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1、ProbabilityonGraphsRandomProcessesonGraphsandLatticesGEOFFREYGRIMMETTStatisticalLaboratoryUniversityofCambridgecG.R.Grimmett1/4/10,17/11/10,23/5/12,5/7/12GeoffreyGrimmettStatisticalLaboratoryCentreforMathematicalSciencesUniversityofCambridgeWilberforceRoadCam
2、bridgeCB30WBUnitedKingdom2000MSC:(Primary)60K35,82B20,(Secondary)05C80,82B43,82C22With44FigurescG.R.Grimmett1/4/10,17/11/10,23/5/12,5/7/12ContentsPrefaceix1Randomwalksongraphs11.1RandomwalksandreversibleMarkovchains11.2Electricalnetworks31.3Flowsandenergy81.4
3、Recurrenceandresistance111.5Polya'stheoremÂ141.6Graphtheory161.7Exercises182Uniformspanningtree212.1De®nition212.2Wilson'salgorithm232.3Weaklimitsonlattices282.4Uniformforest312.5Schramm±LownerevolutionsÈ322.6Exercises373Percolationandself-avoidingwalk393.1Pe
4、rcolationandphasetransition393.2Self-avoidingwalks423.3Coupledpercolation453.4Orientedpercolation453.5Exercises484Associationandin¯uence504.1Holleyinequality504.2FKGinequality534.3BKinequality544.4Hoeffdinginequality56cG.R.Grimmett1/4/10,17/11/10,23/5/12,5/7/
5、12viContents4.5In¯uenceforproductmeasures584.6Proofsofin¯uencetheorems634.7Russo'sformulaandsharpthresholds754.8Exercises785Furtherpercolation815.1Subcriticalphase815.2Supercriticalphase865.3Uniquenessofthein®nitecluster925.4Phasetransition955.5Openpathsinann
6、uli995.6Thecriticalprobabilityintwodimensions1035.7Cardy'sformula1105.8Thecriticalprobabilityviathesharp-thresholdtheorem1215.9Exercises1256Contactprocess1276.1Stochasticepidemics1276.2Couplingandduality1286.3Invariantmeasuresandpercolation1316.4Thecriticalva
7、lue1336.5Thecontactmodelonatree1356.6Space±timepercolation1386.7Exercises1417Gibbsstates1427.1Dependencygraphs1427.2Markov®eldsandGibbsstates1447.3IsingandPottsmodels1487.4Exercises1508Random-clustermodel1528.1Therandom-clusterandIsing/Pottsmodels1528.2Basicp
8、roperties1558.3In®nite-volumelimitsandphasetransition1568.4Openproblems1608.5Intwodimensions1638.6Randomevengraphs1688.7Exercises171cG.R.Grimmett1/4/10,17/11/10,23/5/12,5/7/12Contentsvii9
