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  3. Two-Dimensional Random Walk
  • Cited by 12
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    • Publisher:
      Cambridge University Press
      Publication date:
      December 2020
      March 2021
      ISBN:
      9781108680134
      9781108472456
      9781108459693
      DOI:
      https://doi.org/10.1017/9781108680134
      Dimensions:
      (229 x 152 mm)
      Weight & Pages:
      0.47kg, 224 Pages
      Dimensions:
      (229 x 152 mm)
      Weight & Pages:
      0.327kg, 224 Pages
      • Subjects:
      Mathematics, Abstract Analysis, Probability Theory and Stochastic Processes, General Statistics and Probability, Statistics and Probability
      Series:
      Institute of Mathematical Statistics Textbooks
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    • Selected: Digital
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    Subjects:
    Mathematics, Abstract Analysis, Probability Theory and Stochastic Processes, General Statistics and Probability, Statistics and Probability
    Series:
    Institute of Mathematical Statistics Textbooks
    Information

    Book description

    The main subject of this introductory book is simple random walk on the integer lattice, with special attention to the two-dimensional case. This fascinating mathematical object is the point of departure for an intuitive and richly illustrated tour of related topics at the active edge of research. It starts with three different proofs of the recurrence of the two-dimensional walk, via direct combinatorial arguments, electrical networks, and Lyapunov functions. After reviewing some relevant potential-theoretic tools, the reader is guided toward the relatively new topic of random interlacements - which can be viewed as a 'canonical soup' of nearest-neighbour loops through infinity - again with emphasis on two dimensions. On the way, readers will visit conditioned simple random walks - which are the 'noodles' in the soup - and also discover how Poisson processes of infinite objects are constructed and review the recently introduced method of soft local times. Each chapter ends with many exercises, making it suitable for courses and independent study.

    Reviews

    ‘An excellent and inspiring introduction to simple random walk and random interlacements, in transient and recurrent cases. With its careful and original selection of topics, the reader will soon grasp the general picture and main ideas though to quite advanced material. Each chapter has a great selection of exercises with hints and solutions. This book is primarily designed for self-study, but it can also be used for a graduate course on Markov chains or Poisson processes.'

    Francis Comets - Université de Paris

    ‘… a well-written summary of the subject … Highly recommended.’

    M. Bona Source: Choice Connect

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