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Progress in High-Dimensional Percolation and Random Graphs

  • Markus Heydenreich
  • Remco van der Hofstad

Part of the CRM Short Courses book series (CRMSC)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Introduction to Percolation

    1. Front Matter
      Pages 1-1
    2. Markus Heydenreich, Remco van der Hofstad
      Pages 3-18
    3. Markus Heydenreich, Remco van der Hofstad
      Pages 19-29
    4. Markus Heydenreich, Remco van der Hofstad
      Pages 31-44
  3. Mean-Field Behavior: Differential Inequalities and the Lace Expansion

    1. Front Matter
      Pages 45-45
    2. Markus Heydenreich, Remco van der Hofstad
      Pages 47-54
    3. Markus Heydenreich, Remco van der Hofstad
      Pages 55-64
    4. Markus Heydenreich, Remco van der Hofstad
      Pages 65-75
    5. Markus Heydenreich, Remco van der Hofstad
      Pages 77-99
    6. Markus Heydenreich, Remco van der Hofstad
      Pages 101-124
    7. Markus Heydenreich, Remco van der Hofstad
      Pages 125-137
  4. Mean-Field Behavior: Recent Results

    1. Front Matter
      Pages 139-139
    2. Markus Heydenreich, Remco van der Hofstad
      Pages 141-154
    3. Markus Heydenreich, Remco van der Hofstad
      Pages 155-167
    4. Markus Heydenreich, Remco van der Hofstad
      Pages 169-173
    5. Markus Heydenreich, Remco van der Hofstad
      Pages 175-218
  5. Related and Open Problems

    1. Front Matter
      Pages 219-219
    2. Markus Heydenreich, Remco van der Hofstad
      Pages 221-230
    3. Markus Heydenreich, Remco van der Hofstad
      Pages 231-255
    4. Markus Heydenreich, Remco van der Hofstad
      Pages 257-274
  6. Back Matter
    Pages 275-285

About this book

Introduction

This text presents an engaging exposition of the active field of high-dimensional percolation that will likely provide an impetus for future work. With over 90 exercises designed to enhance the reader’s understanding of the material, as well as many open problems, the book is aimed at graduate students and researchers who wish to enter the world of this rich topic.  The text may also be useful in advanced courses and seminars, as well as for reference and individual study.

Part I, consisting of 3 chapters, presents a general introduction to percolation, stating the main results, defining the central objects, and proving its main properties. No prior knowledge of percolation is assumed. Part II, consisting of Chapters 4–9, discusses mean-field critical behavior by describing the two main techniques used, namely, differential inequalities and the lace expansion. In Parts I and II, all results are proved, making this the first self-contained text discussing high-dimensiona

l percolation.  Part III, consisting of Chapters 10–13, describes recent progress in high-dimensional percolation.   Partial proofs and substantial overviews of how the proofs are obtained are given. In many of these results, the lace expansion and differential inequalities or their discrete analogues are central. Part IV, consisting of Chapters 14–16, features related models and further open problems, with a focus on the big picture.

Keywords

percolation phase transition critical behavior percolation on a tree percolation clusters branching random walk random graphs lace expansion scaling limit finite-size scaling superprocesses mean-field behavior Kesten's incipient infinite cluster critical exponents lace expansion differential inequalities finite size scaling random graphs super-processes

Authors and affiliations

  • Markus Heydenreich
    • 1
  • Remco van der Hofstad
    • 2
  1. 1.Mathematisches InstitutLudwig-Maximilians-Universität MünchenMunichGermany
  2. 2.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-62473-0
  • Copyright Information Springer International Publishing Switzerland 2017
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-62472-3
  • Online ISBN 978-3-319-62473-0
  • Series Print ISSN 2522-5200
  • Series Online ISSN 2522-5219
  • Buy this book on publisher's site
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