LEADER 04726nam 22006615 450 001 9910792491203321 005 20200702051557.0 010 $a3-662-03981-8 024 7 $a10.1007/978-3-662-03981-6 035 $a(CKB)2660000000027159 035 $a(SSID)ssj0001298045 035 $a(PQKBManifestationID)11739054 035 $a(PQKBTitleCode)TC0001298045 035 $a(PQKBWorkID)11242010 035 $a(PQKB)10140078 035 $a(DE-He213)978-3-662-03981-6 035 $a(MiAaPQ)EBC3098573 035 $a(PPN)237913690 035 $a(EXLCZ)992660000000027159 100 $a20130107d1999 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt 182 $cc 183 $acr 200 10$aPercolation$b[electronic resource] /$fby Geoffrey R. Grimmett 205 $a2nd ed. 1999. 210 1$aBerlin, Heidelberg :$cSpringer Berlin Heidelberg :$cImprint: Springer,$d1999. 215 $a1 online resource (XIII, 447 p.) 225 1 $aGrundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics,$x0072-7830 ;$v321 300 $aBibliographic Level Mode of Issuance: Monograph 311 $a3-540-64902-6 311 $a3-642-08442-7 320 $aIncludes bibliographical references and index. 327 $a1 What is Percolation? -- 2 Some Basic Techniques -- 3 Critical Probabilities -- 4 The Number of Open Clusters per Vertex -- 5 Exponential Decay -- 6 The Subcritical Phase -- 7 Dynamic and Static Renormalization -- 8 The Supercritical Phase -- 9 Near the Critical Point: Scaling Theory -- 10 Near the Critical Point: Rigorous Results -- 11 Bond Percolation in Two Dimensions -- 12 Extensions of Percolation -- 13 Percolative Systems -- Appendix I. The Infinite-Volume Limit for Percolation -- Appendix II. The Subadditive Inequality -- List of Notation -- References -- Index of Names. 330 $aPercolation theory is the study of an idealized random medium in two or more dimensions. It is a cornerstone of the theory of spatial stochastic processes with applications in such fields as statistical physics, epidemiology, and the spread of populations. Percolation plays a pivotal role in studying more complex systems exhibiting phase transition. The mathematical theory is mature, but continues to give rise to problems of special beauty and difficulty. The emphasis of this book is upon core mathematical material and the presentation of the shortest and most accessible proofs. The book is intended for graduate students and researchers in probability and mathematical physics. Almost no specialist knowledge is assumed beyond undergraduate analysis and probability. This new volume differs substantially from the first edition through the inclusion of much new material, including: the rigorous theory of dynamic and static renormalization; a sketch of the lace expansion and mean field theory; the uniqueness of the infinite cluster; strict inequalities between critical probabilities; several essays on related fields and applications; numerous other results of significant. There is a summary of the hypotheses of conformal invariance. A principal feature of the process is the phase transition. The subcritical and supercritical phases are studied in detail. There is a guide for mathematicians to the physical theory of scaling and critical exponents, together with selected material describing the current state of the rigorous theory. To derive a rigorous theory of the phase transition remains an outstanding and beautiful problem of mathematics. 410 0$aGrundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics,$x0072-7830 ;$v321 606 $aProbabilities 606 $aMathematical physics 606 $aCombinatorics 606 $aProbability Theory and Stochastic Processes$3https://scigraph.springernature.com/ontologies/product-market-codes/M27004 606 $aTheoretical, Mathematical and Computational Physics$3https://scigraph.springernature.com/ontologies/product-market-codes/P19005 606 $aCombinatorics$3https://scigraph.springernature.com/ontologies/product-market-codes/M29010 615 0$aProbabilities. 615 0$aMathematical physics. 615 0$aCombinatorics. 615 14$aProbability Theory and Stochastic Processes. 615 24$aTheoretical, Mathematical and Computational Physics. 615 24$aCombinatorics. 676 $a530.13 700 $aGrimmett$b Geoffrey R$4aut$4http://id.loc.gov/vocabulary/relators/aut$0265888 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910792491203321 996 $aPercolation$9735622 997 $aUNINA