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  • © 2013

Coarse Geometry and Randomness

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

Authors:

  • Includes many exercises of varying difficulty levels

  • Investigates many open problems

  • Presents topics not covered by any other book

  • Includes supplementary material: sn.pub/extras

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

Part of the book sub series: École d'Été de Probabilités de Saint-Flour (LNMECOLE)

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-vii
  2. Introductory Graph and Metric Notions

    • Itai Benjamini
    Pages 1-18
  3. The Hyperbolic Plane and Hyperbolic Graphs

    • Itai Benjamini
    Pages 23-31
  4. Percolation on Graphs

    • Itai Benjamini
    Pages 33-40
  5. Local Limits of Graphs

    • Itai Benjamini
    Pages 41-51
  6. Random Planar Geometry

    • Itai Benjamini
    Pages 53-58
  7. Critical Percolation on Non-Amenable Groups

    • Itai Benjamini
    Pages 63-68
  8. Percolation Perturbations

    • Itai Benjamini
    Pages 85-95
  9. Percolation on Expanders

    • Itai Benjamini
    Pages 97-105
  10. Harmonic Functions on Graphs

    • Itai Benjamini
    Pages 107-120
  11. Nonamenable Liouville Graphs

    • Itai Benjamini
    Pages 121-124
  12. Back Matter

    Pages 125-132

About this book

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

  • Dept. of Mathematics, The Weizmann Institute of Science, Rehovot, Israel

    Itai Benjamini

Bibliographic Information

Buying options

eBook USD 39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 49.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions