Progress in High-Dimensional Percolation and Random Graphs

• Main Authors: Heydenreich, Markus (Author), van der Hofstad, Remco (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic eBook
• Language: English
• Published: Cham : Springer International Publishing : Imprint: Springer, 2017.
• Series: CRM Short Courses,
• Subjects: Mathematics.; Probabilities.; Statistics.; Probability Theory and Stochastic Processes.; Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences.

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 stude...

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.