Progress in High-Dimensional Percolation and Random Graphs

Description

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. Front Matter ....Pages i-xii Front Matter ....Pages 1-1 Introduction and Motivation (Markus Heydenreich, Remco van der Hofstad)....Pages 3-18 Fixing Ideas: Percolation on a Tree and Branching Random Walk (Markus Heydenreich, Remco van der Hofstad)....Pages 19-29 Uniqueness of the Phase Transition (Markus Heydenreich, Remco van der Hofstad)....Pages 31-44 Front Matter ....Pages 45-45 Critical Exponents and the Triangle Condition (Markus Heydenreich, Remco van der Hofstad)....Pages 47-54 Proof of Triangle Condition: The Infrared Bound (Markus Heydenreich, Remco van der Hofstad)....Pages 55-64 The Derivation of the Lace Expansion via Inclusion–Exclusion (Markus Heydenreich, Remco van der Hofstad)....Pages 65-75 Diagrammatic Estimates for the Lace Expansion (Markus Heydenreich, Remco van der Hofstad)....Pages 77-99 Bootstrap Analysis of the Lace Expansion (Markus Heydenreich, Remco van der Hofstad)....Pages 101-124 Proof that \(\delta =2\) and \(\beta =1\) under the Triangle Condition (Markus Heydenreich, Remco van der Hofstad)....Pages 125-137 Front Matter ....Pages 139-139 The Nonbacktracking Lace Expansion (Markus Heydenreich, Remco van der Hofstad)....Pages 141-154 Further Critical Exponents (Markus Heydenreich, Remco van der Hofstad)....Pages 155-167 Kesten’s Incipient Infinite Cluster (Markus Heydenreich, Remco van der Hofstad)....Pages 169-173 Finite-Size Scaling and Random Graphs (Markus Heydenreich, Remco van der Hofstad)....Pages 175-218 Front Matter ....Pages 219-219 Random Walks on Percolation Clusters (Markus Heydenreich, Remco van der Hofstad)....Pages 221-230 Related Results (Markus Heydenreich, Remco van der Hofstad)....Pages 231-255 Further Open Problems (Markus Heydenreich, Remco van der Hofstad)....Pages 257-274 Back Matter ....Pages 275-285