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1 results

  • 4 - Networks and Discrete Analysis

  • from Part I - Tools and Theory
  • Ariel Yadin, Ben-Gurion University of the Negev, Israel
  • Book:
    Harmonic Functions and Random Walks on Groups
    Published online:
    16 May 2024
    Print publication:
    23 May 2024, pp 117-158

  • Summary

    Here, we dive deeper into the realm of reversible Markov chains, via the perspective of network theory. The notions of conductance and resistance are defined, as well as voltage and current, and the corresponding mathematical theory.The Laplacian and Green function are defined and their relation to harmonic functions explained. The chapter culminates with a proof (using network theory) that recurrence and transience are essentially group properties: these properties remain invariant when changing between different reasonable random walks on the same group (specifically, symmetric and adapted with finite second moment).