Unsolved Problems Concerning Random Walks on Trees

  • Russell Lyons
  • Robin Pemantle
  • Yuval Peres
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 84)


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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Berretti, A. and Sokal, A. D. (1985). New Monte Carlo method for the self-avoiding walk. J. Stat. Physics 40, 483–531.MathSciNetCrossRefGoogle Scholar
  2. Billingsley, P. (1965). Ergodic Theory and Information. Wiley, New York.zbMATHGoogle Scholar
  3. Feller, W. (1970). An Introduction to Probability Theory and Its Applications. Volume II, 2nd ed. Wiley, New York.Google Scholar
  4. Hawkes, J. (1981). Trees generated by a simple branching process, J. London Math. Soc. 24 373–384.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Kesten, H. (1986). Subdiffusive behavior of random walk on a random cluster, Ann. Inst. Henri Poincaré Probab. Statist. 22425–487.MathSciNetzbMATHGoogle Scholar
  6. Lawler, G. F. and Sokal, A. D. (1988). Bounds on the L2 spectrum for Markov chains and Markov processes: A generalization of Cheeger’s inequality. Trans. Amer. Math. Soc. 309 557–589.MathSciNetzbMATHGoogle Scholar
  7. Lyons, R. (1990). Random walks and percolation on trees, Ann. Probab. 18 931–958.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Lyons, R. (1992). Random walks, capacity, and percolation on trees, Ann. Probab. 20 2043–2088.MathSciNetzbMATHCrossRefGoogle Scholar
  9. Lyons, R. (1994). Equivalence of boundary measures on covering trees of finite graphs, Ergodic Theory Dynam. Systems 14,575–597.MathSciNetzbMATHCrossRefGoogle Scholar
  10. Lyons, R., Pemantle, R. and Peres, Y. (1995a). Ergodic theory on Galton-Watson trees: speed of random walk and dimension of har-monic measure. Ergodic Theory Dynamical Systems 15,593–619.MathSciNetCrossRefGoogle Scholar
  11. Lyons, R., PEMANTLE, R. and PERES, Y. (1995b). Biased random walks on Galton-Watson trees, will appear in Probab. Theory Related Fields. Google Scholar
  12. Pemantle, R. and Peres, Y. (1995). Galton-Watson trees with the same mean have the same polar sets, Ann.Probab. 23 1102–1124.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Peres, Y. (1992). Domains of analytic continuation for the top Lyapunov exponent, Ann. Inst. Henri Poincaré Probab. Statist. 28 131–148.MathSciNetzbMATHGoogle Scholar
  14. Randall, D. (1994). Counting in Lattices: Combinatorial Problems from Statistical Mechanics. Ph. D. thesis, University of California, Berkeley.Google Scholar
  15. Ruelle, D. (1979). Analyticity properties of the characteristic exponents of random matrix products, Adv. Math. 3268–80.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Sinclair, A. J. and Jerrum, M. R. (1989). Approximate counting, uniform generation and rapidly mixing Markov chains, Information and Computation 82,93–133.Google Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Russell Lyons
    • 1
  • Robin Pemantle
    • 2
  • Yuval Peres
    • 3
  1. 1.Department of MathematicsIndiana UniversityBloomington
  2. 2.Department of MathematicsUniversity of WisconsinMadison
  3. 3.Department of StatisticsUniversity of CaliforniaBerkeley

Personalised recommendations