Progress in High-Dimensional Percolation and Random Graphs (Hardcover, 2017)

Preface.- 1. Introduction and motivation.- 2. Fixing ideas: Percolation on a tree and branching random walk.- 3. Uniqueness of the phase transition.- 4. Critical exponents and the triangle condition.- 5. Proof of triangle condition.- 6. The derivation of the lace expansion via inclusion-exclusion.- 7. Diagrammatic estimates for the lace expansion.- 8. Bootstrap analysis of the lace expansion.- 9. Proof that d = 2 and beta = 1 under the triangle condition.- 10. The non-backtracking lace expansion.- 11. Further critical exponents.- 12. Kesten's incipient infinite cluster.- 13. Finite-size scaling and random graphs.- 14. Random walks on percolation clusters.- 15. Related results.- 16. Further open problems.- Bibliography.