Perspectives on Spin Glasses

219 Pages·2012·1.166 MB·English

by  Contucci P., Giardin C.

PERSPECTIVES ON SPIN GLASSES
Presenting and developing the theory of spin glasses as a prototype for complex
systems,thisbookisarigorousandup-to-dateintroductiontotheirproperties.
The book combines a mathematical description with a physical insight of spin
glass models. Topics covered include the physical origins of those models and
their treatment with replica theory; mathematical properties such as correlation
inequalitiesandtheiruseinthethermodynamiclimittheory;mainexactsolutions
of the mean field models and their probabilistic structures; and the theory of
the structural properties of the spin glass phase such as stochastic stability and
the overlap identities. Finally, a detailed account is given of the recent numerical
simulationresultsandproperties,includingoverlapequivalence,ultrametricity,and
decayofcorrelations.Thebookisidealformathematicalphysicistsandprobabilists
workingindisorderedsystems.
pierluigi contucci is Professor of Mathematical Physics at the University
of Bologna, and Research Director for the hard sciences section of the Istituto
Cattaneo.Hisresearchinterestsareinstatisticalmechanicsanditsapplicationsto
socio-economicsciences.
cristian giardina` isAssociateProfessorinMathematicalPhysicsattheUni-
versity of Modena and Reggio Emilia, and Visiting Professor in Probability at
NijmegenUniversity.Hisresearchinterestsareinmathematicalstatisticalphysics
andstochasticprocesses.

PERSPECTIVES ON SPIN GLASSES
PIERLUIGI CONTUCCI
UniversityofBologna
CRISTIAN GIARDINA`
UniversityofModenaandReggioEmilia
cambridge university press
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Contucci,Pierluigi,1964–
Perspectivesonspinglasses/PierluigiContucciandCristianGiardina`.
pages cm
Includesbibliographicalreferencesandindex.
ISBN978-0-521-76334-9
1.Spinglasses–Mathematicalmodels. I.Giardina`,Cristian. II.Title.
QC176.8.S68C667 2013
530.4(cid:3)12–dc23 2012032860
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Contents
Preface pagevii
1 Origins,modelsandmotivations 1
1.1 Thespinglassproblem 1
1.2 Randominteractions,finite-dimensionalmodels,
mean-fieldmodels 2
1.3 Quenchedmeasureandrealreplicas 6
1.4 Definitionofamean-fieldspinglass 9
1.5 ReplicamethodfortheSKmodel 13
1.6 Thereplicasymmetricsolution 16
1.7 Theultrametricreplicasymmetrybreakingsolution 19
2 Correlationinequalities 27
2.1 Spinfunctions 27
2.2 FerromagnetismandtheGriffiths–Kelly–Shermaninequalities 31
2.3 SpinglasscorrelationinequalitiesoftypeI 34
2.4 Extensiontoquantumcase 35
2.5 TypeIIinequalities:resultsandcounterexamples 37
2.6 InequalitiesoftypesIandIIontheNishimoriline 39
2.7 Someconsequencesofcorrelationinequalities 44
3 Theinfinite-volumelimit 47
3.1 Introduction 47
3.2 Finite-dimensionalmodelswithGaussianinteractions 48
3.3 Pressureself-averaging 53
3.4 Finite-dimensionalmodelswithcenteredinteractions 56
3.5 Quantummodels 59
3.6 Extensiontonon-centeredinteractions 62
3.7 Independenceofthepressurefromboundaryconditions 64
v
vi Contents
3.8 Surfacepressure 66
3.9 CompletetheoryontheNishimoriline 73
3.10 Thermodynamicallimitformean-fieldmodels,
REMandGREM 78
4 Exactresultsformean-fieldmodels 84
4.1 IntroductiontothePoissonpointprocess 84
4.2 Poissonpointprocesswithexponentialintensity 87
4.3 ThePoisson–Dirichletdistribution 91
4.4 Therandomenergymodel 97
4.5 UpperboundfortheREMpressure 98
4.6 LowerboundfortheREMpressure 103
4.7 StatisticsofenergylevelsfortheREM 107
4.8 Thegeneralizedrandomenergymodel 108
4.9 UpperboundfortheGREMpressure 109
4.10 LowerboundfortheGREMpressure 112
4.11 TheAizenman–Sims–Starrextendedvariationalprinciple 113
4.12 Guerraupperbound 119
4.13 Talagrandtheoremandopenproblems 123
5 Spinglassidentities 125
5.1 Thestabilitymethodandthestructuralidentities 125
5.2 Stochasticstabilityidentities 130
5.3 Strongstochasticstabilityandmarginalindependence 134
5.4 GraphtheoreticalapproachtoAizenman–Contucciidentities 137
5.5 Identitiesfromself-averaging 144
5.6 IdentitiesontheNishimoriline 146
5.7 Interactionflipidentities 151
6 Numericalsimulations 164
6.1 Introduction 164
6.2 Simulationswithrealreplicas 165
6.3 Overlapequivalence 168
6.4 Ultrametricity 176
6.5 Decayofcorrelations 183
6.6 Energyinterfaces 192
References 200
Index 209
Preface
Spinglassesarestatisticalmechanicssystemswithrandominteractions.Thealter-
nating sign of those interactions generates a complex physical behavior whose
mathematical structure is still largely uncovered. The approach we follow in this
book is that of mathematical physics, aiming at the rigorous derivation of their
propertieswiththehelpofphysicalinsight.
Thebookstartswiththetheoreticalphysicsoriginsofthespinglassproblem.The
mainmodelsareintroducedandadescriptionofthereplicaapproachisillustrated
fortheSherrington–Kirkpatrickmodel.
Chapters 2 and 3 contain the starting points of the mathematical rigorous
approachleadingtothecontrolofthethermodynamiclimitforspinglasssystems.
Correlation inequalities are introduced and proved in various settings, including
the Nishimori line. They are then used to prove the existence of the large-volume
limitinbothshort-rangeandmean-fieldmodels.
Chapter 4 deals with exact results which belong to the mean-field case. The
methods and techniques illustrated span from the Ruelle probability cascades to
theAizenman–Sims–Starrvariationalprinciple.InthisframeworktheGuerraupper
boundtheoremforthepressureispresentedandtheTalagrandtheoremisreported.
Chapter5dealswiththestructuralidentitiescharacterizingthespinglassphase.
These are obtained by an extension of the stochastic stability method, i.e. an
invariance property of the system under small perturbations, together with the
self-averagingproperty.
Chapter 6 features some problems which are still out of analytical reach and
areinvestigatedwithnumericalmethods:theequivalenceamongdifferentoverlap
structures,thehierarchicalorganizationofthestates,thedecayofcorrelations,and
theenergyinterfacecost.
Needless to say there are innumerable important issues not covered by the
book.Amongthemarethedynamicalpropertiesofspinglasses(see,forexample,
Sompolinsky and Zippelius (1982); Cugliandolo and Kurchan (1993); Bouchaud
vii
viii Preface
(1992); Ben Arous et al. (2001, 2002); Bovier et al. (2001)). Another very large
topic not covered is that of applications whose ideas originated within spin glass
theoryandsuccessfullyfertilizedotherareas.
Itisapleasuretothanktheco-authorswhoseresearchworkprovidedthefoun-
dationsofthisbook:MichaelAizenman,AlessandraBianchi,MirkoDegliEsposti,
Claudio Giberti, Sandro Graffi, Andreas Knauf, Stefano Isola, Joel Lebowitz,
Satoshi Morita, Hidetoshi Nishimori, Giorgio Parisi, Joe Pule´, Shannon Starr,
FrancescoUnguendoli,andCeciliaVernia.
Usefulconversationswithmanycolleaguesareacknowledged,particularlythose
withLouis-PierreArguin,AdrianoBarra,AntonBovier,EdouardBre´zin,Aernout
van Enter, Silvio Franz, Francesco Guerra, Frank den Hollander, Jorge Kurchan,
EnzoMarinari,MarcMezard,ChuckNewman,DmitryPanchencko,DanielStein,
andMichaelTalagrand.
Last but not least we thank Claudio Giberti, Bernardo D’Auria, Aernout van
Enter,andCeciliaVerniafortheircarefulreadingofthemanuscript.
1
Origins, models and motivations
Abstract
We introduce the basic spin glass models, namely the Edwards–
Anderson model on a finite-dimensional lattice with short-range
interaction and the Sherrington–Kirkpatrick model on the complete
graph. The quenched equilibrium state which is used to describe
the thermodynamical properties of a general disordered system is
defined,togetherwiththeconceptofrealreplicas.Thenotionofmean-
field for a spin glass model is discussed. Finally, the original com-
putations for the Sherrington–Kirkpatrick model based on the replica
method are presented – namely the replica symmetric solution and the
Parisireplicasymmetrybreakingscheme.
1.1 Thespinglassproblem
Spin glass models have been considered in different scientific contexts, includ-
ing experimental condensed matter physics, theoretical physics, mathematical
statistical physics and, more recently, probability. They have also been used to
solve problems in fields as diverse as theoretical computer science (combinato-
rial optimization, traveling salesman, Boolean satisfiability, number partitioning,
randomassignment,errorcorrectingcodes,etc.),biology(Hopfieldmodel),popu-
lationgenetics(hierarchicalcoalescence),andtheeconomy(modelizationoffinan-
cial markets). Thus spin glasses represent a true example of a multi-disciplinary
topic.
The study of spin glasses began after experiments on magnetic alloys, for
instance metals like Fe, Mn and Cr weakly diluted in metals such as Au, Ag and
Cu.Itwasobservedthattheirthermodynamicalbehaviorwasnotcompatiblewith
the theory of ferromagnetism and showed peculiar dynamical out-of-equilibrium
1