文档介绍：该【Zeros of Gaussian Weyl–Heisenberg Functions and Hyperuniformity of Charge 2022 Antti Haimi 】是由【十二官】上传分享，文档一共【41】页，该文档可以免费在线阅读，需要了解更多关于【Zeros of Gaussian Weyl–Heisenberg Functions and Hyperuniformity of Charge 2022 Antti Haimi 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:.
JournalofStatisticalPhysics(2022)187:22
/s10955-022-02917-3
ZerosofGaussianWeylHeisenbergFunctionsand
HyperuniformityofCharge
AnttiHaimi1·GüntherKoliander2·JoséLuisRomero1,2
Received:8July2021/Accepted:21March2022/Publishedonline:15April2022
©TheAuthor(s)2022
Abstract
WestudyGaussianrandomfunctionsonthecomplexplanewhosestochasticsareinvariant
undertheWeylHeisenberggroup(twistedstationarity).Thetheoryismodeledontransla-
tioninvariantGaussianentirefunctions,butallowsfornon-analyticexamples,inwhichcase
setsofsuchfunctions,bothwhenconsideredaspointsontheplane,oraschargesaccord-
,chargesareshowntobeinacertainaverage
equilibriumindependentlyoftheparticularcovariancestructure(universalscreening).We
investigatethecorrespondingfluctuations,andshowthatinmanycasestheyaresuppressed
atlargescales(hyperuniformity).Thismeansthatuniversalscreeningisempiricallyobserv-
mainapplication,weobtainstatisticsforthezerosetsoftheshort-timeFouriertransformof
complexwhitenoisewithgeneralwindows,andalsoprovethefollowinguncertaintyprinci-
ple:theexpectednumberofzerosperunitareaisminimized,amongallwindowfunctions,
-entirefunctionssuchas
covariantderivativesofGaussianentirefunctions.
CommunicatedbySalvatoreTorquato.
AnttiHaimi,GüntherKolianderandJoséLuisRomerogratefullyacknowledgesupportfromAustrian
ScienceFund(FWF):Y1199,P31153,andP29462,andfromtheWWTFGrantINSIGHT(MA16-053).
PreliminaryversionsofthisworkwerepresentedbyAnttiHaimiatthemeetingofthemathematicsgroupof
thanktheinstitutemembersfortheirhelpfulcomments.
BJoséLuisRomero
.******@
AnttiHaimi
antti.******@
GüntherKoliander
******@
1FacultyofMathematics,UniversityofVienna,Oskar-Morgenstern-Platz1,1090Vienna,Austria
2AcousticsResearchInstitute,AustrianAcademyofSciences,Wohllebengasse12-14,1040Vienna,
Austria
123:.
.
KeywordsGaussianWeylHeisenbergfunction·Zeroset·Charge·Twistedconvolution·
Short-timeFouriertransform·Hyperuniformity
MathematicsSubjectClassiflcation60G15·60G55·94A12·42A61
1IntroductionandResults
TheinvestigationofzerosofrandomfunctionswithGaussiandistributionisaclassical
endeavorinstatisticalphysics,ingreatpartmotivatedbythegoaltoderivegenericmodels
forthedistributionofquantumchaoticsystems[7,8].Thisarticleisconcernedwithcomplex-
valuedrandomfunctionsonthecomplexplane,inwhichcasezerosaretypicallydiscrete,
andcorrespondtophasesingularities[9].WhilerandomfunctionsontheEuclideanplane
areoftenstudiedundertheassumptionofstationarity,thatis,stochasticinvarianceunder
Euclideanshifts,westudyaformofinvariancecompatiblewiththecomplexstructureofthe
plane,whichwecalltwistedstationarity.
Amodelcaseforourtheoryarerandompowerserieswithproperlyscaledindependent
Gaussianentirefunctions(TI-GEF),because,eventhoughtheyarenotstochasticallyinvariant
underEuclideanshifts,theirzerosare[24,29].Infact,duetotheso-calledCalabirigidity,
thesearetheonlyexamplesofGaussiananalyticfunctionsontheplanewithstationaryzero
sets[29].
ThenotionoftwistedstationaritythatweintroduceabstractssomepropertiesofTI-GEF
suchasstationarityofzeros,while,crucially,allowingfornon-,
ourmainmotivationisthestudyofcertainpossiblynon-analyticrandomfunctionssuchas
thecorrelationofwhitenoisewiththetime-frequencyshiftsofagivenreferencewindow
function(short-timeFouriertransformorcrossradarambiguity[17,][21,]).
Whilethefreedomtochooseareferencewindowfunctionisveryvaluableinapplications,
onlyonespeciflcchoiceleadstoGaussianentirefunctions[16,],[5,6].Further
motivationcomesfromthefactthatanoperationasbasicascomputingaderivativeofa
TI-GEF(inthesenseofcomplexgeometry)
istheidentiflcationofacommonelementinthepreviousexamples:stochasticinvariance
underacertainrepresentationoftheWeyl
unifledmodelforsuchsituation,andthecorrespondingrandomfunctionsarecalledGaussian
WeylHeisenbergfunctions(GWHF).
Poissonprocess,theyexhibitrepulsion,thatis,negativecorrelationsimilartothatofcharged
particlesofequalsign,andtheyarehyperuniform,inthesensethatthevarianceofthenumber
ofpointsinalargeobservationwindowisasymptoticallysmallerthanthecorresponding
expectedvalue[20,30].Inthenon-analyticsettingofGWHFweshallflndthenewelement
ofasignedcharge,since,asisthecasewithnon-analyticrandomwaves,zerosmayhave
negativewindingnumbers[9].
Wenowintroducethemainmathematicalobjectsandresults,andprovidecontextontheir
signiflcance.
123:.
ZerosofGaussianWeyl–HeisenbergFunctions...Page3of4122
–HeisenbergFunctions
WestudyzerosetsofGaussiancircularlysymmetricrandomfunctionsontheplaneF:C→
Cwhosecovariancekernelisgivenbytwistedconvolution:
EF(z)·F(w)=H(z−w)·ei(zw),z,w∈C.()
Here,H:C→
z1,...,zn∈C,(F(z1),...,F(zn))-
cularitymeansthatF∼eiθF,forallθ∈R,andimpliesthatFhasvanishingexpectation
andpseudo-covariance,.,E[F(z)]=0,E[F(z)F(w)]=0,forallz,w∈,the
stochasticsofFarecompletelyencodedinthetwistedkernel().
Whilethecovariancestructure()withoutthecomplexexponentialfactorwouldmean
thatFisstationary,thepresenceoftheoscillatoryfactormeansthatFistwistedstationary:
Eei(zζ)·F(z−ζ)·ei(wζ)·F(w−ζ)=EF(z)·F(w),z,w,ζ∈C.()
Inotherwords,thestochasticsofFareinvariantundertwistedshifts:
F(z)→ei(zζ)·F(z−ζ),ζ∈C.()
WecallsucharandomfunctionFaGaussianWeylHeisenbergfunction(GWHF),asthe
operators()generatethe(reduced)WeylHeisenberggroup[17,].Letusmention
somemotivatingexamples().
(Gaussianentirefunctions)LetFbeaGWHFwithtwistedkernelH(z)=
−1|z|21|z|2
e2andsetG(z)=e2F(z).Then
EG(z)·G(w)=exp1|z|2+1|w|2−1|z−w|2+i(zw)¯=ezw¯.
222
Hence,GisaGaussianentirefunctionontheplanewithcorrelationkernelgivenbythe
Bargmann-Fockkernel[35],anditszerosetiswell-studied[24].IntermsofG,thetwisted
stationaritypropertyofF()isaninstanceoftheprojectiveinvarianceproperty[29],and,
indeed,reflectstheinvarianceofthestochasticsofGunderBargmann-Fockshifts:
−1|ζ|2+zζ
G(z)→e2·G(z−ζ),ζ∈C.()
(Theshort-timeFouriertransformofcomplexwhitenoise)
Givenawindowfunctiong∈S(R),theshort-timeFouriertransformofafunction
f:R→Cis
Vgf(x,y)=f(t)g(t−x)e−2πitydt,(x,y)∈R2.()
R
Theshort-timeFouriertransformisawindowedFouriertransform,andthevalueVgf(x,y)
,oneoftenchooses
Gaussian,or,moregenerally,Hermitefunctions,astheseoptimizeseveralmeasuresrelated
toHeisenbergsuncertaintyprinciple.
Insignalprocessing,theshort-timeFouriertransformisoftenusedtoanalyzefunctions
(calledsignals)
importantroleinmanymodernalgorithms,forexampleinthedynamicsofcertainnon-
linearprocedurestosharpenspectrograms[16,]orinthedesignoffllteringmasks,
wherelandmarksarechosenguidedbythestatisticsofzerosets[15].
123:.
.
Ofparticularinterestarethezerosoftheshort-timeFouriertransformofcomplexwhite
GaussiannoiseVgN[16,].Whilemanyapplicationsdemandtheuseofdifferent
windowfunctionsgsee,.,[16,]zero-statisticsfortheSTFTarecurrently
onlyunderstoodforGaussianwindows[5,6],asthesefacilitatetheapplicationofthetheory
ofGaussianentirefunctions().Onemainmotivationforthisarticleistoobtain
zero-statisticsforgeneralwindowsg,includingforexampleHermitefunctions.(Somerelated
numericscanbefoundin[16,].)
Withanadequatedistributionalinterpretation,theSTFTofcomplexwhiteGaussiannoise
withrespecttoaSchwartzwindowfunctiongdeflnesasmoothcircularlysymmetricGaussian
−ixy√√
F(x+iy):=e·VgNx/π,−y/π,()
which,,
theinvarianceofthestochasticsofcomplexwhitenoiseundertime-frequencyshifts
f(t)→e2πibtf(t−a),(a,b)∈R2.
Basicquestionsaboutzerosetsofshort-timeFouriertransformsalsounderlieproblemsabout
thespanningpropertiesofthetime-frequencyshiftsofagivenfunction(Gaborsystems)[23,
26],oraboutBerezinsquantization[22].Thestudyofrandomcounterpartsprovidesaflrst
formofaveragecaseanalysisforsuchproblems.
(DerivativesofGEF)ThecovariantderivativeofanentirefunctionG:C→C
is
∂¯∗G(z)=¯zG(z)−∂G(z),
anditisdistinguishedamongotherdifferentialoperatorsoforder1becauseitcommutes
withtheBargmannFockshifts().Asaconsequence,ifGisaGaussianentirefunction,
,thestochasticsof∂¯∗GarealsoinvariantunderBargmannFockshifts,
andthetransformation
−1|z|2¯∗
F(z)=e2∂G(z)()
.
Zerosofcovariantderivativesareinstrumentalinthedescriptionofvanishingordersof
analyticfunctions[11,13].Theyarealsoimportantinthestudyofweightedmagnitudesof
−1|z|2
,theamplitudeA(z)=e2|G(z)|ofanentirefunction
Gsatisfles
−1|z|2∗
∇A=e2∂¯G.()
Thus,thecriticalpointsoftheamplitudeofaGaussianentirefunctionGareexactlythezeros
oftheGWHF()seealso[12,14].ThesquaredamplitudeA2(z)isalsoofinterest,asit
correspondsafternormalizationtothespectrogramofcomplexwhitenoisewithaGaussian
window(.,thesquaredabsolutevalueoftheSTFT())[5];seealso[6,].
(Gaussianpoly-entirefunctions)Iteratedcovariantderivativesofananalytic
functionG0,
G=(∂¯∗)q−1G,()
0
123:.
ZerosofGaussianWeyl–HeisenbergFunctions...Page5of4122
arenotthemselvesanalytic,butsatisfyahigherorderCauchy-Riemanncondition
∂¯qG=0,()
knownaspoly-analyticity[4].InVasilevskisparlance[33],()isatrueorpurepoly-entire
function,whilethemoregeneralsolutionto(),
q−1
1¯∗k
G=√(∂)Gk,()
k=0k!
withG0,...,Gq−1entire,isafullypoly-entirefunction.
RandomGaussianpoly-entirefunctionsofeitherpureoffulltypearedeflnedby()and
(),lettingG0,...,Gq−1beindependentGaussianentirefunctions.
Poly-entirefunctionsareimportantinstatisticalphysics,intheanalysisofhighenergy
systemsofparticles[2],andweexpecttheirrandomanalogstoalsobeusefulinthatfleld.
Thepositivesemi-deflnitenessofthecovariancekernelofaGWHFFreadsasfollows:1
H(zk−zj)·ei(zkzj)≥0forallz1,...,zn∈C.()
j,k=1,...,n
Asaconsequence,H(−z)=H(z)andH(0)≥,weassumethat
H(0) =0,since,otherwise,
normalization
H(0)=1.()
Wealsoassumethat
|H(z)|<1,z∈C\{0},()
whichmeansthatnotwosamplesF(z),F(w)withz =waredeterministicallycorrelated,as
1H(z−w)ei(zw)¯
()amountstotheinvertibilityofthejointcovariancematrixH(z−w)e−i(zw)¯1.
Wealsoassumeacertainregularityofthetwistedkernel:
HisC2intherealsense,()
anddenotethecorrespondingderivativeswithsupraindices;.,H(1,1)(x+iy)=
∂x∂yH(x+iy).Finally,wewillalwaysassumethatFhasC2pathsintherealsense:
AlmosteveryrealizationofFisaC2(R2)function.()
Thisisthecase,forexample,ifH∈C6(R2)intherealsense[19,Theorem5],butalsoother
weakerassumptionssufflce(see[1,]).
(twistedkernel)H:C→Cissaidtosatisfythestandingassump-
tionsi