching Processes. The IMA Volumes in Mathematics and its Applications, vol 84. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1862-3_18
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DOI: https://doi.org/10.1007/978-1-4612-1862-3_18
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