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Random recursive trees and the elephant random walk.

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Elephant random walks, models of anomalous diffusion, connect to bond percolation on random recursive trees. This study derives exact results for cluster sizes in these novel random walk and percolation models.

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Area of Science:

  • Physics
  • Mathematics
  • Computer Science

Background:

  • Elephant random walks are models for anomalous diffusion with infinite memory.
  • Bond percolation on random recursive trees is a graph theory problem.

Purpose of the Study:

  • To establish a connection between elephant random walks and bond percolation on random recursive trees.
  • To derive exact mathematical expressions for key quantities in both models.

Main Methods:

  • A coupling technique was employed to link elephant random walks with bond percolation.
  • Analytical calculations were performed to determine moments of cluster sizes.

Main Results:

  • Exact expressions for the first and second moments of root cluster size were derived.
  • Exact expressions for the number of nodes in child clusters were calculated.
  • The first and second moments of a new model, the skew elephant random walk, were determined.

Conclusions:

  • The study reveals a significant, previously unrecognized link between anomalous diffusion models and percolation theory.
  • The findings provide new analytical tools and exact results for understanding complex random processes on trees.