Percolation Theory for Mathematicians

1. Front Matter

   Pages i-viii

   PDF

2. Introduction and Summary

   • Harry Kesten

   Pages 1-9

3. Which Graphs Do We Consider?

   • Harry Kesten

   Pages 10-39

4. Periodic Percolation Problems

   • Harry Kesten

   Pages 40-68

5. Increasing Events

   • Harry Kesten

   Pages 69-80

6. Bounds for the Distribution of # W

   • Harry Kesten

   Pages 81-125

7. The Russo-Seymour-Welsh Theorem

   • Harry Kesten

   Pages 126-167

8. Proofs of Theorems 3.1 and 3.2

   • Harry Kesten

   Pages 168-197

9. Power Estimates

   • Harry Kesten

   Pages 198-237

10. The Nature of the Singularity at pH

    • Harry Kesten

    Pages 238-254

11. Inequalities for Critical Probabilities

    • Harry Kesten

    Pages 255-334

12. Resistance of Random Electrical Networks

    • Harry Kesten

    Pages 335-379

13. Unsolved Problems

    • Harry Kesten

    Pages 380-385

14. Back Matter

    Pages 386-423

    PDF

Quite apart from the fact that percolation theory had its orlgln in an honest applied problem (see Hammersley and Welsh (1980)), it is a source of fascinating problems of the best kind a mathematician can wish for: 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 problems are of interest to or proposed by statistical physicists and not dreamt up merely to demons~te ingenuity. Progress in the field has been slow. Relatively few results have been established rigorously, despite the rapidly growing literature with variations and extensions of the basic model, conjectures, plausibility arguments and results of simulations. It is my aim to treat here some basic results with rigorous proofs. This is in the first place a research monograph, but there are few prerequisites; one term of any standard graduate course in probability should be more than enough. Much of the material is quite recent or new, and many of the proofs are still clumsy. Especially the attempt to give proofs valid for as many graphs as possible led to more complications than expected. I hope that the Applications and Examples provide justifi­ cation for going to this level of generality.

• Simula
• Statistica
• Variation
• field
• graphs
• minimum
• model
• probability
• proof
• simulation
• time

• Department of Mathematics, Cornell University, Ithaca, USA

  Harry Kesten

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