Overview
- 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
- Includes supplementary material: sn.pub/extras
Part of the book series: CRM Short Courses (CRMSC)
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Table of contents (16 chapters)
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Front Matter
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Introduction to Percolation
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Front Matter
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Mean-Field Behavior: Differential Inequalities and the Lace Expansion
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Front Matter
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Mean-Field Behavior: Recent Results
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Front Matter
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Related and Open Problems
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Front Matter
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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
About this book
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.Reviews
Authors and Affiliations
About the authors
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.
Bibliographic Information
Book Title: Progress in High-Dimensional Percolation and Random Graphs
Authors: Markus Heydenreich, Remco van der Hofstad
Series Title: CRM Short Courses
DOI: https://doi.org/10.1007/978-3-319-62473-0
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing Switzerland 2017
Hardcover ISBN: 978-3-319-62472-3Published: 04 December 2017
Softcover ISBN: 978-3-319-87321-3Published: 27 May 2018
eBook ISBN: 978-3-319-62473-0Published: 22 November 2017
Series ISSN: 2522-5200
Series E-ISSN: 2522-5219
Edition Number: 1
Number of Pages: XII, 285
Number of Illustrations: 9 b/w illustrations, 1 illustrations in colour
Topics: Probability Theory and Stochastic Processes, Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences