Isomonodromy transformations of linear systems of

arXiv:math/0209144v2[math.CA]3Oct2002ISOMONODROMYTRANSFORMATIONSOFLINEARSYSTEMSOFDIFFERENCEEQUATIONSAlexeiBorodinAbstract.Weintroduceandstudy“isomonodromy”transformationsofthematrixlineardifferenceequationY(z+1)=A(z)Y(z)withpolynomial(orrational)A(z).OurmainresultisaconstructionofanisomonodromyactionofZm(n+1)−1onthespaceofcoefficientsA(z)(heremisthesizeofmatricesandnisthedegreeofA(z)).The(birational)actionofcertainranknsubgroupscanbedescribedbydifferenceanalogsoftheclassicalSchlesingerequations,andweprovethatforgenericinitialconditionsthesedifferenceSchlesingerequationshaveauniquesolution.WealsoshowthatboththeclassicalSchlesingerequationsandtheSchlesingertransforma-tionsknownintheisomonodromytheory,canbeobtainedaslimitsofouractionintwodifferentlimitregimes.Similarlytothecontinuouscase,form=n=2thedifferenceSchlesingerequa-tionsandtheirq-analogsyielddiscretePainlev´eequations;examplesincludedPII,dPIV,dPV,andq-PVI.IntroductionInrecentyearstherehasbeenconsiderableinterestinanalyzingacertainclassofdiscreteprobabilisticmodelswhichinappropriatelimitsconvergetowell-knownmodelsofRandomMatrixTheory.Thesourcesofthesemodelsarequitediverse,theyincludeCombinatorics,RepresentationTheory,PercolationTheory,RandomGrowthProcesses,tilingmodelsandothers.Onequantityofinterestinbothdiscretemodelsandtheirrandommatrixlimitsisthegapprobability–theprobabilityofhavingnoparticlesinagivenset.Itisknown,duetoworksofmanypeople,see[JMMS],[Me],[TW],[P],[HI],[BD],thatinthecontinuous(randommatrixtype)setuptheseprobabilitiescanbeexpressedthroughasolutionofanassociatedisomonodromyproblemforalinearsystemofdifferentialequationswithrationalcoefficients.Thegoalofthispaperistodevelopageneraltheoryof“isomonodromy”trans-formationsforlinearsystemsofdifferenceequationswithrationalcoefficients.Thissubjectisofinterestinitsownright.Asanapplicationofthetheory,weshowinasubsequentpublicationthatthegapprobabilitiesinthediscretemodelsmen-tionedaboveareexpressiblethroughsolutionsofisomonodromyproblemsforsuchsystemsofdifferenceequations.Inthecaseofone-intervalgapprobabilitythishasbeendone(inadifferentlanguage)in[Bor],[BB].OneexampleoftheprobabilisticmodelsinquestioncanbefoundattheendofthisIntroduction.ConsideramatrixlineardifferenceequationY(z+1)=A(z)Y(z).(1)TypesetbyAMS-TEX1HereA(z)=A0zn+A1zn−1+···+An,Ai∈Mat(m,C),isamatrixpolynomialandY:C→Mat(m,C)isamatrixmeromorphicfunction.1WeassumethattheeigenvaluesofA0arenonzeroandthattheirratiosarenotreal.Then,withoutlossofgenerality,wemayassumethatA0isdiagonal.ItisafundamentalresultprovedbyBirkhoffin1911,thattheequation(1)hastwocanonicalmeromorphicsolutionsYl(z)andYr(z),whichareholomorphicandinvertibleforℜz≪0andℜz≫0respectively,andwhoseasymptoticsatz=∞inanyleft(right)half-planehasacertainform.BirkhofffurthershowedthattheratioP(z)=(Yr(z))−1Yl(z),whichmustbeperiodicforobviousreasons,is,infact,arationalfunctioninexp(2πiz).ThisrationalfunctionhasjustasmanyconstantsinvolvedastherearematrixelementsinA1,...,An.LetuscallP(z)themonodromymatrixof(1).OtherresultsofBirkhoffshowthatforanyperiodicmatrixPofaspecificform,thereexistsanequationoftheform(1)withprescribedA0,whichhasPasthemonodromymatrix.Furthermore,iftwoequationswithcoefficientsA(z)andeA(z),eA0=A0,havethesamemonodromymatrix,thenthereexistsarationalmatrixR(z)suchthateA(z)=R(z+1)A(z)R−1(z).(2)Thefirstresultofthispaperisaconstruction,forgenericA(z),ofahomomor-phismofZm(n+1)−1intothegroupofinvertiblerationalmatrixfunctions,suchthatthetransformation(2)foranyR(z)intheimage,doesnotchangethemonodromymatrix.Ifwedenotebya1,...,amntherootsoftheequationdetA(z)=0(calledeigen-valuesofA(z))andbyd1,...,dncertainuniquelydefinedexponentsoftheasymp-toticbehaviorofacanonicalsolutionY(z)of(1)atz=∞,thentheactionofZm(n+1)−1isuniquelydefinedbyintegralshiftsof{ai}and{dj}withthetotalsumofallshiftsequaltozero.(Weassumethatai−aj/∈Zanddi−dj/∈Zforanyi6=j.)ThematricesR(z)dependrationallyonthematrixelementsof{Ai}ni=1and{ai}mni=1(A0isalwaysinvariant),anddefinebirationaltransformationsofthevari-etiesof{Ai}withgiven{ai}and{dj}.ThereexistremarkablesubgroupsZn⊂Zm(n+1)−1whichdefinebirationaltrans-formationsonthespaceofallA(z)(withfixedA0andwithnorestrictionsontherootsofdetA(z)),buttoseethatweneedtoparameterizeA(z)differently.Todefinethenewcoordinates,wesplittheeigenvaluesofA(z)intongroupsofmnumberseach:{a1,...,amn}={a(1)1,...,a(1)m}∪···∪{a(n)1,...,a(n)m}.Thesplittingmaybearbitrary.ThenwedefineBitobetheuniquelydetermined(remember,everythingisgeneric)elementofMat(m,C)witheigenvaluesna(i)jomj=1,suchthatz−BiisarightdivisorofA(z):A(z)=(A0zn−1+A′1zn−1+···+A′n−1)(z−Bi).1ChangingY(z)to(Γ(z))kY(z)readilyreducesarationalA(z)toapolynomialone.2Thematrixelementsof{Bi}ni=1arethenewcoordinatesonthespaceofA(z).TheactionofthesubgroupZnmentionedaboveconsistsofshiftingtheeigen-valuesinanygroupbythesameintegerassignedtothisgroup,andalsoshiftingtheexponents{di}bythesameinteger(whichisequaltominusthesumofthegroupshifts).Ifwedenoteby{Bi(k1,...,kn)}theresultofapplyingk∈Znto{Bi},thenthefollowingequationsaresatisfied:Bi(...)−Bi(...,kj+1,...)=Bj(...)−Bj(...,ki+1,...),(3)B