Overview

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.

Full Product Details

Author:   Markus Heydenreich ,  Remco van der Hofstad
Publisher:   Springer International Publishing AG
Imprint:   Springer International Publishing AG
Edition:   Softcover reprint of the original 1st ed. 2017
Dimensions:   Width: 15.50cm , Height: 1.60cm , Length: 23.50cm
Weight:   4.569kg
ISBN:  

9783319873213

ISBN 10:   3319873210
Pages:   285
Publication Date:   27 May 2018
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Manufactured on demand  
We will order this item for you from a manufactured on demand supplier.

Table of Contents

Preface.- 1. Introduction and motivation.- 2. Fixing ideas: Percolation on a tree and branching random walk.- 3. Uniqueness of the phase transition.- 4. Critical exponents and the triangle condition.- 5. Proof of triangle condition.- 6. The derivation of the lace expansion via inclusion-exclusion.- 7. Diagrammatic estimates for the lace expansion.- 8. Bootstrap analysis of the lace expansion.- 9. Proof that δ = 2 and β = 1 under the triangle condition.- 10. The non-backtracking lace expansion.- 11. Further critical exponents.- 12. Kesten's incipient infinite cluster.- 13. Finite-size scaling and random graphs.- 14. Random walks on percolation clusters.- 15. Related results.- 16. Further open problems.- Bibliography.

Reviews

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 Moench, Mathematical Reviews, July, 2019)

Author Information

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.

Tab Content 6

Author Website:  

Customer Reviews

Recent Reviews

No review item found!

Add your own review!

Countries Available

All regions