Coarse Geometry and Randomness

École d’Été de Probabilités de Saint-Flour XLI – 2011

  • Itai Benjamini

Part of the Lecture Notes in Mathematics book series (LNM, volume 2100)

Also part of the École d'Été de Probabilités de Saint-Flour book sub series (LNMECOLE, volume 2100)

Table of contents

  1. Front Matter
    Pages i-vii
  2. Itai Benjamini
    Pages 1-18
  3. Itai Benjamini
    Pages 23-31
  4. Itai Benjamini
    Pages 33-40
  5. Itai Benjamini
    Pages 41-51
  6. Itai Benjamini
    Pages 53-58
  7. Itai Benjamini
    Pages 63-68
  8. Itai Benjamini
    Pages 85-95
  9. Itai Benjamini
    Pages 97-105
  10. Itai Benjamini
    Pages 107-120
  11. Itai Benjamini
    Pages 121-124
  12. Back Matter
    Pages 125-132

About this book

Introduction

These lecture notes study the interplay between randomness and geometry of graphs. The first part of the notes reviews several basic geometric concepts, before moving on to examine the manifestation of the underlying geometry in the behavior of random processes, mostly percolation and random walk.

The study of the geometry of infinite vertex transitive graphs, and of Cayley graphs in particular, is fairly well developed. One goal of these notes is to point to some random metric spaces modeled by graphs that turn out to be somewhat exotic, that is, they admit a combination of properties not encountered in the vertex transitive world. These include percolation clusters on vertex transitive graphs, critical clusters, local and scaling limits of graphs, long range percolation, CCCP graphs obtained by contracting percolation clusters on graphs, and stationary random graphs, including the uniform infinite planar triangulation (UIPT) and the stochastic hyperbolic planar quadrangulation (SHIQ).

Keywords

82B43,82B41,05C81,05C10,05C80 Coarse geometry Graphs Percolation Random walk Unimodular random graph and sparse graph limits

Authors and affiliations

  • Itai Benjamini
    • 1
  1. 1.Dept. of MathematicsThe Weizmann Institute of ScienceRehovotIsrael

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-02576-6
  • Copyright Information Springer International Publishing Switzerland 2013
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-02575-9
  • Online ISBN 978-3-319-02576-6
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • About this book