Stochastic processes and random matrices : lecture notes of the les houches summer school first edit

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StochasticProcessesandRandomMatrices

LectureNotesoftheLesHouches

StochasticProcessesandRandom Matrices

Editedby

Gr´egorySchehr,AlexanderAltland,YanV.Fyodorov, NeilO’Connell,LeticiaF.Cugliandolo

GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom

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Previoussessions

I1951Quantummechanics.Quantumfieldtheory

II1952Quantummechanics.Statisticalmechanics.Nuclearphysics

III1953Quantummechanics.Solidstatephysics.Statistical mechanics.Elementaryparticlephysics

IV1954Quantummechanics.Collisiontheory.Nucleon-nucleon interaction.Quantumelectrodynamics

V1955Quantummechanics.Nonequilibriumphenomena.Nuclear reactions.Interactionofanucleuswithatomicandmolecular fields

VI1956Quantumperturbationtheory.Lowtemperaturephysics. Quantumtheoryofsolids.Ferromagnetism

VII1957Scatteringtheory.Recentdevelopmentsinfieldtheory.Nuclear andstronginteractions.Experimentsinhighenergyphysics

VIII1958Themanybodyproblem

IX1959Thetheoryofneutralandionizedgases

X1960Elementaryparticlesanddispersionrelations

XI1961Lowtemperaturephysics

XII1962Geophysics;theearthsenvironment

XIII1963Relativitygroupsandtopology

XIV1964Quantumopticsandelectronics

XV1965Highenergyphysics

XVI1966Highenergyastrophysics

XVII1967Manybodyphysics

XVIII1968Nuclearphysics

XIX1969Physicalproblemsinbiologicalsystems

XX1970Statisticalmechanicsandquantumfieldtheory

XXI1971Particlephysics

XXII1972Plasmaphysics

XXIII1972Blackholes

XXIV1973Fluidsdynamics

XXV1973Molecularfluids

XXVI1974Atomicandmolecularphysicsandtheinterstellarmatter

XXVII1975Frontiersinlaserspectroscopy

XXVIII1975Methodsinfieldtheory

XXIX1976Weakandelectromagneticinteractionsathighenergy

XXX1977Nuclearphysicswithheavyionsandmesons

XXXI1978Illcondensedmatter

XXXII1979Membranesandintercellularcommunication

XXXIII1979Physicalcosmology

XXXIV1980Laserplasmainteraction

XXXV1980Physicsofdefects

XXXVI1981Chaoticbehaviorofdeterministicsystems

XXXVII1981Gaugetheoriesinhighenergyphysics

XXXVIII1982Newtrendsinatomicphysics

XXXIX1982Recentadvancesinfieldtheoryandstatisticalmechanics

XL1983Relativity,groupsandtopology

XLI1983Birthandinfancyofstars

XLII1984Cellularandmolecularaspectsofdevelopmentalbiology

XLIII1984Criticalphenomena,randomsystems,gaugetheories

XLIV1985Architectureoffundamentalinteractionsatshortdistances

XLV1985Signalprocessing

XLVI1986Chanceandmatter

XLVII1986Astrophysicalfluiddynamics

XLVIII1988Liquidsatinterfaces

XLIX1988Fields,stringsandcriticalphenomena

L1988Oceanographicandgeophysicaltomography

LI1989Liquids,freezingandglasstransition

LII1989Chaosandquantumphysics

LIII1990Fundamentalsystemsinquantumoptics

LIV1990Supernovae

LV1991Particlesinthenineties

LVI1991Stronglyinteractingfermionsandhigh Tc superconductivity

LVII1992Gravitationandquantizations

LVIII1992Progressinpictureprocessing

LIX1993Computationalfluiddynamics

LX1993Cosmologyandlargescalestructure

LXI1994Mesoscopicquantumphysics

LXII1994Fluctuatinggeometriesinstatisticalmechanicsandquantum fieldtheory

LXIII1995Quantumfluctuations

LXIV1995Quantumsymmetries

LXV1996Fromcelltobrain

LXVI1996Trendsinnuclearphysics,100yearslater

LXVII1997Modelingtheearthsclimateanditsvariability

LXVIII1997ProbingtheStandardModelofparticleinteractions

LXIX1998Topologicalaspectsoflowdimensionalsystems

LXX1998Infraredspaceastronomy,todayandtomorrow

LXXI1999Theprimordialuniverse

LXXII1999Coherentatomicmatterwaves

LXXIII2000Atomicclustersandnanoparticles

LXXIV2000Newtrendsinturbulence

LXXV2001Physicsofbio-moleculesandcells

LXXVI2001Unityfromduality:Gravity,gaugetheoryandstrings

LXXVII2002Slowrelaxationsandnonequilibriumdynamicsincondensed matter

viii Previoussessions

LXXVIII2002Accretiondiscs,jetsandhighenergyphenomenain astrophysics

LXXIX2003Quantumentanglementandinformationprocessing

LXXX2003Methodsandmodelsinneurophysics

LXXXI2004Nanophysics:Coherenceandtransport

LXXXII2004MultipleaspectsofDNAandRNA

LXXXIII2005Mathematicalstatisticalphysics

LXXXIV2005ParticlephysicsbeyondtheStandardModel

LXXXV2006Complexsystems

LXXXVI2006Particlephysicsandcosmology:thefabricofspacetime

LXXXVII2007Stringtheoryandtherealworld:Fromparticlephysicsto astrophysics

LXXXVIII2007Dynamos

LXXXIX2008Exactmethodsinlow-dimensionalstatisticalphysicsand quantumcomputing

XC2008Long-rangeinteractingsystems

XCI2009Ultracoldgasesandquantuminformation

XCII2009Newtrendsinthephysicsandmechanicsofbiologicalsystems

XCIII2009ModernperspectivesinlatticeQCD:quantumfieldtheory andhighperformancecomputing

XCIV2010Many-bodyphysicswithultra-coldgases

XCV2010Quantumtheoryfromsmalltolargescales

XCVI2011Quantummachines:Measurementcontrolofengineered quantumsystems

XCVII2011TheoreticalphysicstofacethechallengeofLHC

SpecialIssue2012Advanceddataassimilationforgeosciences

XCVIII2012Softinterfaces

XCIX2012Stronglyinteractingquantumsystemsoutofequilibrium

C2013Post-Planckcosmology

CI2013QuantumOpticsandNanophotonics

SpecialIssue2013StatisticalPhysics,Optimization,Inference andMessage-PassingAlgorithms

CII2014FromMoleculestoLivingOrganisms:AnInterplayBetween BiologyandPhysics

CIII2014TopologicalAspectsofCondensedMatterPhysics

CIV2015StochasticProcessesandRandomMatrices

Publishers –SessionVIII:Dunod,Wiley,Methuen –SessionsIXandX:Herman,Wiley –SessionXI:GordonandBreach,PressesUniversitaires –SessionsXII–XXV:GordonandBreach –SessionsXXVI–LXVIII:NorthHolland –SessionLXIX–LXXVIII:EDPSciences,Springer –SessionLXXIX–LXXXVIII:Elsevier –SessionLXXXIX–:OxfordUniversityPress

Preface

ThefieldofstochasticprocessesandRMThasbeenarapidlyevolvingsubjectduring thelastfifteenyearswherethecontinuousdevelopmentanddiscoveryofnewtools, connectionsandideashaveledtoanavalancheofnewresults.Thesebreakthroughs havebeenmadepossiblethanks,toalargeextent,totherecentdevelopmentofvarious newtechniquesinRandomMatrixTheory(RMT).Matrixmodelshavebeenplaying animportantroleintheoreticalphysicsforalongtimeandtheyarecurrentlyalsoa veryactivedomainofresearchinmathematics.Anemblematicexampleoftheserecent advancesconcernsthetheoryofgrowthphenomenaintheKardar-Parisi-Zhang(KPZ) universalityclasswherethejointeffortsofphysicistsandmathematiciansduringthe lasttwentyyearshaveunveiledthebeautifulconnectionsbetweenthisfundamental problemofstatisticalmechanicsandthetheoryofrandommatrices,namelythefluctuationsofthelargesteigenvalueofcertainensembleofrandommatrices.Theselecture notesnotonlycoverthistopicindetailbutalsopresentmorerecentdevelopmentsthat haveemergedfromthesediscoveries,forinstanceinthecontextoflowdimensional heattransport(onthephysicsside)orincontextofintegrableprobability(onthe mathematicalside).Morewidely,ourgoal,inorganizingthisschoolinLesHouches, wastopresentthelatestdevelopmentsonthesetopicsattheinterfacebetweentheoreticalphysicsandmathematics,withaspecialemphasisonthelargespectrumof techniquesandapplicationsofRMT.Byfollowingthelecturesinthisvolumethe readerwillsurelybeabletoappreciatethebreadthandbeautyofthesubject.

TheschoolwasheldinJuly2015.Itstartedwithahistoricalintroductorylectures ontheapplicationsofRMTgivenbyH.Weidenm¨ullerandconsistedofmorethan fifty90-minuteslectures,coveringawiderangeoftopics.Theschoolcomprised,on theonehand,fivelongcourses(fivelectureseach)onrathergeneralsubjects,and,on theotherhand,tenshortercourses(twotothreelectureseach)onmorespecialized topics.Thisvolumepresentsthelecturenotespreparedbythespeakers.

Thisvolumestartswiththelecturenotesofthreeofthefivelonglectures:A. BorodinonIntegrableProbability,A.GuionnetonFreeProbabilityandH.Spohn ontheKardar-Parisi-Zhangequation(weregretthatwewerenotabletoincludethe lecturesnotesbyP.LeDoussalandB.Viraginthisvolume).Theyarefollowedup bythelecturenotesonmorespecializedtopics:G.AkemannonRMTandquantum chromodynamics,J.-Ph.BouchaudonRMTand(Big)dataanalysis,B.Eynardon Randommatricesandloopequations,J.P.KeatingonRandommatricesandnumber theory,A.L.MoustakasontheapplicationsofRMTtomoderntelecommunications, H.Schomerusonrandommatrixapproachestoopenquantumsystems,Y.Tourigny (andA.Comtet)onimpuritymodelsandproductsofrandommatrices,V.Vargas (andR.Rhodes)GaussianmultiplicativechaosandLiouvilleQuantumGravityand A.ZabrodinonQuantumspinchainsandclassicalintegrablesystems.Wealsohad

x Preface thepleasuretolistentoaseminarbyJ.-P.EckmannonM.Hairer’sworkandthe Kardar-Parisi-Zhangequation.

Thestudents,halfofthemphysicistsandtheotherhalfmathematicians,werefull ofenthusiasmbothinsideandoutsidetheclassroom.Manyofthemhadtheopportunitytopresenttheirworkduringtwopostersessions.ThisschoolinLesHouches hascertainlybeenaverygoodopportunityforthemtointeractscientificallywith otherstudentsaswellaswiththelecturers,whowereallofthemextremelyopento discussionswiththestudentsduringtheirstayinLesHouches.

Wearedeeplygratefultothelecturersworkinpreparingthelecturenotes,which willbeusefulinthefuturetothewholecommunity(bothphysicistsandmathematicians)workingonstochasticprocessesandRMT.WealsowishtothankPiotrWarchol forhispicturesofthelecturersandSunˇcanaDuli´cforherbeautifuldrawingsofthe mountainsaroundLesHouches.Finally,wewanttowarmlythankthestaffofthe SchoolofPhysicsinLesHouches,whomadeafantasticworkduringthisschool, whichwasreallyagreatmomentforallofus.

DrawingsbySunˇcanaDuli´ c

Listofparticipantsxxiii

1History—anoverview

OriolBOHIGASandHansA.WEIDENM ¨ ULLER1

1.1Bohr’sconceptofthecompoundnucleus3

1.2Spectralproperties4 1.3Data8

1.4Many-bodytheory9 1.5Chaos10

1.6Numbertheory11

1.7Scatteringtheory12

1.8Replicatrickandsupersymmetry15

1.9Disorderedsolids18

1.10Interactingfermionsandfieldtheory20 Acknowledgements20 References20

2Integrableprobability:stochasticvertexmodelsand symmetricfunctions

AlexeiBORODINandLeonidPETROV26

2.1Introduction28

2.1.1Preface28

2.1.2Ourmodelinaquadrant30

2.1.3Themainresult31

2.1.4Symmetricrationalfunctions32

2.1.5Cauchyidentities33

2.1.6Organizationofthechapter34

2.2Vertexweights34

2.2.1Higherspinsix-vertexmodel34

2.2.2Vertexweights35

2.2.3Motivation36

2.2.4Conjugatedweightsandstochasticweights37

2.3TheYang–Baxterequation38

2.3.1TheYang–Baxterequationincoordinatelanguage38

2.3.2TheYang–Baxterequationinoperatorlanguage39

2.3.3Attachingverticalcolumns41

2.4Symmetricrationalfunctions42

2.4.1Signatures43

2.4.2Semi-infiniteoperators A and B anddefinitionof symmetricrationalfunctions43

2.4.3Semi-infiniteoperator D 46

2.4.4Cauchy-typeidentitiesfromtheYang–Baxter commutationrelations48

2.4.5Symmetrizationformulas51

2.5Stochasticweightsandfusion59

2.5.1Stochasticweights Lu 59

2.5.2Fusionofstochasticweights60

2.5.3Principalspecializationsofskewfunctions63

2.6Markovkernelsandstochasticdynamics66

2.6.1ProbabilitymeasuresassociatedwiththeCauchyidentity66

2.6.2FourMarkovkernels69

2.6.3Specializations71

2.6.4Interactingparticlesystems73

2.6.5Degenerationtothesix-vertexmodelandtheASEP76

2.6.6Degenerationto q -Hahnand q -Bosonsystems80

2.7Orthogonalityrelations85

2.7.1Spatialbiorthogonality85

2.7.2Plancherelisomorphismsandcompleteness91

2.7.3Anintegralrepresentationfor Gμ 95

2.8 q -correlationfunctions96

2.8.1ComputingobservablesviatheCauchyidentity97

2.8.2Computationof Gν ( ,w1 ,...,wk )98

2.8.3Extractingtermsbyintegratingover wi 101

2.8.4 q -correlationfunctions104

2.8.5Remark.Fromobservablestoduality,andback109

2.9 q -momentsoftheheightfunction111

2.9.1Heightfunctionandits q -moments111

2.10Degenerationsofmomentformulas123

2.10.1Momentformulasforthestochasticsix-vertexmodel andtheASEP124

2.10.2Momentformulasfor q -Hahnand q -Bosonsystems126 Acknowledgements128 References128

3Freeprobability

AliceGUIONNET132

3.1Introduction134

3.2Freeprobabilitysetting135

3.2.1Noncommutativeprobabilityspace136

3.2.2Weak*-topology140

3.2.3Freeness140

3.3Asymptoticfreenessofrandommatrices141

3.3.1IndependentGUEmatricesareasymptoticallyfree141

3.3.2MatricesconjugatedbyHaarunitarymatricesare asymptoticallyfree146

3.4Freeconvolution150

3.4.1The R-transform150

3.4.2Thefreecentrallimittheorem153

3.5Nonnormaloperators154

3.5.1Thecircularlaw157

3.5.2Thesingleringtheorem157

3.6Loopequationsandtopologicalexpansions157

3.6.1Thefreedifferencequotientandtheconjugatevariable158

3.6.2Matrixmodelandintegrationbyparts158

3.6.3Topologicalexpansions159

3.6.4Formaltopologicalexpansions160

3.6.5Loopequationsandasymptoticexpansions165

3.6.6ExtensiontoHaardistributedmatrices171

3.7Applicationstoloopmodels171

3.7.1Applicationtoplanaralgebrasandloopmodels171

3.7.2Loopmodelsandsubfactors174 References175

4TheKardar–Parisi–Zhangequation:astatisticalphysics perspective HerbertSPOHN177

4.1Stable-metastableinterfacedynamics179

4.2Scalingproperties,KPZequationasweakdrivelimit182

4.3Eden-typegrowthmodels184

4.4TheKPZuniversalityclass185

4.4.1One-pointdistributions187

4.4.2Multipointdistributions188

4.4.3Stationarycovariance189

4.5Directedpolymersinarandommedium190

4.6Replicasolutions192

4.7Statisticalmechanicsoflineensembles194

4.8Noisylocalconservationlaws195

4.9One-dimensionalfluidsandanharmonicchains196

4.10DiscretenonlinearSchr¨odingerequation202

4.11LinearizedEulerequations204

4.12Linearfluctuatinghydrodynamics207

4.13Second-orderexpansion,nonlinearfluctuatinghydrodynamics211

4.14Couplingcoefficients,dynamicalphasediagram212

4.15Low-temperatureDNLS214

4.16Mode-couplingtheory216

4.16.1Decouplinghypothesis216

4.16.2One-loop,diagonal,andsmalloverlapapproximations217

4.17Moleculardynamicsandothermissedtopics222 Acknowledgements222 References223

5Randommatrixtheoryandquantumchromodynamics

GernotAKEMANN228

5.1Introductionandmotivation230

5.2OrthogonalpolynomialapproachtotheDiracoperatorspectrum233

5.2.1Theeigenvaluemodelanddefinitionofitscorrelation functions234

5.2.2Propertiesoforthogonalpolynomialswithgeneralweights237

5.2.3Allcorrelationfunctionswithmassesintermsof Laguerrepolynomials242

5.2.4Thelarge-N limitatthehardedge247

5.3SymmetriesofQCDanditsrelationtoRMT251

5.3.1TheDiracoperatorandglobalsymmetriesofQCD252

5.3.2Chiralsymmetrybreakingandchiralperturbationtheory254

5.3.3ThelimittoRMT:theepsilonregime257

5.4Recentdevelopments262

5.4.1RMTandQCDatfinitelatticespacing—theWilson Diracoperator262

5.4.2RMTandQCDwithchemicalpotential267

5.5Summary276 Acknowledgements277 References277

6Randommatrixtheoryand(big)dataanalysis

Jean-PhilippeBOUCHAUD283

6.1Introduction285

6.2Eigenvaluespectrumandeigenvectoroverlaps286

6.2.1Relationamongresolvent,spectrum,andoverlaps287

6.2.2Freeadditivenoise288

6.2.3Freemultiplicativenoiseandempiricalcovariancematrices289

6.3Rotationallyinvariantestimators291

6.3.1Theoracleestimatorandtheoverlaps291

6.3.2Optimalrotationalinvariantestimator293

6.4Rectangularcorrelationmatrices295

6.5Conclusionand(some)openproblems297

6.6AppendixA:ReminderontransformsinRMT298

6.7AppendixB:Resolventsandreplicas299

6.7.1Thereplicamethod299

6.7.2Freeadditivenoise300 References302

7Randommatricesandloopequations

BertrandEYNARD304

7.1Introduction306

7.1.1Randommatrices306

7.1.2Adetailedexample:randompartitionsasrandommatrices312

7.2Coulombgasandsaddle-pointapproximation317

7.2.1Coulombgasofeigenvalues317

7.2.2Saddle-pointapproximation318

7.2.3Functionalsaddle-point322

7.2.41-cutcase,Joukovskitransformation325

7.2.5Examples326

7.3Loopequations330

7.3.1Basisofmomentsandobservables330

7.3.2Integratingbyparts332

7.3.3Perturbativetopologicalexpansion336

7.3.4Higherorders340 References347

8Randommatricesandnumbertheory:somerecentthemes JonP.KEATING348

8.1TheRiemannzetafunction350

8.1.1Definitionandbasicproperties350

8.1.2Explicitformulae351

8.1.3 L-functions353

8.1.4Countingzeros354

8.1.5Generalexplicitformula355

8.1.6Connectionwithrandommatrices:Montgomery’s pair-correlationconjecture356

8.2Valuedistributionofthezetafunctiononitscriticalline360

8.2.1Selberg’stheorem360

8.2.2Momentsof ζ 1 2 + it 361

8.2.3Characteristicpolynomialsofrandommatrices362

8.2.4Extremevalues365

8.3FunctionFields369

8.3.1Background370

8.3.2Zetafunctionsofcurves371

8.3.3Spectralinterpretation372

8.3.4Spectralstatistics372

8.3.5Arithmeticstatisticsinfunctionfields373 Acknowledgements377 References377

9Moderntelecommunications:aplaygroundforphysicists? ArisL.MOUSTAKAS382 9.1Introduction384

9.1.1Outline385

9.2Informationtheorybasics385

9.2.1Informationcapacity385

9.2.2Linearprecoders386

9.3Wirelesscommunications:replicasandmobility387

9.3.1Capacityofcorrelatedantennas388

9.3.2Effectofmobilityonenergyconsumption390

9.3.3Discussion393 9.4Opticalcommunications:momentsandtails394 9.4.1Characterexpansionsincommunications395 9.4.2Tailsofthemutualinformation399

9.4.3Discussion404 9.5Conclusionsandoutlook405 Acknowledgments405 References405

10Randommatrixapproachestoopenquantumsystems

HenningSCHOMERUS409

10.1Introduction411 10.1.1Welcome411

10.1.2Primer412 10.1.3Opensystems413 10.1.4Preview414 10.2Foundationsofrandom-matrixtheory415 10.2.1RandomHamiltoniansandGaussianHermitianensembles415 10.2.2Time-reversalsymmetryandtheWigner–Dysonensembles416 10.2.3Chiralsymmetry418 10.2.4Charge-conjugationsymmetry420 10.2.5Randomtime-evolutionoperatorsandcircularensembles421 10.2.6Positive-definitematricesandWishart–Laguerreensembles423 10.2.7Jacobiensembles424 10.2.8Non-Hermitianmatrices425 10.3Thescatteringmatrix427 10.3.1Pointsofinterest427 10.3.2Definitionofthescatteringmatrix428 10.3.3Preliminaryanswers428 10.3.4Effectivescatteringmodels432 10.3.5Stroboscopicscatteringapproach433 10.3.6Continuous-timescatteringtheory436 10.3.7Merits438 10.4Decay,dynamics,andtransport440 10.4.1Scatteringpoles441 10.4.2Modenonorthogonality444 10.4.3Delaytimes446 10.4.4Transport450 10.5Localization,thermalization,andentanglement454 10.5.1Andersonlocalization455 10.5.2Thermalizationandlocalizationinmany-bodysystems457 10.6Conclusions460 10.7AppendixA:Eigenvaluedensitiesofmatriceswithlargedimensions461 10.7.1GaussianHermitianensembles461 10.7.2AppendixA.1:Wishart–Laguerreensembles462

10.7.3AppendixA.2:Jacobiensembles462

10.7.4AppendixA.3:Ginibreensembles463 Acknowledgements464 References465

11Impuritymodelsandproductsofrandommatrices

AlainCOMTETandYvesTOURIGNY474

11.1Introduction476

11.1.1Productofmatrices476

11.1.2Disorderedsystems476

11.1.3Outline477

11.1.4Recommendedreading477

11.2Someimpuritymodels478

11.2.1ThevibratingstringandDyson’srandomchainmodel478

11.2.2TheFrisch–Lloydmodel480

11.2.3TheAndersonmodel482

11.2.4Furthermotivations482 11.3Thespectralproblem483

11.3.1Thespectralmeasure485

11.3.2Theintegrateddensityofstatesperunitlength486

11.3.3TheWeylcoefficient486

11.3.4TheRiccatiequation487

11.3.5Classificationintermsoflimit-circleandlimit-pointtypes490

11.3.6Dyson’sdisorderedstringsofTypeI494

11.4ThecomplexLyapunovexponent496

11.4.1Theimaginarypart498

11.4.2Somedeterministicexamples498

11.4.3Adisorderedexample:Kotani’sTypeIIstring501

11.4.4CalculationoftheLyapunovexponent503

11.4.5TherelationshipbetweenΩ(λ)and w (λ)504

11.4.6ApplicationtotheFrisch–Lloydmodel506

11.4.7Notation506

11.5FurtherremarksontheFrisch–Lloydmodel508

11.5.1Pr¨ufervariables:thephaseformalism508

11.5.2Riccatianalysis:aqualitativepicture511

11.5.3TheRiceformula513

11.6Thewhitenoiselimit515

11.6.1L´evyprocesses515

11.6.2TheLyapunovexponent518

11.6.3Riccatianalysis519

11.7LifshitztailsandLifshitzsingularities520

11.7.1Short-range,repulsivepotentials521

11.7.2StringsofTypeI521

11.7.3Supersymmetricpotentials522

11.7.4Furtherreading523

11.8Distributionofthegroundstateenergyandstatisticsof energylevels523 11.9Scatteringandhyperbolicgeometry525 11.9.1Hyperbolicgeometry526 11.9.2Decayofthetransmissioncoefficient528 11.9.3Distributionofthereflexionphase529 11.10TheLyapunovexponentofaproductofrandommatrices530 11.10.1TheCohen–Newmanexample532 11.10.2ApplicationtotheFrisch–Lloydmodel533 11.11Furstenberg’sformulafortheLyapunovexponent534 11.11.1Theprojectivespace534 11.11.2AMarkovchainanditsstationarydistribution535 11.11.3Thecase d =2536 11.12Furstenberg’sTheorem539 11.13Concludingremarks544 Acknowledgements544 References545

12GaussianmultiplicativechaosandLiouvillequantumgravity R´emiRHODESandVincentVARGAS548 12.1Introduction550 12.1.1Notations551 12.2Gaussianmultiplicativechaos551

12.2.1ReminderonGaussianvectorsandprocesses551 12.2.2ConstructionoftheGMCmeasures552

12.2.3MainpropertiesoftheGMCmeasures555 12.3LiouvillequantumgravityontheRiemannsphere558

12.3.1ElementaryRiemanniangeometryonthesphere558 12.3.2AnintroductiontoCFTontheRiemannsphere559 12.3.3IntroductiontoLQFTontheRiemannsphere560 12.3.4ConstructionofLQFT563

12.3.5Propertiesofthetheory566

12.3.6TheLiouvillemeasures567 12.3.7Conjecturedrelationwithplanarmaps569 12.3.8OntheIsingmodelatcriticaltemperature571

12.3.9Finalremarksandconclusion573 12.4Appendix574

12.4.1Theconformalstructureonplanarmaps574 Acknowledgments575 References575

13Quantumspinchainsandclassicalintegrablesystems AntonZABRODIN578 13.1Introduction580

13.1.1Organizationofthepaper583 13.1.2Thenotation583 13.2ThemasterT-operatorforspinchains584 13.2.1Quantum R-matrices584

13.2.2InhomogeneousXXXspinchains586

13.2.3ThehigherT-operators590

13.2.4TheconstructionofthemasterT-operator592

13.2.5ThemasterT-operatorandthemKPhierarchy593

13.3FromthemasterT-operatortotheclassicalRSmodelandback596

13.3.1EigenvaluesofthespinchainHamiltoniansas velocitiesoftheRSparticles596

13.3.2LaxpairfortheRSmodelfromdynamicsofpoles597

13.3.3TheBAfunctionandthemasterT-operator599

13.4SpectrumofthespinchainHamiltoniansfromtheclassicalRSmodel601

13.4.1TwistparametersaseigenvaluesoftheLaxmatrix601

13.4.2TheQCcorrespondence602

13.4.3Algebraicequationsforthespectrum603

13.5TheQCcorrespondencevianestedBetheansatz605

13.5.1ThenestedBetheansatzsolution605

13.5.2TheQCcorrespondence:adirectproof606

13.6Concludingremarks608

13.7AppendixA:ThehigherT-operatorsthroughcharacters608

13.8AppendixB:HamiltonianformulationoftheRSmodel609 Acknowledgements610 References610

Listofparticipants Organizers

SCHEHRGr´egory

Universit´eParis-Sud,France

ALTLANDAlexander UniversityofK¨oln,Germany

FYODOROVYanV.

King’sCollegeLondon,DepartmentofMathematics,LondonWC2R2LS, UnitedKingdom

O’CONNELL

UniversityofBristol,SchoolofMathematics,HowardHouse,Queen’sAve,Bristol BS81SD,UnitedKingdom

CUGLIANDOLOLeticiaE. SorbonnesUniversit´es,France

Lecturers

AKEMANNGernot

BielefeldUniversity,Germany

BORODINAlexei

MIT,Cambridge,MA,USA

BOUCHAUDJean-Philippe

CFM/Ecolepolytechnique,Paris,France

ECKMANNJean-Pierre UniversityofGeneva,Switzerland

EYNARDBertrand

IPHTCEASaclay,GifsurYvette,France

GUIONNETAlice

MIT,Cambridge,MA,USA

Universit´edeLyon,EcoleNormaleSup´erieure,Lyon,France

KEATINGJon UniversityofBristol,UK

LEDOUSSALPierre

ENS,Paris,France

MOUSTAKASAris

UniverstiyofAthens,Greece

Listofparticipants Organizers

SCHEHRGr´egory

Universit´eParis-Sud,France

ALTLANDAlexander UniversityofK¨oln,Germany

FYODOROVYanV.

King’sCollegeLondon,DepartmentofMathematics,LondonWC2R2LS, UnitedKingdom

O’CONNELL

UniversityofBristol,SchoolofMathematics,HowardHouse,Queen’sAve,Bristol BS81SD,UnitedKingdom

CUGLIANDOLOLeticiaE. SorbonnesUniversit´es,France

Lecturers

AKEMANNGernot

BielefeldUniversity,Germany

BORODINAlexei

MIT,Cambridge,MA,USA

BOUCHAUDJean-Philippe

CFM/Ecolepolytechnique,Paris,France

ECKMANNJean-Pierre UniversityofGeneva,Switzerland

EYNARDBertrand

IPHTCEASaclay,GifsurYvette,France

GUIONNETAlice

MIT,Cambridge,MA,USA

Universit´edeLyon,EcoleNormaleSup´erieure,Lyon,France

KEATINGJon UniversityofBristol,UK

LEDOUSSALPierre

ENS,Paris,France

MOUSTAKASAris

UniverstiyofAthens,Greece

xxiv Listofparticipants

SCHOMERUSHenning

LancasterUniversity,UK

SPOHNHerbert

TechnicalUniversityMunich,Germany

TOURIGNYYves

UniversityofBristol,UK

VARGASVincent

ENS,Paris,France

VIRAGBalint

UniversityofToronto,Canada

WEIDENMULLERHans

MaxPlanckInstitut,Heidelberg,Germany

ZABRODINAnton

ITEP,Moscow,Russia

Participants

ALLEGRANicolas

Universit´edeLorraine,Nancy,France

ANDRAUSSergio

UniversityofTokyo,Japan

ANOKHINAAlexandra

ITEP,Moscow,Russia

ASSIOTISTheodoros

UniversityofWarwick,Coventry,UK

AUGERIFanny

Universit´ePaulSabatier,Toulouse,France

BALDWINChristopher UniversityofWashington,Seattle,WA,USA

BENITO-MATIASEnrique

CFMAC,Madrid,Spain

BENOISTTristan

McGillUniversity,Montreal,Canada

BUTEZRapha¨el

Universit´eParis-Dauphine,Paris,France

CHECINSKITomasz

UniversityofBielefeld,Germany

CUNDENFabioDeelan

UniversityofBristol,UK

DeNARDISJacopo UniversityofAmsterdam,NL

DOERAENEAntoine

Universit´eCatholiquedeLouvain,Belgium

DUCLUTCharlie

LPTMC,Paris,France

EMRAHElnur UniversityofWisconsin,Madison,WI,USA

FAHSBenjamin UniversityofLouvain,Louvain-la-Neuve,Belgium

FUKAIYosuke SanoLaboratory,UniversityofTokyo,Japan

GROUXBenjamin LaboratoiredeMath´ematiquesdeVersailles,France

HOLCOMBDiane UniversityofArizona,Tucson,AZ,USA

HUANGJiaoyang

HarvardUniversity,Cambridge,MA,USA

IPSENJesper BielefeldUniversity,Germany

JANJIGIANChristopher UniversityofWisconsin,Madison,WI,USA

JOYNERChristopher QueenMaryUniversity,London,UK

JUNNILAJanne UniversityofHelsinki,Finland

KIEFFMax ColumbiaUniversity,NewYork,USA

LAMBERTGaultier

KTHStockholmSweden

LIYiting BrandeisUniversity,Waltham,MA,USA

LIUDang-Zheng UniversityofChina,Hefei,China

MARINORicardo LPTMS,Universit´eParis-Sud,Paris,France

MUDUTE-NDUMBESteve

ImperialCollege,London,UK

NEMISHYuriy

Universit´ePaulSabatier,Toulouse,France

Listofparticipants xxv

xxvi Listofparticipants

NGUYENVu-Lan

LPMA,Paris,France

NOCKAndr´ e

QueenMaryUniversity,London,UK

OCHABJeremi

JagiellonianUniversity,Krakow,Poland

PAQUETTEElliot

WeizmannInstituteofScience,Rehovot,Israel

PEREZCASTILLOIsaac

UNAM,Mexico

POPLAVSKYIMihail

UniversityofWarwick,Coventry,UK

PRATTony

LaboratoireKastler-Brossel,Paris,France

REYNOLDSAlexi

UniversityofBristol,UK

RODRIGUEZ-LOPEZPablo

LPTMS—Universit´eParisSud,Orsay,France

SIMMNicholas

UniversityofLondon,UK

SLOWMANAlexander

SUPA,UniversityofEdinburgh,UK

SWIECHArtur

UniversityofCologneGermany

TARNOWSKIWojciech

JagiellonianUniversity,Krakow,Poland

TARPINMalo

LPMMC,Grenoble,France

THIERYThimoth´ee

LPT-ENS,Paris,France

TUISKUPetri

UniversityofHelsinki,Finland

TURUNENJoonas

UniversityofHelsinki,Finland

VADAKKEVEETTILPrasad

RamanResearchInstitute,Bangalore,India

WARCHOLPiotr

JagielloninaUniversity,Krakow,Poland

XUYuanyuan

UniversityofCalifornia,Davis,CA,USA

1

History—anoverview

Oriol Bohigas1 andHansA. Weidenm ¨ uller2

1 LPTMS,CNRS,Univ.Paris-Sud,Universit´eParis-Saclay,91405Orsay,France; deceased

2 Max-Planck-Institutf¨urKemphysik,Heidelberg,P.O.Box103980,69029 Heidelberg,Germany

Bohigas,O.andWeidenm¨uller,H.A.,‘History–anoverview’firstpublishedin TheOxford HandbookofRandomMatrixTheory. Edited:GernotAkemann,JinhoBaik,PhilippeDiFrancesco, OxfordUniversityPress(2011). c OxfordUniversityPress2011.Reproducedas‘History’in Stochastic ProcessesandRandomMatrices. Editedby:Gr´egorySchehretal,OxfordUniversity Press(2017). c OxfordUniversityPress2017.DOI10.1093/oso/9780198797319.003.0001

1.1 Bohr’sconceptofthecompoundnucleus3 1.2 Spectralproperties4 1.3 Data8 1.4 Many-bodytheory9

1.5 Chaos10

1.6 Numbertheory11

1.7 Scatteringtheory12

1.8 Replicatrickandsupersymmetry15 1.9 Disorderedsolids18 1.10 Interactingfermionsandfieldtheory20

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whenever we stopped Mansing pathetically conversed with the dog, who seemed almost to understand all the coolie was telling him.

It was rather a puzzle to me why this dog followed us so long, for we had so little food that we could but seldom spare him any. He slept near us at night with his head on the lap of one of us, and during the march he showed quite sporting instincts by chasing antelopes and kiang (wild horse) when we encountered herds of hundreds of them. Curiously enough, when we entered a Tibetan encampment he always avoided being seen in our company. It seemed almost as if he realised that we were not welcome guests in the country, and feared the consequences. Possibly [139]he only temporarily left us to see what he could pick up in the way of food, but whenever we came across him in the encampment, he never would show signs of recognition, much less of affection, as was the case when he would rejoin us some miles beyond on the march, when he made ample efforts to reingratiate himself. He seemed almost to want to express: “Sorry I had to cut you in the encampment, but I really had to!”

A Tibetan Camp of Black Tents

At last the day came when we were captured, and underwent several kinds of tortures, as I have already described in In the Forbidden Land. The dog had vanished, and, to tell the truth, we did not give him much of a thought, as we were somewhat concerned about ourselves.

One day, when Mansing and I were stretched, or rather suspended, on a primitive kind of rack, and we were for some time left to ourselves—the soldiers and Lamas having retired some distance off into the huge tent of the Pombo, a high official—the dog sadly walked towards us, sniffing us, and rubbing himself against Mansing and me. He was particularly affectionate to Mansing, whose face he licked several times; then with a pathetic movement of his head as if to express his sorrow, he gave us a parting sad look, turned his back, [140]and walked slowly and sorrowfully away. That was his last mark of friendship and the last we saw of him.

Tibetan encampments have no great interest except for the peculiar shape of the black tents—a pattern of shelter most suitable for the climate of their country. The two sides of the tent are separate, and when the tent is put up it leaves an aperture all along its highest ridge. This is for various reasons. First, because the Tibetans light fires inside their tents, and an opening is necessary to let the smoke out; also as a means of ventilation, the cold air not penetrating so quickly as when it comes in at the sides, owing to the warmed atmosphere inside. The black tents are woven of a coarse and waterproof fabric of yak hair. Through the slit at the top generally protrude the props of the matchlocks bundled against one of the tent poles.

Every man in Tibet owns one of these weapons, and is considered a soldier in time of war.

The inside of a large Tibetan tent is quaint enough when you have reached it by skipping over masses of dirt and refuse which surround its outside. Only, when you peep in, the odour is rather strong of the people, old and young, all since [141]birth innocent of washing, and the smell of badly-prepared skins, and stores of chura (cheese). Nor must I forget to mention the wall of yak-dung erected

Interior of a Tibetan Tent, showing Churn for mixing Tea with Butter

right round the tent inside to serve the double purpose of protection against the wind where the tent meets the ground, and of fuel, being gradually demolished to feed the double mud-stove erected in the centre of the tent. Mud alone is also occasionally used for the inside wall.

As you know, dung is practically the only fuel obtainable in the highest parts of Tibet, although occasionally a few low shrubs are to be found. The fuel is constantly collected and conveyed from one camp to the next, when changing in order to find more suitable grazing for the sheep and yaks.

The centre mud-stove is built according to the most practical notions to make it draw properly, and upon it can nearly always be seen one or two large raksangs, brass vessels in which brick-tea is being stewed and stirred with a long brass spoon. But the operation of tea-making is rather complicated in Tibet. After the leaves have been stewed long enough the liquid is poured into a dongbo, or cylindrical wooden churn, in which have been deposited several balls of butter with copious [142]sprinkling of salt. A piston which passes through the movable lid is then vigorously set in action, and when well stirred and steaming the mixture is served all round and avidly drunk in wooden bowls, one of which every one carries about the person. Tsamba, a kind of oatmeal, is frequently mixed with the tea in the bowls, where it is made into a paste with the fingers.

No matter how much non-Tibetan folks may find merriment in the idea of tea being brewed with butter and salt, there is no doubt that for a climate like Tibet it is “the drink” par excellence. It warms, nourishes, and is easily digested. I very often indulged in the luxury myself, when I could obtain butter, only, my digestion working rather rapidly owing to the amount of roughing we daily endured, I left out the salt so that I should not digest the mixture too quickly.

The richer owners of tents generally have a sort of folding shrine, with one or more images of Buddha, which occupies the place of honour in the tent. Numerous brass bowls and ornaments are displayed in front of these images

A Little Boy learning to Pray

and also offerings of tsamba and butter. Wicks, burning in butter, are occasionally lighted around and upon the shrine. Decrepit old women seem to [143]spend most of their time revolving their prayer-wheels and muttering prayers in front of these altars, and when occasion arises thus teaching little children to do the same. The younger folk, too, are very religious, but not to the fanatical extent of the older ones.

It is quite amusing to see little mites—children are always quaint in every country —try to master the art of revolving the prayer-wheel. It must be revolved from left to right, to pray in the proper fashion,—not that if you revolved it the other way you would necessarily be swearing, only, according to the laws of Tibetan Buddhism, prayers spun in the wrong direction would have no effect and bring no benefit. In a similar way circumambulations, either round hills for pilgrimages, or round a tent, or round a sacred lake, must always follow a similar direction to the revolving of the prayer-wheel.

In Lhassa and many other sacred places fanatical pilgrims make these circumambulations, sometimes for miles and miles, and for days together, covering the entire distance lying flat upon their bodies, then placing the feet where the head was and stretching themselves full length. Inside temples a central enclosure is provided, round [144]which these circumambulations are performed, special devotions being offered before Buddha and many of the other gilt or high-coloured images which adorn the walls of the temple.

As can be seen by the coloured plate illustrating one of these scenes, from the ceiling of the temple hang hundreds of long strips, Katas, offered by pilgrims to the temple and becoming so many flying prayers when hung up—for mechanical praying in every way is prominent in Tibet. There is, after all, no reason why praying should not be made easy like everything else. Thus, instead of having to learn by heart long and varied prayers, all you have to do is to stuff the entire prayer-book (written on a roll in Tibet) into the prayer-wheel, and revolve it while repeating as fast as you can go these four words: “Om mani padme hum,”— words of Sanscrit origin and referring to the reincarnation of Buddha from a lotus flower, literally “O God, the gem emerging from the lotus flower.”

Worshippers circumambulating the inner enclosure lying flat full length

The temples of Tibet, except in Lhassa itself, are not beautiful in any way—in fact, they are generally very tawdry and dirty. The attention of the pilgrims is directed to a large box, or often a big bowl, where they may deposit whatever [145]offerings they can spare, and it must be said that their religious ideas are so

Interior of Tibetan Temple

strongly developed that they will dispose of a considerable portion of their money in this fashion.

Large monasteries, of red or yellow Lamas, are attached to these temples, where proselytes are also educated. These Lamas, whatever their colour, are very clever in many ways, and have a great hold over the entire country. They are, ninety per cent of them, unscrupulous scamps, depraved in every way, and given to every sort of vice. So are the women Lamas. They live and sponge on the credulity and ignorance of the crowds; and it is to maintain this ignorance, upon which their luxurious life depends, that foreign influence of every kind is strictly kept out of the country. Their abnormal powers have been grossly exaggerated. They practise, it is true, hypnotism, but that is all. They can perform no more marvellous feats than any one can do in England who is able to mesmerize. As for the Mahatmas, who, our spiritualistic friends tell us, live in Tibet, they are purely imaginary, and do not exist. The Tibetans have never heard of them nor about their doings.

Personally—and I am glad that the few men [146]who know Tibet from personal knowledge and not from political rivalry agree with me—I believe that the intrigues of the Lamas with Russia are absolute nonsense. Tibet, it must be remembered, was not forbidden to Englishmen only, but to everybody from every side, whether native or white, certain Nepalese and Chinamen, only, having the privilege of entering the country. It was a fight against Western ways in general which the Lamas were carrying on, quite successfully owing to the geographical position of their country, and the natural difficulties of reaching it, and not a fight against one race more than another. The accounts of the Lhassa Mission to the Czar were possibly the best diplomatic practical jokes which have been played upon this credulous country; and the mythical and much-feared Dorjeff is possibly—at least as far as power is concerned—nothing more than the creation of hysterical Anglo-Indian officials who, everybody knows, seem to see the treacherous hand of Russia in everything.

Perhaps no other country but England would be so rash as to go and sink millions of pounds sterling good money on a country that is, for all practical purposes, absolutely useless and worthless. [147]This does not detract from its pictorial, nor from its geographical or ethnological interest; from these points it is most interesting indeed.

Agriculturally, as I have stated, nothing grows there; no very wealthy mines have so far been discovered, the only mines that are plentiful being of borax, which has not sufficient market value to pay for the expensive carriage from Tibet to the coast. Regarded as a climate for a sanatorium for our sick soldiers in India— for which Tibet is frequently recommended by Anglo-Indian papers—I believe that such an establishment would be a very quick way of disposing altogether of all the sick men sent there. And as for such gigantic schemes as the construction of railways, say from India to the upper waters of the YangtzeKiang, or to Pekin, the expense of taking a railway over the Himahlya range and keeping it in working order during the wintry months—nine out of twelve—would, I think, never be remunerative. In Tibet itself the construction of a railway would be comparatively easy, as great stretches of the country are almost flat. Stations

Tibetan Women weaving

of imported fuel would have to be provided for the entire distance across Tibet, and the engines would have to be constructed specially to suit the great altitude. [148]

For trade and commerce with the natives themselves, the population of the country is so small, so deplorably poor and so lacking in wants, and the country is so large that, personally, I do not see how any large commercial venture in such a country can turn out successful. It is very difficult to get money where there is none. Small native traders, of course, can make small profits and be satisfied. Besides, the intercourse between Tibet and the neighbouring countries, particularly those to the south, can only take place with comfort during three months of the summer when the high snow-passes are open.

So that, much as I would like to see Tibet open in a proper way to travellers, I cannot quite understand the necessity of the Government spending millions of money and butchering thousands of helpless and defenceless natives in a manner most repulsive to any man who is a man, and of which we can but be ashamed—and all this to obtain a valueless commercial treaty. It is true, the Tibetans had been very impudent in every way on our frontier, but for this we only have to blame ourselves and our incompetent officials. If, instead of giving way to their bluff, we had kept a firm hand, matters would have been different.

Tibetan Women cleaning Wool [149]

Even in the case of my capture and torture on my first expedition into Tibet I never had a feeling of resentment towards the Tibetans for what they did to me. It was very exciting and interesting for me to endeavour to reach their sacred city, but I did so at my own risk and against their repeated warnings and threats, and I got nothing more for it in the end than I expected, in fact, bad as it was, considerably less. Highly amusing as it was to me to give them endless trouble, it was undoubtedly equally enjoyable to them to torture me, when once they succeeded in effecting my capture. Possibly, if I now have any feelings at all towards the Tibetans, it is a feeling of gratitude towards them for sparing my life in the end, which, by the way, they came within an ace of taking as they had promised to do.

As a punishment for what they did to me—because, after all, my men and I suffered a great deal more than the average man could stand—the Government of India practically ceded, as we have said, all the rights to Tibet of an immense district of British territory at the frontier. Can you blame the Tibetans for doing worse if they had a chance? [150]

[Contents]

CHAPTER XIV

In heart and soul the Tibetan is a sportsman; but if you look for grace in his movements you will be sorely disappointed. Indeed, more fervour and clumsiness combined are hardly to be paralleled anywhere. Perhaps the Tibetan is seen to advantage on his pony, and some of his feats on the saddle I will here describe.

A Lama Standard-Bearer

Horse races are quite a favourite form of amusement, and are run in a sensible manner. Only two ponies at a time go round the course, the final race being run between the winners of the two best heats. Praying is usually combined, in

some form or other, with everything people do in Tibet, and so even races are run round the foot of an isolated hill or around an encampment of tents; for, as you know, circumambulation of any kind, if in the right direction, is equivalent to prayer, and pleases God. Thus, just as with their prayer-wheels, a rotatory [151]motion is kept up from left to right, so races are run in the same way from the standpoint of the spectator

A Tibetan race would astonish an English crowd—the means adopted by the well-matched couples being very effective, if somewhat primitive. Such simple devices as seizing one’s opponent’s reins, or lashing him in the face to keep him back, or pushing or pulling him off his saddle, are considered fair and legal means in order to win the race. The last heat is usually the most exciting, especially for the spectators, for blows with the lash are exchanged in bewildering profusion by both riders taking part in it, their respective ponies sharing unsparingly in the punishment. Occasionally the race becomes a regular hippic wrestling match, when both riders, clinging tightly together, tumble over and roll to the ground. When the ponies are recaptured, the bruised horsemen remount and continue the race as if nothing had happened.

The heavy sheepskin coats worn by the Tibetans are some protection when the lash is applied, and the pain inflicted is not always in proportion to the noise made by the blow; but such is not the case when they catch one another across the face. [152]

The winner is presented with a kata by the umpire—a high Lama or a military officer, a most picturesque creature in a brilliant red coat and fluffy hat, who has a peculiar standard with hundreds of long, vari-coloured strips of cloth, or flying prayers. Sitting on a handsome pony, with gaudy harness of green leather inlaid with brass, a valuable Chinese rug upon the saddle, and many tinkling bells round the pony’s neck, the umpire and his pony certainly produce a gay ensemble. This gentleman takes himself very seriously, and seldom condescends to smile.

The kata, or “scarf of love and friendship,” which is given to the winner is a long piece of silk-like gauze, the ends of which have been trimmed into a fringe. As I have elsewhere described at greater length, these katas play quite an important part in the social intercourse of Tibetans. They can be purchased or obtained from the Lamas of any monastery, or where no monastery exists the natives manufacture them themselves, for they are constantly needed. No gift can be

sent nor accepted without “a veil of friendship” accompanying it, and no stranger ever enters a tent without offering, with outstretched hands, a kata, which he quickly lays at the feet of [153]his host. Diminutive katas are enclosed in letters; sweethearts exchange katas on every possible occasion—until they are actually married. Polyandry being prevalent in Tibet, when one of the several husbands returns to his wife after the customary absence, he never fails to bring a kata with him. Not to offer a kata to an honoured visitor is as palpable a breach of manners, and as great a slight as can possibly be offered in the Forbidden Land.

Necessarily, when a kata has been blessed by the Lamas, or is won in a race before high officials, it has additional value, and these simple folks value it more than a gift of money or food. It is stored away in the tent among the heirlooms, and is handed down to posterity

A slightly more difficult feat, very common in a similar form in most countries, is the picking up of a kata by horsemen at full gallop. One horseman, a high official, revolves the kata seven times in the wind, and then darts full gallop in one direction, followed by twenty, thirty, or more horsemen riding wildly, and each trying to push his neighbours out of the way. The official, some thirty yards ahead, flies the kata in the wind, and when fancy takes him lets it drop out of his hand. The kata eventually settles on the ground, and the horde [154]of riders gallops away from it, yelling and quarrelling. At a signal from the officer the horsemen turn round and make a dash for the scarf, towards which all the ponies are converging. Clinging to the saddle with one hand and hanging over, each rider attempts to pick up the kata without dismounting. Collisions and nasty falls are numerous, and this sport generally partakes of the character of an allround fight among the ponies’ legs. Somebody, however, always succeeds in picking up the scarf and getting clear of the others, when he triumphantly rides round the camp fluttering the prize in the wind.

Some of the younger fellows are clever at this sport, and when one rider at a time does the feat, he seldom misses picking up the kata at the first swoop.

A Race for the Kata

An interesting and more difficult feat of horsemanship I witnessed in Tibet was the loading and firing of a matchlock while at full gallop—a performance which requires a firmer seat on the saddle than appears. The heavy and cumbersome weapons had to be unslung from the shoulders, the props let down, the fuse lighted by flint and steel, some gunpowder placed and kept in the small side receptacle, and last, but not least, the shot fired off—that [155]is to say, when it would go off! The full use of both hands was required in this exercise, and therefore the horsemen held the reins with their teeth. When firing they lay almost flat on the ponies’ backs in order to prevent being thrown by the sudden bucking of the frightened ponies.

Another exercise consists in bodily lifting a person on the saddle while the pony is at full gallop. The pedestrian is seized as low near the waist as possible, and the impetus of the pony’s flight, not the rider’s actual strength, is utilised in raising the person on the saddle.

The women seemed particularly interested in this sport, because a practical application of this exercise is used by enterprising lads of Tibet to overcome the scruples of reluctant maids who do not reciprocate their love. At a suitable

opportunity the doomed young lady is abducted bodily in that fashion, and conveyed in all haste to the suitor’s tent, with the honourable intention, of course, of making her his happy bride.

Women are scarce in Tibet, and actual raiding parties, I was told, occasionally take place against neighbouring tribes in order to obtain a fresh supply of wives.

Taking things all round, there are few men and [156]women in Tibet who cannot ride well, yet there are few who can claim exceptional skill in that line. The Tibetan generally values his bones too much to indulge in fancy tricks upon his pony. Some young fellow, more ambitious than others, will master the art of standing erect upon the saddle while going full speed, his feet being inserted into the stirrups, which have for the purpose been shortened as high as they could go. By pressing with his ankles against the saddle he manages to maintain his balance, in the familiar way of the Cossacks and tribes of Central Asia, who all excel in this game.

Tibetan saddles, as you know, are in appearance not unlike a cross between a Cossack saddle and a rude Mexican saddle, and as good as neither, but quite suited to the country where they are used. Men and women ride astride, with exceptionally short stirrups, so that the leg is bent at the knee at a right or even an acute angle. In order to maintain one’s equilibrium when riding fast some additional stability is obtained by stretching out the arms sideways.

Taking all things into consideration, there is no doubt that in a rugged, mountainous country like Tibet, and for a Tibetan, his is the most practical [157]and useful type of saddle, and his fashion of riding the most sensible— evidently the outcome of practical experience. When riding in caravans, driving herds of laden yaks or ponies, the advantages of legs doubled up high upon the saddle are soon apparent, avoiding the danger of crushing one’s lower limbs or having them partly torn off. In the English way of riding, when among obstacles, one’s legs are always in the way; in the Tibetan fashion they are always out of the way, or, at any rate, can easily and quickly be moved over from one side to the other of the saddle. Also, when tired of riding in one position, altering one’s position to side-saddle is quite convenient and easy

The blocks of the saddles are of wood imported mainly from India, Nepal, or China, with bindings of hammered iron or brass, often inlaid with silver and gold. Lizard skin and coloured leather adorn the front and back of the saddles, and a substantial pad covers the central part and the otherwise very angular seat. For extra comfort rugs—occasionally valuable and always decorative in blue and red tints—are spread, while to leather laces behind the saddle are slung double bags containing tsamba, chura, or cheese, a brick of compressed tea, and whatever sundry articles may be used on a journey. [158]The last, but not least

Tibetan Soldier at Target Practice

attachment on a Tibetan saddle is a long coiled rope of yak hair with a wooden peg at the end for tethering the pony at night.

Whatever one may say of Tibetans, the best-inclined could not compliment them on their shooting. Their matchlocks—their only firearms, made in Lhassa and Shigatz—are weapons so clumsy and heavy and badly made, that when fired it is truly more dangerous to be behind them than in front of their muzzle. During my captivity in Tibet in 1897, indeed, I was fired upon twice—by distinguished marksmen who took accurate aim only a few paces from me—but neither time was I hit. Nor in all my experience of Tibet have I any remembrance of ever seeing a Tibetan hit with a projectile from his matchlock anything which he intended, although the range was never more than twenty or thirty yards. Few are the matchlocks in the Forbidden Land which will carry as far as fifty or a hundred yards.

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