Progress in High-Dimensional Percolation and Random Graphs

Häftad, Engelska, 2018

Av Markus Heydenreich, Remco van der Hofstad, Remco Van Der Hofstad, Remco Van Der Hofstad

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The text may also be useful in advanced courses and seminars, as well as for reference and individual study. Part I, consisting of 3 chapters, presents a general introduction to percolation, stating the main results, defining the central objects, and proving its main properties.

Produktinformation

Utgivningsdatum2018-05-27
Mått155 x 235 x 17 mm
Vikt458 g
FormatHäftad
SpråkEngelska
SerieCRM Short Courses
Antal sidor285
FörlagSpringer International Publishing AG
ISBN9783319873213

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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.

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.

“Text appeals to a wide audience, be it postgraduate students, researchers aspiring to work in the field, or experts in percolation. It is written in a very accessible style, comprising many excellent exercises, and may be used as a basis for postgraduate courses on various aspects of percolation. … it is a must read for anybody seriously interested in percolation.” (Christian Mönch, Mathematical Reviews, July, 2019)