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Favorite Sites of a Persistent Random Walk.

Arka Ghosh1, Steven Noren2, Alexander Roitershtein3

  • 1Department of Statistics, Iowa State University, Ames, IA.

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Summary
This summary is machine-generated.

The study examines favorite sites of a symmetric persistent random walk. Researchers found the number of most visited sites is eventually at most three, generalizing prior work on simple random walks.

Keywords:
60G5060J1060J55correlated random walksdiscrete Ray-Knight theoremsfavorite siteslocal timemost visited sites

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Area of Science:

  • Probability theory
  • Stochastic processes
  • Statistical mechanics

Background:

  • Random walks are fundamental stochastic processes used to model various phenomena.
  • Persistent random walks exhibit memory, unlike simple random walks.
  • The study of favorite sites relates to long-term behavior and site recurrence.

Purpose of the Study:

  • To analyze the number of most visited sites for a symmetric persistent random walk on integers.
  • To generalize and extend previous results concerning favorite sites of random walks.
  • To contribute to the understanding of long-term properties of correlated random walks.

Main Methods:

  • Mathematical analysis of a symmetric persistent random walk model.
  • Derivation of bounds on the cardinality of the set of favorite sites.
  • Comparison with existing results for simple random walks.

Main Results:

  • The cardinality of the set of favorite sites for a symmetric persistent random walk is eventually at most three.
  • This result generalizes a known theorem for simple random walks.
  • The findings offer insights into the impact of directional history correlation on site visitation patterns.

Conclusions:

  • The number of frequently visited sites in a symmetric persistent random walk is bounded.
  • This work extends the understanding of random walk behavior in the presence of memory.
  • The study provides a foundation for further research into correlated random walks and their properties.