Progress in High-Dimensional Percolation and Random Graphs: CRM Short Courses
Autor Markus Heydenreich, Remco van der Hofstaden Limba Engleză Hardback – 4 dec 2017
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.
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| Springer International Publishing – 4 dec 2017 | 422.19 lei 38-44 zile |
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Specificații
ISBN-13: 9783319624723
ISBN-10: 3319624725
Pagini: 214
Ilustrații: XII, 285 p. 10 illus., 1 illus. in color.
Dimensiuni: 155 x 235 x 24 mm
Greutate: 0.63 kg
Ediția:1st ed. 2017
Editura: Springer International Publishing
Colecția Springer
Seria CRM Short Courses
Locul publicării:Cham, Switzerland
ISBN-10: 3319624725
Pagini: 214
Ilustrații: XII, 285 p. 10 illus., 1 illus. in color.
Dimensiuni: 155 x 235 x 24 mm
Greutate: 0.63 kg
Ediția:1st ed. 2017
Editura: Springer International Publishing
Colecția Springer
Seria CRM Short Courses
Locul publicării:Cham, Switzerland
Cuprins
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.
Recenzii
“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)
Notă biografică
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.
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.
Textul de pe ultima copertă
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.
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.
Caracteristici
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