Classical and Modern Branching Processes

Volume 84 of the series The IMA Volumes in Mathematics and its Applications pp 223-237

Unsolved Problems Concerning Random Walks on Trees

  • Russell LyonsAffiliated withDepartment of Mathematics, Indiana University
  • , Robin PemantleAffiliated withDepartment of Mathematics, University of Wisconsin
  • , Yuval PeresAffiliated withDepartment of Statistics, University of California

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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?

Key words

Galton-Watson random walk speed rate of escape