LEADER 04340nam 22005775 450 001 9910254297803321 005 20200629210259.0 010 $a3-319-62473-3 024 7 $a10.1007/978-3-319-62473-0 035 $a(CKB)4340000000223263 035 $a(DE-He213)978-3-319-62473-0 035 $a(MiAaPQ)EBC6313121 035 $a(MiAaPQ)EBC5578106 035 $a(Au-PeEL)EBL5578106 035 $a(OCoLC)1013823004 035 $z(PPN)258872179 035 $a(PPN)221254021 035 $a(EXLCZ)994340000000223263 100 $a20171122d2017 u| 0 101 0 $aeng 135 $aurnn#008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aProgress in High-Dimensional Percolation and Random Graphs /$fby Markus Heydenreich, Remco van der Hofstad 205 $a1st ed. 2017. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2017. 215 $a1 online resource (XII, 285 p. 10 illus., 1 illus. in color.) 225 1 $aCRM Short Courses,$x2522-5200 311 $a3-319-62472-5 327 $aPreface -- 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 ? = 2 and ? = 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. 330 $aThis text presents an engaging exposition of the active field of high-dimensional percolation that will likely provide an impetus for future work. With over 90 exercises designed to enhance the reader?s understanding of the material, as well as many open problems, the book is aimed at graduate students and researchers who wish to enter the world of this rich topic. The text may also be useful in advanced courses and seminars, as well as for reference and individual study. Part I, consisting of 3 chapters, presents a general introduction to percolation, stating the main results, defining the central objects, and proving its main properties. No prior knowledge of percolation is assumed. Part II, consisting of Chapters 4?9, discusses mean-field critical behavior by describing the two main techniques used, namely, differential inequalities and the lace expansion. In Parts I and II, all results are proved, making this the first self-contained text discussing high-dimensiona l percolation. Part III, consisting of Chapters 10?13, describes recent progress in high-dimensional percolation. Partial proofs and substantial overviews of how the proofs are obtained are given. In many of these results, the lace expansion and differential inequalities or their discrete analogues are central. Part IV, consisting of Chapters 14?16, features related models and further open problems, with a focus on the big picture. 410 0$aCRM Short Courses,$x2522-5200 606 $aProbabilities 606 $aStatistics 606 $aProbability Theory and Stochastic Processes$3https://scigraph.springernature.com/ontologies/product-market-codes/M27004 606 $aStatistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences$3https://scigraph.springernature.com/ontologies/product-market-codes/S17020 615 0$aProbabilities. 615 0$aStatistics. 615 14$aProbability Theory and Stochastic Processes. 615 24$aStatistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences. 676 $a530.13 700 $aHeydenreich$b Markus$4aut$4http://id.loc.gov/vocabulary/relators/aut$0767507 702 $avan der Hofstad$b Remco$4aut$4http://id.loc.gov/vocabulary/relators/aut 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910254297803321 996 $aProgress in High-Dimensional Percolation and Random Graphs$92039120 997 $aUNINA