Percolation

  • Geoffrey¬†Grimmett
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 321)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Geoffrey Grimmett
    Pages 1-31
  3. Geoffrey Grimmett
    Pages 32-52
  4. Geoffrey Grimmett
    Pages 53-76
  5. Geoffrey Grimmett
    Pages 77-86
  6. Geoffrey Grimmett
    Pages 87-116
  7. Geoffrey Grimmett
    Pages 117-145
  8. Geoffrey Grimmett
    Pages 146-196
  9. Geoffrey Grimmett
    Pages 197-231
  10. Geoffrey Grimmett
    Pages 232-253
  11. Geoffrey Grimmett
    Pages 254-280
  12. Geoffrey Grimmett
    Pages 281-348
  13. Geoffrey Grimmett
    Pages 349-377
  14. Geoffrey Grimmett
    Pages 378-396
  15. Back Matter
    Pages 397-447

About this book

Introduction

Percolation theory is the study of an idealized random medium in two or more dimensions. It is a cornerstone of the theory of spatial stochastic processes with applications in such fields as statistical physics, epidemiology, and the spread of populations. Percolation plays a pivotal role in studying more complex systems exhibiting phase transition. The mathematical theory is mature, but continues to give rise to problems of special beauty and difficulty. The emphasis of this book is upon core mathematical material and the presentation of the shortest and most accessible proofs. The book is intended for graduate students and researchers in probability and mathematical physics. Almost no specialist knowledge is assumed beyond undergraduate analysis and probability. This new volume differs substantially from the first edition through the inclusion of much new material, including: the rigorous theory of dynamic and static renormalization; a sketch of the lace expansion and mean field theory; the uniqueness of the infinite cluster; strict inequalities between critical probabilities; several essays on related fields and applications; numerous other results of significant. There is a summary of the hypotheses of conformal invariance. A principal feature of the process is the phase transition. The subcritical and supercritical phases are studied in detail. There is a guide for mathematicians to the physical theory of scaling and critical exponents, together with selected material describing the current state of the rigorous theory. To derive a rigorous theory of the phase transition remains an outstanding and beautiful problem of mathematics.

Keywords

Lattice Mathematica Percolation cubic lattice fields mathematical physics physics probability renormalization vertices

Authors and affiliations

  • Geoffrey¬†Grimmett
    • 1
  1. 1.Statistical LaboratoryUniversity of CambridgeCambridgeUK

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-662-03981-6
  • Copyright Information Springer-Verlag Berlin Heidelberg 1999
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-08442-3
  • Online ISBN 978-3-662-03981-6
  • Series Print ISSN 0072-7830
  • About this book