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1 results

  • 10 - Markov Chains

  • Luca Leuzzi, National Research Council of Italy, Enzo Marinari, Sapienza Università di Roma, Giorgio Parisi, Sapienza Università di Roma
  • Book:
    Probability Theory for Quantitative Scientists
    Published online:
    24 July 2025
    Print publication:
    14 August 2025, pp 249-288

  • Summary

    As we realize that random walks, chain reactions, and recurrent events are all Markov chains, i.e., correlated processes without memory, in this chapter we derive a general theory, including classification and properties of the single states and of chains. In particular, we focus on the building blocks of the theory, i.e., irreducible chains, presenting and proving a number of fundamental and useful theorems. We end up deriving the balance equation for the limit probability and the approach to the limit for long times, developing and applying the Perron–Frobenius theory for non-negative matrices and the spectral decomposition for non-Hermitian matrices. Among the applications of the theory, we underline the sorting of Web pages by search engines.