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Extremal paths on a random cayley tree

Majumdar1, Krapivsky

  • 1Laboratoire de Physique Quantique, CNRS UMR No. C5626, Universite Paul Sabatier, 31062 Toulouse Cedex, France and Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|January 4, 2001
PubMed
Summary
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This study analyzes shortest and longest paths in random Cayley trees. A critical transition is found for minimal path length, affecting its growth with tree height.

Area of Science:

  • Statistical physics
  • Probability theory
  • Network science

Background:

  • Investigates extremal path statistics in random Cayley trees.
  • Considers random edge lengths and branching factors.

Purpose of the Study:

  • Derive exact results for shortest and longest path lengths.
  • Analyze the behavior of minimal path length with varying parameters.

Main Methods:

  • Exact analytical derivations for arbitrary edge length distributions.
  • Focus on binary 0,1 distribution for detailed analysis.
  • Utilizes front selection mechanisms to determine velocities.

Main Results:

  • Identifies an unbinding transition for minimal path length at a critical branching probability.

Related Experiment Videos

  • Minimal path length saturates in the localized phase and grows linearly in the moving phase.
  • Maximal path length always grows linearly; a duality relation exists between minimal and maximal velocities.

    • Conclusions:

    • The study provides a comprehensive understanding of extremal path statistics in random trees.
    • The discovered transition and duality relations offer insights into random growth processes.