Introduction to the theory of Gibbs point processes

0.31 MB·

by  David Dereudre

Introduction to the theory of Gibbs point
processes
DEREUDREDavid
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2 Abstract The Gibbs point processes (GPP) constitute a large class of point pro-

cesseswithinteractionbetweenthepoints.Theinteractioncanbeattractive,repul-
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R sive, dependingon geometricalfeatures whereas the null interaction is associated
P totheso-calledPoissonpointprocess.Inafirstpartofthismini-course,wepresent
. several aspects of finite volume GPP defined on a bounded window in Rd. In a
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secondpart,weintroducethemorecomplicatedformalismofinfinitevolumeGPP
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a defined on the full space Rd. Existence, uniqueness and non-uniqueness of GPP
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arenon-trivialquestionswhichwetreatherewithcompletelyself-containedproofs.
[ TheDLRequations,theGNZequationsandthevariationalprinciplearepresented

1 as well. Finally, in a last part, we investigate the estimation of parameters. The
v main standard estimators (MLE, MPLE, Takac-Fiksel and variational estimators)
5 arepresentedandweprovetheirconsistency.Forsakeofsimplicity,duringallthe
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mini-course,weconsideronlythecaseoffiniterangeinteractionandthesettingof
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markedpointsisnotpresented.
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0 Introduction
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: Thespatialpointprocessesarewellstudiedobjectsinprobabilitytheoryandstatis-
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i tics for modelling and analysing spatial data which appear in several disciplines
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asstatisticalmechanics,materialscience,astronomy,epidemiology,plantecology,
r seismology, telecommunications, and others [1, 11]. There exist many models of
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suchrandompointsconfigurationsinspaceandthemostpopularoneissurelythe
Poissonpointprocess.Itcorrespondstothenaturalwaytoproduceindependentlo-
cationsofpointsinspacewithoutinteraction.Fordependentrandomstructure,we
can mentionfor instance the Cox processes, determinantalpointprocesses, Gibbs
pointprocesses,etc.Noneofthemisestablishedasthemostrelevantmodelforap-
DEREUDREDavid,UniversityLille1,e-mail:[email protected]
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plications.Actuallythechoiceofthemodeldependsonthenatureofthedataset,the
knowledgeof(physicalorbiological)mechanismsproducingthepattern,theaimof
thestudy(theoretical,appliedornumerical).
Inthismini-course,wefocusonGibbspointprocesses(GPP)whichconstitutea
largeclassofpointsprocesses,abletofitseveralkindsofpatternsandwhichprovide
aclearinterpretationoftheinteractionbetweenthepoints,suchasattractionorre-
pulsiondependingontheirrelativeposition.Notethatthisclassisparticularlylarge
sinceseveralpointprocessescanberepresentedasGPP(see[23,32]forinstance).
ThemaindisadvantageofGPPisthecomplexityofthemodelduetoanintractable
normalizedconstantwhichappearinthelocalconditionaldensities.Thereforetheir
analyticalstudiesareingeneralbasedonimplicitequilibriumequationswhichlead
to complicated and delicate analysis. Moreover the theoretical results, which are
needed to investigate the Gibbs point process theory, are scattered across several
publications or books. The aim of this mini-course is to provide a solid and self-
containedtheoreticalbaseforunderstandingdeeplytheGibbspointprocesstheory.
Theresultsareingeneralnotexhaustivebutthemainideasandtoolsarepresented
infollowingmodernandrecentdevelopments.Themainstrongrestrictionherein-
volvestherangeoftheinteractionwhichisassumedtobefinite.Theinfiniterange
interactionrequirestheintroductionoftemperedconfigurationspacesandforsake
of simplicity we decidedto avoid this levelof complexity.The mini-courseis ad-
dressedforMasterandPhdstudentsandalsoforresearcherswhichwanttodiscover
or investigatethe domain.The manuscriptis based on a minicourse givenduring
theconferenceofGDR3477ge´ome´triestochastique,atuniversityofNantesinApril
2016.
In a first section, we introduce the finite volume GPP on a bounded window
L Rd. They are simply definedas pointprocesses inL whose the distributions
⊂
areabsolutelycontinuouswithrespecttothePoissonpointprocessdistribution.The
unnormalized densities have the following form zNe b E where z and b are posi-
−
tiveparameters(calledrespectivelyactivityandinversetemperature),Nthenumber
ofpointsandE anenergyfunction.Clearlythesedistributionsfavour(orpenalize)
configurationswithlow(orhigh)energyE.Thisdistortionstrengthensasb islarge.
Theparameterzallowstotunethemeannumberofpoints.Thissettingisrelatively
simplesincealltheobjectsaredefinedexplicitly.Howevertheintractablenormal-
izationconstantiseveraproblemandmostofquantitiesarenotcomputable.Several
standard notions (DLR and GNZ equations, Ruelle’s estimates, etc) are treated in
thisfirstsectionasapreparationforthemorecomplicatedsettingofinfinitevolume
GPP developed in the second section. Note that we do not present the setting of
markedGibbspointprocessesinordertokeepthenotationsassimpleaspossible.
Howeveralltheresultscanbeeasilyextendedinthiscase.
Inasecondsection,we presentthetheoryofinfinitevolumeGPPinRd.There
areseveralmotivationsforstudyingsuchinfinitevolumeregime.First,theGPPare
thestandardmodelsinstatisticalphysicsformodellingsystemswithalargenumber
of interactingparticles(around1023 accordingto the Avogadro’snumber).There-
fore the case, where the number of particles is infinite, is an idealization of this
setting and furnishes microscopic descriptions of gas, liquid or solid. The macro-
IntroductiontothetheoryofGibbspointprocesses 3
scopicquantitiesasthedensityofparticles,thepressure,themeanenergyarecon-
sequentlyeasilydefinedbymeanvaluesorlawsoflargenumbers.Secondly,inthe
spatialstatisticcontext,theasymptoticpropertiesofestimatorsortestsareobtained
whentheobservationwindowtendstothefullspaceRd.Thisstrategyrequiresthe
existenceofinfinitevolumemodels.Finally,sincetheinfinitevolumeGPParesta-
tionary(shiftinvariant)inRd,severalpowerfultools,astheergodictheoremorthe
centrallimitTheoremformixingfield,areavailableinthisinfinitevolumeregime.
TheinfinitevolumeGibbsmeasuresaredefinedbyacollectionofimplicitDLR
equations (Dobrushin, Lanford and Ruelle). The existence, uniqueness and non-
uniqueness are non trivial questions which we treat deeply with self-contained
proofs in this second section. The phase transition between uniqueness and non
uniquenessis one ofthe mostdifficultconjecturesin statistical physics.Thisphe-
nomenon is expected to occur for all standard interactions although it is proved
rigorouslyonlyforfewmodels.Theareainteractionisoneofsuchmodelandthe
completeproofofits phasetransitionisgivenin themini-course.TheGNZequa-
tions,thevariationalprinciplearediscussedaswell.
Inthelast section,we investigatetheestimationof parameterswhichappearin
the distribution of GPP. For sake of simplicity we deal only with the activity pa-
rameter z and the inverse temperature b . We present several standard procedures
(MLE,MPLE,Takacs-Fikselprocedure)andanewvariationalprocedure.Weshow
theconsistencyofestimators,whichhighlightsthatmanytheoreticalresultsarepos-
sibleinspiteoflackofexplicitcomputations.WewillseethattheGNZequations
playacrucialroleinthistask.Forsakeofsimplicitytheasymptoticnormalityisnot
presentedbutsomereferencesaregiven.
Letusfinishthisintroductionbygivingstandardreferences.HistoricallytheGPP
have been introduced for statistical mechanics considerations and an unovoidable
reference is the book by Ruelle [46]. Important theoretical contributions are also
developedintwo LectureNotes[19,45]byGeorgiiandPreston.Fortherelations
between GPP and the stochastic geometry, we can mention the book [8] by Chiu
etal.andforspatialstatisticandnumericalconsiderations,thebookbyMøllerand
Waagepetersen[41]isthestandardreference.Letusmentionalsothebook[50]by
vanLieshoutontheapplicationsofGPP.

Contents
IntroductiontothetheoryofGibbspointprocesses ................... 1
DEREUDREDavid
1 FinitevolumeGibbspointprocesses ......................... 6
1.1 Poissonpointprocess .............................. 6
1.2 Energyfunctions .................................. 7
1.3 FiniteVolumeGPP ................................ 10
1.4 DLRequations.................................... 12
1.5 GNZequations.................................... 13
1.6 TheRuelle’sestimates ............................. 15
2 InfinitevolumeGibbspointprocesses ........................ 16
2.1 Thelocalconvergencesetting ....................... 17
2.2 AnaccumulationpointPz,b ......................... 18
2.3 ThefiniteRangeproperty........................... 20
2.4 DLRequations.................................... 21
2.5 GNZequations.................................... 23
2.6 Variationalprinciple ............................... 25
2.7 Anuniquenessresult............................... 27
2.8 Anon-uniquenessresult............................ 30
3 Estimationofparameters. .................................. 35
3.1 Maximumlikelihoodestimator ...................... 35
3.2 Takacs-Fikselestimator ............................ 37
3.3 Maximumpseudo-likelihoodestimator ............... 40
3.4 Solvinganunobservableissue....................... 41
3.5 Avariationalestimator ............................. 43
References..................................................... 46
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6 Contents
1 FinitevolumeGibbs pointprocesses
Inthisfirstsection wepresentthe theoryofGibbspointprocessona boundedset
L Rd.AGibbspointprocess(GPP)isapointprocesswithinteractionsbetween
⊂
thepointsdefinedviaanenergyfunctionalonthespaceofconfigurations.Roughly
speakingtheGPPproducesrandomconfigurationsforwhichtheconfigurationswith
low energy have more chance to appear than the configurationswith high energy
(seedefinition2).InafirstSection1.1werecallsuccinctlysomedefinitionsofpoint
processtheoryandweintroducedthereferencePoissonpointprocess.Theenergy
functions are discussed in Section 1.2 and the definiton of finite volume GPP is
giveninSection1.3.Somefirstpropertiesarepresentedaswell.ThecentralDLR
equationsandGNZequationsaretreatedinSections1.4and1.5.Finallywefinish
the first section by giving the Ruelle’s estimates in the setting of superstable and
lowerregularenergyfunctions.
1.1 Poissonpointprocess
In this first section, we describe succinctlythe setting of pointprocesstheoryand
weintroducethereferencePoissonpointprocess.Wegiveonlythemaindefinitions
andconceptsandweadvice[11,36]forageneralpresentation.
ThespaceofconfigurationsC isdefinedasthesetoflocallyfinitesubsetsinRd;
C = g Rd,gL :=g L isfiniteforanyboundedsetL Rd .
{ ⊂ ∩ ⊂ }
Notethatweconsideronlythesimplepointconfigurations,whichmeansthatthe
pointsdonotoverlap.WedenotebyC thespaceoffiniteconfigurationsinC and
f
CL isthespaceoffiniteconfigurationsinsideL Rd.
The space C is equipped with the sigma-fiel⊂d FC generating by the counting
functionsNL forallboundedmeasurableL Rd,whereNL :g #gL .Soapoint
processG issimplyameasurablefunctionfr⊂omanyprobabilitys7→pace(W ,F,P)to
(C,FC). Asusual,thedistribution(orthelaw)ofapointprocessG isdefinedby
theimageofPto(C,FC)bytheapplicationG .WesaythatG hasafiniteintensity
ifforanyboundedsetL ,thefollowingexpectationm (L ):=E(NL (G ))isfinite.In
thiscase,m isasigma-finitemeasurecalledintensitymeasureofG .Whenm =zl d,
wherel d istheLebesguemeasureonRd andz 0apositivereal,wesaysimply
thatG hasafiniteintensityz . ≥
ThemainclassofpointprocessesisthefamilyofPoissonpointprocesseswhich
furnishthenaturalwaytoproduceindependentpointsinspace.Let m beasigma-
finitemeasureinRd.APoissonpointprocesswithintensitym isapointprocessG
suchthatforanyboundedL inRd,thesebothpropertiesoccur
TherandomvariableNL (G )isdistributedfollowingaPoissondistributionwith
• parameterm (L ).
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Giventheevent NL (G )=n thenpointsinGL areindependentanddistributed
• followingthedis{tributionm L }/m (L ).
The distribution of such Poisson pointprocess is denotedby p m . When the in-
tensity m =zl d, we say that the Poisson point process is stationary (or homoge-
neous)with intensityz >0.Itsdistributionisdenotedbyp z .Foranymeasurable
setL Rd, we denote by p z the distribution of a Poisson pointprocess with in-
L
tensity⊂zl d whichisalsothedistributionofastationaryPoissonpointprocesswith
L
intensityz restrictedtoL .Forsakeofbrevity,p andp L denotethedistributionof
Poissonpointprocesseswithintensityz =1.
1.2 Energy functions
Inthissection,wepresenttheenergyfunctionswiththestandardassumptionswhich
weassumeduringalltheminicourse.Thechoicesofenergyfunctionscomefrom
two main motivations. First, the GPP are natural models in statistical physics for
modellingcontinuuminteractingparticlessystems. In general,in these setting the
energyfunctionisasumoftheenergycontributionofallpairsofpoints(seeexpres-
sion(1)).TheGPParealsousedinspatialstatistictofitthebestaspossiblethereal
datasets. So, in a first step, the energyfunctionis chosen by the user with respect
tothecharacteristicsofthedataset.Thentheparametersareestimatedinasecond
step.
Definition1. Anenergyfunctionisameasurablefunction
H:C R +¥
f
7→ ∪{ }
suchthatthefollowingassumptionshold
H isnon-degenerate:
• H(0/)<+¥ .
Hishereditary:foranyg C andx g then
f
• ∈ ∈
H(g )<+¥ H(g x )<+¥ .
⇒ \{ }
H isstable:thereexistsaconstantAsuchthatforanyg C
f
• ∈
H(g ) ANRd(g ).
≥
The stability implies that the energyis superlinear. If the energy function H is
positivethenthechoiceA=0worksbutintheinterestingcases,theconstantAis
negative.Thehereditarymeansthatthesetofallowedconfigurations(configurations
withfiniteenergy)isstable whenpointsareremoved.Thenon-degeneracyisvery
natural.Withoutthisassumption,theenergywouldbeequaltoinfinityeverywhere
8 Contents
(byhereditary).
1)Pairwise interaction.Letusstartbythemostpopularenergyfunctionwhich
isbasedonafunction(calledpairpotential)
j :R+ R +¥ .
→ ∪{ }
Thepairwiseenergyfunctionisdefinedforanyg C by
f
∈
H(g )= (cid:229) j (x y). (1)
| − |
x,y g
{ }⊂
Note that such energy function is trivially hereditary and non-degenerate. The
stability is more delicate and we refer to general results in [46]. However if j is
positivetheresultisobvious.
A standard example coming from statistical physics is the so-called Lennard-
Jones pair potential where j (r)=ar 12+br 6 with a>0 and b R. In the in-
− −
teresting case b<0, the pair potentialj (r) is positive (repulsive)f∈or small r and
negative(attractive)forlarger.ThestabilityisnotobviousandisprovedinPropo-
sition3.2.8in[46].
TheStraussinteractioncorrespondsto thepairpotentialj (r)=1 (r) where
[0,R]
R>0isasupportparameter.Thisinteractionexhibitsaconstantrepulsionbetween
theparticlesasdistancesmallerthanR.Thissimplemodelisverypopularinspatial
statistics.
Themulti-Straussinteractioncorrespondstothepairpotential
k
j (r)=(cid:229) a1 ,
i ]Ri 1,Ri]
i=1 −
where (a) is a sequence of real numbersand 0=R <R <...<R a se-
i 1 i k 0 1 k
quence of in≤c≤reasing real numbers. Clearly, the pair potential exhibits a constant
attractionorrepulsionatdifferentscales.TheStabilityoccursprovidedthatthepa-
rametera islargeenough[46].
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2)Energyfunctionscomingfrom geometricalobjects.Severalenergyfunctions
are based on local geometrical characteristics. The main motivation is to provide
random configurations such that special geometrical features appear with higher
probabilityundertheGibbsprocessesthantheoriginalPoissonpointprocess.Inthis
paragraphwe give examplesrelated to the Delaunay-Voronoidiagram. Obviously
othergeometricalgraphstructurescouldbeconsidered.
Letusrecallthatforanyx g C theVoronoicellC(x,g )isdefinedby
f
∈ ∈
C(x,g )= w Rd, suchthat y g x w x y .
∈ ∀ ∈ | − |≤| − |
n o
TheDelaunaygraphwithverticesg isdefinedinconsideringtheedges
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D(g )= x,y g suchthatC(x,g ) C(y,g )=0/ .
{ }⊂ ∩ 6
n o
See[39]forageneralpresentationontheDelauany-Voronoitessellations.
Afirstgeometricenergyfunctioncanbedefinedby
H(g )= (cid:229) 1C(x,g)isboundedj (C(x,g )), (2)
x g
∈
wherej isanyfunctionfromthespaceofpolytopesinRd toR.Examplesofsuch
functionsj aretheArea,the(d 1)-Hausdorffmeasureoftheboundary,thenum-
−
beroffaces,etc...Clearlytheseenergyfunctionsarenon-degenerateandhereditary.
Thestabilityholdsassoonasthefunctionj isboundedfrombelow.
Another kind of geometric energy function can be constructed via a pairwise
interactionalongtheedgesoftheDelaunaygraph.Letusconsiderafinitepairpo-
tentialj :R+ R.Thentheenergyfunctionisdefinedby
7→
H(g )= (cid:229) j (x y) (3)
| − |
x,y D(g)
{ }⊂
whichisagainclearlynon-degenerateandhereditary.TheStabilityoccursindi-
mension d =2 thanks to the Euler’s formula. Indeed the number of edges in the
Delaunaygraphislinear withrespectto thenumberof vertices.Thereforethe en-
ergy function is stable as soon as the pair potential j is bounded from below. In
higherdimensiond>2,thestabilityismorecomplicatedandnotveryunderstood.
Obviously,ifj ispositive,thestabilityoccurs.
Let us give a last exampleof geometricenergyfunctionwhich is notbased on
theDelaunay-Voronoidiagrambutonagerm-grainstructure.ForanyradiusR>0
wedefinethegerm-grainstructureofg C by
∈
L (g )= B(x,R),
R
x g
[∈
where B(x,R) is the closed ball centredat x with radius R. Severalinteresting en-
ergyfunctionsarebuiltfromthisgerm-grainstructure.FirsttheWidom-Rowlinson
interactionissimplydefinedby
H(g )=Area(L (g )), (4)
R
wherethe”Area”issimplytheLebesguemeasurel d.Thismodelisverypopular
since itisoneofafew modelsforwhichthephasetransitionresultisproved(see
Section 2.8).Thisenergyfunctionis sometimescalled Area-interaction[4, 51]. If
theAreafunctionalisreplacedbyanylinearcombinationoftheMinkowski’sfunc-
tionalsweobtaintheQuermassinteraction[13].
Anotherexampleistherandomclusterinteractiondefinedby
H(g )=Ncc(L (g )), (5)
R
10 Contents
where Ncc denotes the functionalwhich counts the number of connected compo-
nents.Thisenergyfunctionisintroducedfirstin[7]foritsrelationswiththeWidom-
Rowlinsonmodel.Seealso[16]forageneralstudyintheinfinitevolumeregime.
1.3 FiniteVolumeGPP
LetL Rd suchthat0<l d(L )<+¥ .Inthissectionwedefinethefinitevolume
GPPo⊂nL andwegiveitsfirstproperties.
Definition2. The finite volume Gibbs measure on L with activity z>0, inverse
temperatureb 0andenergyfunctionH isthedistribution
≥
PLz,b = Z1z,b zNL e−b Hp L , (6)
L
whereZLz,b ,calledpartitionfunction,isthenormalizationconstant zNL e−b Hdp L .
A finite volumeGibbspointprocess(GPP) onL with activity z>0, inversetem-
R
perature b 0 and energy function H is a point process on L with distribution
Pz,b . ≥
L
Note that Pz,b is well-defined since the partition function Zz,b is positive and
L L
finite.Indeed,thankstothenondegeneracyofH
ZLz,b p L (0/)e−b H({0/})=e−l d(L )e−b H({0/})>0
≥
andthankstothestabilityofH
ZLz,b e−l d(L )+(cid:229) ¥ (ze−b Al d(L ))n =el d(L )(ze−bA−1)<+¥ .
≤ n!
n=0
Inthe case b =0, we recoverthatPz,b isthe Poisson pointprocessp z. So the
L L
activityparameterzisthemeannumberofpointsperunitvolumewhentheinterac-
tionisnull.Whentheinteractionisactive(b >0),Pz,b favourstheconfigurations
L
withlowenergyandpenalizestheconfigurationswithhighenergy.Thisdistortion
strengthensasb islarge.
Therearemanymotivationsfortheexponentialformofthedensityin(6).Histor-
icallyitisduetothefactthatthefinitevolumeGPPsolvesthevariationalprinciple
ofstatisticalphysics.IndeedPz,b istheuniqueprobabilitymeasurewhichrealizethe
L
minimumofthefreeexcessenergyequalstothemeanenergyplustheentropy.Itex-
pressesthecommonideathattheequilibriumstatesinstatisticalphysicsminimize
theenergyandmaximizesthe”disorder”.Thisresultispresentedin thefollowing
proposition.RecallfirstthattherelativeentropyofaprobabilitymeasurePonCL
withrespecttothePoissonpointprocessp z isdefinedby
L