Progress in High-Dimensional Percolation and Random Graphs / by Markus Heydenreich, Remco van der Hofstad.

Format:

Book

Author/Creator:

Heydenreich, Markus., Author.

van der Hofstad, Remco., Author.

Series:

CRM Short Courses, 2522-5200

Language:

English

Subjects (All):

Probabilities.

Statistics.

Probability Theory and Stochastic Processes.

Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences.

Local Subjects:

Probability Theory and Stochastic Processes.

Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences.

Physical Description:

1 online resource (XII, 285 p. 10 illus., 1 illus. in color.)

Edition:

1st ed. 2017.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2017.

Summary:

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.

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.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

3-319-62473-3

OCLC:

1013823004