Large deviations and metastability

530 Pages·2005·2.484 MB·English

by  Enzo Olivieri, Maria Eulália Vares

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Largedeviationsandmetastability
The book provides a general introduction to the theory of large deviations and a wide
overviewofthemetastablebehaviourofstochasticdynamics.Withonlyminimalprereq-
uisites,thebookcoversallthemainresultsandbringsthereadertothemostrecentde-
velopments.ParticularemphasisisgiventothefundamentalFreidlin–Wentzellresultson
smallrandomperturbationsofdynamicalsystems.Metastabilityisfirstdescribedonphys-
ical grounds, following which more rigorous approaches are developed. Many relevant
examplesareconsideredfromthepointofviewoftheso-calledpathwiseapproach.The
firstpartofthebookdevelopstherelevanttools,includingthetheoryoflargedeviations,
whicharethenusedtoprovideaphysicallyrelevantdynamicaldescriptionofmetastabil-
ity.Writtentobeaccessibletograduatestudents,thisbookprovidesanexcellentrouteinto
contemporaryresearch.
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Large deviations and metastability
ENZO OLIVIERI
Universita` diRomaTorVergata
MARIA EULA´LIA VARES
CentroBrasileirodePesquisasF´ısicas,RiodeJaneiro
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Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
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Published in the United States of America by Cambridge University Press, New York
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First published in print format 2005
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ToAnnaandDaniela,Juliana,LucasandVladas

CONTENTS
Preface page xi
1 Largedeviations:basicresults 1
Introduction 1
1.1 Crame´r–ChernofftheoremonR 3
1.2 Abstractformulation 20
1.3 Sanovtheoremforfinite‘alphabets’ 28
1.4 Crame´rtheoremonRd 39
1.5 Crame´rtheoremininfinitedimensionalspaces 44
1.6 Empiricalmeasures.Sanovtheorem 56
2 Smallrandomperturbationsofdynamicalsystems.Basic
estimatesofFreidlinandWentzell 64
2.1 Brownianmotion 65
2.2 Itoˆ’sintegral 77
2.3 Itoˆ’sequations 87
2.4 BasicFreidlinandWentzellestimates 90
2.5 FreidlinandWentzellbasicestimates.Variablediffusion
coefficients 98
2.6 Exitfromadomain 104
3 Largedeviationsandstatisticalmechanics 118
3.1 Largedeviationsfordependentvariables.Ga¨rtner–Ellistheorem 118
3.2 LargedeviationsforMarkovchains 125
3.3 Abriefintroductiontoequilibriumstatisticalmechanics 134
3.4 LargedeviationsforGibbsmeasures 176
vii
viii Contents
4 Metastability.Generaldescription.Curie–Weissmodel.
Contactprocess 198
4.1 ThevanderWaals–Maxwelltheory 198
4.2 Thepathwiseapproach:basicdescription 225
4.3 TheCurie–WeissMarkovchain 226
4.4 TheHarriscontactprocess 242
4.5 Notesandcomments 277
5 Metastability.ModelsofFreidlinandWentzell 287
5.1 ThecouplingmethodofLindvallandRogers 289
5.2 Thecaseofadoublewellpotential 294
5.3 Generalcase.Asymptoticmagnitudeoftheescapetime 302
5.4 Cyclesandasymptoticunpredictabilityoftheescapetime 319
5.5 Notesandcomments 329
6 ReversibleMarkovchainsintheFreidlin–Wentzellregime 335
Introduction 337
6.1 Definitionsandnotation 338
6.2 Mainresults 348
6.3 Restricteddynamics 367
6.4 Conditionalergodicproperties 370
6.5 Escapefromtheboundaryofatransientcycle 372
6.6 Asymptoticexponentialityoftheexittime 375
6.7 Theexittube 378
6.8 Decompositionintomaximalcycles 385
6.9 Renormalization 388
6.10 Reductionandrecurrence 395
6.11 Asymptoticsinprobabilityoftunnellingtimes 397
7 Metastablebehaviourforlatticespinmodelsatlow
temperature 399
Introduction 399
7.1 ThestandardstochasticIsingmodelintwodimensions 401
7.2 Thelocalminima 406
7.3 Subcriticalandsupercriticalrectangles 410
7.4 Subcriticalconfigurationsandglobalsaddles 412
7.5 Alternativemethodtodetermine(cid:2)(−1,+1) 424
7.6 Extensionsandgeneralizations 427
7.7 TheanisotropicferromagneticIsingmodelintwodimensions 428
7.8 Alternativeapproachtostudyofthe J J model 433
1 2
7.9 TheferromagneticIsingmodelwithnearestandnextnearest
neighbourinteractionsintwodimensions 438
7.10 TheferromagneticIsingmodelwithalternatingfield 443