Progress in High-Dimensional Percolation and Random Graphs

1. Front Matter

   Pages i-xii

   PDF

2. Introduction to Percolation

   1. Front Matter

      Pages 1-1

      PDF

   2. Introduction and Motivation

      • Markus Heydenreich, Remco van der Hofstad

      Pages 3-18

   3. Fixing Ideas: Percolation on a Tree and Branching Random Walk

      • Markus Heydenreich, Remco van der Hofstad

      Pages 19-29

   4. Uniqueness of the Phase Transition

      • 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

      PDF

   2. Critical Exponents and the Triangle Condition

      • Markus Heydenreich, Remco van der Hofstad

      Pages 47-54

   3. Proof of Triangle Condition: The Infrared Bound

      • Markus Heydenreich, Remco van der Hofstad

      Pages 55-64

   4. The Derivation of the Lace Expansion via Inclusion–Exclusion

      • Markus Heydenreich, Remco van der Hofstad

      Pages 65-75

   5. Diagrammatic Estimates for the Lace Expansion

      • Markus Heydenreich, Remco van der Hofstad

      Pages 77-99

   6. Bootstrap Analysis of the Lace Expansion

      • Markus Heydenreich, Remco van der Hofstad

      Pages 101-124

   7. Proof that \(\delta =2\) and \(\beta =1\) under the Triangle Condition

      • Markus Heydenreich, Remco van der Hofstad

      Pages 125-137

4. Mean-Field Behavior: Recent Results

   1. Front Matter

      Pages 139-139

      PDF

   2. The Nonbacktracking Lace Expansion

      • Markus Heydenreich, Remco van der Hofstad

      Pages 141-154

   3. Further Critical Exponents

      • Markus Heydenreich, Remco van der Hofstad

      Pages 155-167

   4. Kesten’s Incipient Infinite Cluster

      • Markus Heydenreich, Remco van der Hofstad

      Pages 169-173

   5. Finite-Size Scaling and Random Graphs

      • Markus Heydenreich, Remco van der Hofstad

      Pages 175-218

5. Related and Open Problems

   1. Front Matter

      Pages 219-219

      PDF

   2. Random Walks on Percolation Clusters

      • Markus Heydenreich, Remco van der Hofstad

      Pages 221-230

   3. Related Results

      • Markus Heydenreich, Remco van der Hofstad

      Pages 231-255

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-dime

nsional 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.

• 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

“Text appeals to a wide audience, be it postgraduate students, researchers aspiring to work in the field, or experts in percolation. It is written in a very accessible style, comprising many excellent exercises, and may be used as a basis for postgraduate courses on various aspects of percolation. … it is a must read for anybody seriously interested in percolation.” (Christian Mönch, Mathematical Reviews, July, 2019)

• Mathematisches Institut, Ludwig-Maximilians-Universität München, Munich, Germany

  Markus Heydenreich

• Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands

  Remco van der Hofstad

Markus Heydenreich is a professor of Applied Mathematics at Ludwig-Maximilians-Universität München. Professor Heydenreich works in Probability theory, he investigates random spatial structures.

Remco van der Hofstad is a professor in Mathematics at Eindhoven University of Technology and scientific director of Eurandom. He received the Prix Henri Poincaré 2003 jointly with Gordon Slade and the Rollo Davidson Prize in 2007. He works on high-dimensional statistical physics,  random graphs as models for complex networks, and applications of probability to related fields such as electrical engineering, computer science and chemistry.

Policies and ethics