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An Introduction to Random Interlacements

  • Alexander Drewitz
  • Balázs Ráth
  • Artëm Sapozhnikov

Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Table of contents

  1. Front Matter
    Pages i-x
  2. Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov
    Pages 1-9
  3. Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov
    Pages 11-18
  4. Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov
    Pages 19-29
  5. Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov
    Pages 31-35
  6. Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov
    Pages 37-50
  7. Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov
    Pages 51-60
  8. Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov
    Pages 61-73
  9. Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov
    Pages 75-86
  10. Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov
    Pages 87-95
  11. Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov
    Pages 97-113
  12. Back Matter
    Pages 115-120

About this book

Introduction

This book gives a self-contained introduction to the theory of random interlacements. The intended reader of the book is a graduate student with a background in probability theory who wants to learn about the fundamental results and methods of this rapidly emerging field of research. The model was introduced by Sznitman in 2007 in order to describe the local picture left by the trace of a random walk on a large discrete torus when it runs up to times proportional to the volume of the torus. Random interlacements is a new percolation model on the d-dimensional lattice. The main results covered by the book include the full proof of the local convergence of random walk trace on the torus to random interlacements and the full proof of the percolation phase transition of the vacant set of random interlacements in all dimensions. The reader will become familiar with the techniques relevant to working with the underlying Poisson Process and the method of multi-scale renormalization, which helps in overcoming the challenges posed by the long-range correlations present in the model. The aim is to engage the reader in the world of random interlacements by means of detailed explanations, exercises and heuristics. Each chapter ends with short survey of related results with up-to date pointers to the literature.

Keywords

Percolation Potential theory Random interlacements Random walk Random walk on torus Renormalization

Authors and affiliations

  • Alexander Drewitz
    • 1
  • Balázs Ráth
    • 2
  • Artëm Sapozhnikov
    • 3
  1. 1.Department of MathematicsColumbia UniversityNew York CityUSA
  2. 2.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  3. 3.Max-Planck Institute of Mathematics in the SciencesLeipzigGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-05852-8
  • Copyright Information The Author(s) 2014
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-05851-1
  • Online ISBN 978-3-319-05852-8
  • Series Print ISSN 2191-8198
  • Series Online ISSN 2191-8201
  • Buy this book on publisher's site
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