Markus Heydenreich is a professor of Applied Mathematics at Ludwig-Maximilians-Universität München. Professor Heydenreich works in Probability theory, he investigates random spatial structures. Remco van der Hofstad is a professor in Mathematics at Eindhoven University of Technology and scientific director of Eurandom. He received the Prix Henri Poincaré 2003 jointly with Gordon Slade and the Rollo Davidson Prize in 2007. He works on high-dimensional statistical physics,  random graphs as models for complex networks, and applications of probability to related fields such as electrical engineering, computer science and chemistry.

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• 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 δ = 2 and β = 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.

Markus Heydenreich is a professor of Applied Mathematics at Ludwig-Maximilians-Universität München. Professor Heydenreich works in Probability theory, he investigates random spatial structures. Remco van der Hofstad is a professor in Mathematics at Eindhoven University of Technology and scientific director of Eurandom. He received the Prix Henri Poincaré 2003 jointly with Gordon Slade and the Rollo Davidson Prize in 2007. He works on high-dimensional statistical physics,  random graphs as models for complex networks, and applications of probability to related fields such as electrical engineering, computer science and chemistry.

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• 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 δ = 2 and β = 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.

Markus Heydenreich is a professor of Applied Mathematics at Ludwig-Maximilians-Universität München. Professor Heydenreich works in Probability theory, he investigates random spatial structures. Remco van der Hofstad is a professor in Mathematics at Eindhoven University of Technology and scientific director of Eurandom. He received the Prix Henri Poincaré 2003 jointly with Gordon Slade and the Rollo Davidson Prize in 2007. He works on high-dimensional statistical physics,  random graphs as models for complex networks, and applications of probability to related fields such as electrical engineering, computer science and chemistry.

Select Guide Rating

• 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 δ = 2 and β = 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.