Percolation

1. Front Matter

   Pages i-xi

   PDF

2. What Is Percolation?

   • Geoffrey Grimmett

   Pages 1-24

3. Some Basic Techniques

   • Geoffrey Grimmett

   Pages 25-43

4. The Uniqueness of the Critical Point

   • Geoffrey Grimmett

   Pages 44-71

5. The Number of Open Clusters per Vertex

   • Geoffrey Grimmett

   Pages 72-81

6. The Subcritical Phase

   • Geoffrey Grimmett

   Pages 82-108

7. The Supercritical Phase

   • Geoffrey Grimmett

   Pages 109-147

8. Near the Critical Point: Scaling Theory

   • Geoffrey Grimmett

   Pages 148-168

9. Near the Critical Point: Rigorous Results

   • Geoffrey Grimmett

   Pages 169-185

10. Bond Percolation in Two Dimensions

    • Geoffrey Grimmett

    Pages 186-235

11. A Miscellany of Random Processes

    • Geoffrey Grimmett

    Pages 236-265

12. Back Matter

    Pages 266-296

    PDF

Quite apart from the fact that percolation theory had its ongm in an honest applied problem, it is a source of fascinating problems of the best kind for which a mathematician can wish: problems which are easy to state with a minimum of preparation, but whose solutions are apparently difficult and require new methods. At the same time, many of the prob­ lems are of interest to or proposed by statistical physicists and not dreamed up merely to demonstrate ingenuity. Much progress has been made in recent years, and many of the open problems of ten years aga have been solved. With such solutions we have seen the evolution of new techniques and questions; the consequent knowledge has shifted the ground under percolation, and it is time to examine afresh the mathematics of the subject. The quantity of literature related to percolation seems to grow hour by hour, mostly in the physics journals. It is becoming increasingly diffi­ cult to get to know the subject from scratch, and one of the principal purposes of this book is to remedy this. This book is about the mathematics of percolation theory, with the emphasis upon presenting the shortest rigorous proofs of the main facts.

• Mathematica
• cluster
• mathematics
• phase
• physics
• probability
• renormalization

• School of Mathematics, University of Bristol, Bristol, England

  Geoffrey Grimmett

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