Progress in High-Dimensional Percolation and Random Graphs
Authors: Heydenreich, Markus, van der Hofstad, Remco
Free Preview- Contains over 90 exercises designed to enhance the reader’s understanding of the material
- Presents an introduction to percolation, as well as sources for textbooks that mainly focus on percolation
- Gives a self-contained proof of mean-field behavior for high-dimensional percolation
- Discusses recent extensions and additions to classical results
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- About this Textbook
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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. - About the authors
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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.
- Reviews
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“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)
- Table of contents (16 chapters)
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Introduction and Motivation
Pages 3-18
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Fixing Ideas: Percolation on a Tree and Branching Random Walk
Pages 19-29
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Uniqueness of the Phase Transition
Pages 31-44
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Critical Exponents and the Triangle Condition
Pages 47-54
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Proof of Triangle Condition: The Infrared Bound
Pages 55-64
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Table of contents (16 chapters)
- Download Preface 1 PDF (57 KB)
- Download Sample pages 2 PDF (172.9 KB)
- Download Table of contents PDF (94.1 KB)
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Bibliographic Information
- Bibliographic Information
-
- Book Title
- Progress in High-Dimensional Percolation and Random Graphs
- Authors
-
- Markus Heydenreich
- Remco van der Hofstad
- Series Title
- CRM Short Courses
- Copyright
- 2017
- Publisher
- Springer International Publishing
- Copyright Holder
- Springer International Publishing Switzerland
- eBook ISBN
- 978-3-319-62473-0
- DOI
- 10.1007/978-3-319-62473-0
- Hardcover ISBN
- 978-3-319-62472-3
- Softcover ISBN
- 978-3-319-87321-3
- Series ISSN
- 2522-5200
- Edition Number
- 1
- Number of Pages
- XII, 285
- Number of Illustrations
- 9 b/w illustrations, 1 illustrations in colour
- Topics
*immediately available upon purchase as print book shipments may be delayed due to the COVID-19 crisis. ebook access is temporary and does not include ownership of the ebook. Only valid for books with an ebook version. Springer Reference Works and instructor copies are not included.
