Location of Repository

Random walk on the range of random walk

By David A. Croydon


We study the random walk X on the range of a simple random walk on ℤ d in dimensions d≥4. When d≥5 we establish quenched and annealed scaling limits for the process X, which show that the intersections of the original simple random walk path are essentially unimportant. For d=4 our results are less precise, but we are able to show that any scaling limit for X will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions. Furthermore, we demonstrate that when d=4 similar logarithmic corrections are necessary in describing the asymptotic behavior of the return probability of X to the origin

Topics: QA
Publisher: Springer New York LLC
OAI identifier: oai:wrap.warwick.ac.uk:2264

Suggested articles



  1. (1980). A self-avoiding random walk, doi
  2. (1994). afer, Self-avoiding walks in four dimensions: logarithmic corrections, doi
  3. (1989). An invariance principle for reversible Markov processes. Applications to random motions in random environments, doi
  4. (2005). Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs, doi
  5. (2003). Cut points and di®usive random walks in random environment, doi
  6. (1976). Field theoretical approach to critical phenomena, Phase transitions and critical phenomena, doi
  7. (1962). First passage times and sojourn times for Brownian motion in space and the exact Hausdor® measure of the sample path, doi
  8. (2007). Functional CLT for random walk among bounded random conductances, doi
  9. (1980). Gennes, Scaling concepts in polymer physics, doi
  10. (2009). Hausdor® measure of arcs and Brownian motion on Brownian spatial trees, doi
  11. (2004). Heat kernel estimates and parabolic Harnack inequalities on graphs and resistance forms, doi
  12. (2008). Heat kernel estimates for strongly recurrent random walk on random media, doi
  13. Heat kernel for random walk trace on Z3 and Z4,
  14. (1991). Intersections of random walks, Probability and its Applications, BirkhÄ auser Boston Inc.,
  15. (1999). Loop-erased random walk, Perplexing problems in probability, doi
  16. (1986). Polymer chains in four dimensions, doi
  17. Probability on trees and networks, doi
  18. (1984). Random walks and electric networks, doi
  19. (1983). Resistance of random walks, doi
  20. (1960). Some intersection properties of random walk paths, doi
  21. (1960). Some problems concerning the structure of random walk paths, doi
  22. (1984). Structure of clusters generated by random walks, doi
  23. (1990). The continuum random tree. II. An overview, Stochastic analysis doi
  24. (2006). The Einstein relation for random walks on graphs, doi
  25. (2006). The lace expansion and its applications, doi
  26. (1993). The logarithmic correction for loop-erased walk in four dimensions,
  27. (2003). The speed of biased random walk on percolation clusters, doi
  28. (1996). times for simple random walk, doi

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.