RANDOM PLANAR LATTICES AND INTEGRATED SUPERBROWNIAN EXCURSION

Chassaing

Description

Niveau: Supérieur, Doctorat, Bac+8
RANDOM PLANAR LATTICES AND INTEGRATED SUPERBROWNIAN EXCURSION PHILIPPE CHASSAING AND GILLES SCHAEFFER Abstract. In this paper, a surprising connection is described between a spe- cific brand of random lattices, namely planar quadrangulations, and Aldous' Integrated SuperBrownian Excursion (ISE). As a consequence, the radius rn of a random quadrangulation with n faces is shown to converge, up to scaling, to the width r = R?L of the support of the one-dimensional ISE, or precisely: n?1/4rn law?? (8/9)1/4 r. More generally the distribution of distances to a random vertex in a random quadrangulation is described in its scaled limit by the random measure ISE shifted to set the minimum of its support in zero. The first combinatorial ingredient is an encoding of quadrangulations by trees embedded in the positive half-line, reminiscent of Cori and Vauquelin's well labelled trees. The second step relates these trees to embedded (discrete) trees in the sense of Aldous, via the conjugation of tree principle, an analogue for trees of Vervaat's construction of the Brownian excursion from the bridge. From probability theory, we need a new result of independent interest: the weak convergence of the encoding of a random embedded plane tree by two contour walks (e(n), W (n)) to the Brownian snake description (e, W ) of ISE.

• random planar


• random quadrangulations


• continuum limit


• freely embedded trees


• qn-valued random variable


• discrete trees


• trees


• aldous' prescription


• let ln

Sujets

• Ise Ekiti

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 8
• Langue: English

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RANDOMPLANARLATTICESANDINTEGRATED
SUPERBROWNIANEXCURSION

PHILIPPECHASSAINGANDGILLESSCHAEFFER

Abstract.
Inthispaper,asurprisingconnectionisdescribedbetweenaspe-
cificbrandofrandomlattices,namelyplanarquadrangulations,andAldous’
IntegratedSuperBrownianExcursion(ISE).Asaconsequence,theradius
r
n
ofarandomquadrangulationwith
n
facesisshowntoconverge,uptoscaling,
tothewidth
r
=
R
−
L
ofthesupportoftheone-dimensionalISE,orprecisely:
n
−
1
/
4
r
n
l
−
aw
→
(8
/
9)
1
/
4
r.
Moregenerallythedistributionofdistancestoarandomvertexinarandom
quadrangulationisdescribedinitsscaledlimitbytherandommeasureISE
shiftedtosettheminimumofitssupportinzero.
Thefirstcombinatorialingredientisanencodingofquadrangulationsby
treesembeddedinthepositivehalf-line
,reminiscentofCoriandVauquelin’s
welllabelledtrees.Thesecondsteprelatesthesetreestoembedded(discrete)
treesinthesenseofAldous,viathe
conjugationoftreeprinciple
,ananalogue
fortreesofVervaat’sconstructionoftheBrownianexcursionfromthebridge.
Fromprobabilitytheory,weneedanewresultofindependentinterest:the
weakconvergenceoftheencodingofarandomembeddedplanetreebytwo
contourwalks(
e
(
n
)
,W
ˆ
(
n
)
)totheBrowniansnakedescription(
e,W
ˆ)ofISE.
Ourresultssuggesttheexistenceofa
ContinuumRandomMap
describing
intermofISEthescaledlimitofthedynamicaltriangulationsconsideredin
two-dimensionalpurequantumgravity.

1.
Introduction
Fromadistantperspective,thisarticleuncoversasurprising,andhopefullydeep,
relationbetweentwofamousmodels:
randomplanarmaps
,asstudiedincombina-
toricsandquantumphysics,and
Browniansnakes
,asstudiedinprobabilitytheory
andstatisticalphysics.Moreprecisely,ourresultsconnectsomedistance-related
functionalsof
randomquadrangulations
withfunctionalsofAldous’
IntegratedSu-
perBrownianExcursion
(ISE)indimensionone.
Quadrangulations.
Ontheonehand,quadrangulationsarefiniteplanegraphs
with4-regularfaces(seeFigure1andSection2forprecisedefinitions).Random
quadrangulations,likerandomtriangulations,randompolyhedra,orthe
φ
4
-models
ofphysics,areinstancesofageneralfamilyofrandomlatticesthathasreceived
considerableattentionbothincombinatorics(underthename
randomplanarmaps
,
followingTutte’sterminology[35])andinphysics(underthename
Euclideantwo-
dimensionaldiscretisedquantumgeometry
,orsimply
dynamicaltriangulations
or
fluidlattices
[3,10,18]).
Manyprobabilisticpropertiesofrandomplanarmapshavebeenstudied,that
are
localproperties
likevertexorfacedegrees[7,17],or0
−
1lawsforproperties
expressibleinfirstorderlogic[8].Otherwelldocumentedfamiliesofpropertiesare
relatedtoconnectednessandconstantsizeseparators[6],alsoknownasbranchings
1

2

CHASSAINGANDSCHAEFFER

Figure1.
Randomquadrangulations,inplanarorsphericalrepresentation.
intobabyuniverses[21].Inthisarticleweconsideranotherfundamentalaspectof
thegeometryofrandommaps,namely
globalpropertiesofdistances
.The
profile
(
H
kn
)
k
≥
0
and
radius
r
n
ofarandomquadrangulationwith
n
facesaredefinedin
analogywiththeclassicalprofileandheightoftrees:
H
kn
isthenumberofverticesat
distance
k
fromabasepoint,while
r
n
isthemaximaldistancereached.Ithadbeen
acceptedinphysicssincethe80’sthatforsuchdiscretegeometries
theinternal
Hausdorffdimensionis4
,butexactcomputationsabouttheprofile(withtrian-
gulationsinsteadofquadrangulations)werefirstobtainedbyphysicistsWatabiki,
Ambjørnet
al.
(see[4,36]andref.therein).Understrong(unproven)smoothness
assumptionsonthebehavioroftheprofile,theyobtainthattheonlypossiblescal-
ingisindeed
k
∼
n
1
/
4
,andtheycomputeatransformoftheprofileinthescaling
limit
1
.Althoughnonrigourous(duetoapproximationsandlimitsinterchanges),
theseresultsareremarkablypreciseandbeautifullyderived.Independentlythe
conjecturethat
E
(
r
n
)
∼
cn
1
/
4
wasproposedbySchaeffer[31].
IntegratedSuperBrownianExcursion.
Ontheotherhand,ISEwasintroduced
byAldousasamodelofrandomdistributionsofmasses[1].Heconsidersrandom
embeddeddiscretetreesasobtainedbythefollowingtwosteps:firstanabstract
tree
t
,sayaCayleytreewith
n
nodes,istakenfromtheuniformdistributionand
eachedgeof
t
isgivenlength1;then
t
isembeddedintheregularlatticeon
Z
d
,
withtherootattheorigin,andedgesofthetreerandomlymappedonedgesofthe
lattice.Assigningmassestoleavesofthetree
t
yieldsarandomdistributionofmass
on
Z
d
.Uponscalingthelatticeto
n
−
1
/
4
Z
d
,theserandomdistributionsofmass
admit,for
n
goingtoinfinity,acontinuumlimit
J
whichisarandomprobability
measureon
R
d
calledISE.
DerbezandSladeprovedthatISEdescribesindimensionlargerthan8the
continuumlimitofamodeloflatticetrees[16],whileHaraandSladeobtainedthe
samelimitfortheincipientinfiniteclusterinpercolationindimensionlargerthan
6[19].Asopposedtotheseworks,weshallconsiderISEindimensiononeandour
embeddeddiscretetreesshouldbethoughtofasfoldedonaline.Thesupportof
ISEisthenarandominterval(
L,R
)of
R
thatcontainstheorigin.
FromquadrangulationstoISE.
Thepurposeofthispaperistodrawarela-
tionbetween,ontheonehand,randomquadrangulations,and,ontheotherhand,
1
Morepreciselytheyobtainthatthesingularbehaviorofthefunction

n
≥
k
H
k
(
n
)
z
n
nearits
dominantsingularity
z
0
is
ct
3
/
4
cosh(
t
1
/
4
k
)
/
sinh(
t
1
/
4
k
)with
c,c

constantsand
t
=
c

(1
−
z/z
0
).

ERANDOMMAPSANDISE3
Aldous’ISE:uponproperscaling,theprofileofarandomquadrangulationsisde-
scribedinthelimitbyISEtranslatedtohavesupport(0
,R
−
L
).Thisrelation
impliesinparticularthattheradius
r
n
ofrandomquadrangulations,againupon
scaling,weaklyconvergestothewidthofthesupportofISEinonedimension,that
isthecontinuousrandomvariable
r
=
R
−
L
.Weshallindeedprove(Corollary3)
tahtn
−
1
/
4
r
n
l
−
a
→
w
(8
/
9)
1
/
4
r,
aswellastheconvergenceofmoments.Thisprovestheconjecture
E
(
r
n
)
∼
cn
1
/
4
.
Inthefirstversionofthispaper,thevalueoftheconstant
c
wasquotedasunknown.
Inthemeantime,Delmas[14]hascomputedamomentgeneratingfunctionof
R
andthefirstandsecondmomentof
r
,givinginparticular
2
3
/
4
Γ(5
/
4)
(
r
)=6
√
.
πWeshallmoregenerallyprove(Corollary4)thatthenormalizedcumulatedpro-
file,thatisthefractionofthenumberofverticesatdistanceatmost
xn
1
/
4
ofthe
basepoint,convergestothefractionofthemassoftheISEintheinterval(0
,x
),
whenISEisshiftedtohavesupport(0
,R
−
L
).
ThepathfromquadrangulationstoISEconsistsofthreemainsteps,thefirsttwo
ofcombinatorialnatureandthelastwithamoreprobabilisticflavor.Ourfirststep,
Theorem1,revisitsacorrespondenceofCoriandVauquelin[13]betweenplanar
mapsandsome
welllabelledtrees
,thatcanbeviewedasplanetreesembeddedin
thepositivehalf-line.Thankstoanalternativeconstruction[31,Ch.7],weshow
thatunderthiscorrespondencetheprofilecanbemappedtothemassdistribution
onthehalf-line.Inparticular,theradius
r
n
ofarandomquadrangulationisequal
inlawtothemaximallabel
µ
n
ofarandomwelllabelledtree.
Safeforthepositivitycondition,welllabelledtreeswouldbeconstructedexactly
accordingtoAldous’prescriptionforembeddeddiscretetrees.Welllabelledtrees
arethustoAldous’embeddedtreeswhattheBrownianexcursionistotheBrownian
bridge,andweseekananalogueofVervaat’srelation.Atthediscretelevela
classicalelegantexplanationofsuchrelationsisbasedonDvoretskyandMotzkin’s
cyclicshiftsandcyclelemma(seealsoOtter).Oursecondcombinatorialstep,
Theorem3,consistsintheadaptationoftheseideastoembeddedtrees.More
precisely,viathe
conjugationoftreeprinciple
of[31,Chap.2],weboundthe
discrepancybetweenthemassdistributionofourconditionedtreesonthepositive
half-lineandatranslatedmassdistributionoffreelyembeddedtrees.Inparticular
weconstructacouplingbetweenwelllabelledtreesandfreelyembeddedtreessuch
thatthelargestlabel
µ
n
,andthustheradius
r
n
,iscoupledtothewidthofthe
support(
L
n
,R
n
)ofrandomfreelyembeddedtrees:
|
r
n
−
(
R
n