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Invariance principles for random bipartite planar maps
Jean-François Marckert, Grégory Miermont
To cite this version:
Jean-François Marckert, Grégory Miermont. Invariance principles for random bipartite planar maps. Annals of Probability, Institute of Mathematical Statistics, 2007, 35 (5), pp.1642–1705.
�10.1214/009117906000000908�. �hal-00004645v2�
ccsd-00004645, version 2 - 20 Mar 2006
Jean-FrançoisMarkert
∗
, Grégory Miermont
†
20th Marh 2006
Abstrat
Randomplanarmapsareonsideredin thePhysisliteratureasthedisreteounterpartof
randomsurfaes.Itisonjeturedthatproperlyresaledrandomplanarmaps,whenonditioned
to have a large number of faes, should onverge to a limiting surfae whose law does not
depend,uptosalingfators,ondetailsofthelassofmapsthatare sampled. Previousworks
on the topi, starting with Chassaing & Shaeer, have shown that the radius of a random
quadrangulation with n faes, i.e. the maximal graph distane on suh a quadrangulation to a xed referenepoint, onvergesin distribution one resaledby n1/4 to the diameter of the
Browniansnake,upto asalingonstant.
Using a bijetion due to Bouttier, di Franeso & Guitter between bipartiteplanar maps
and a family of labeled trees, we show the orresponding invariane priniple for a lass of
random maps that follow a Boltzmann distribution putting weight qk on faes of degree 2k:
the radius of suh maps, onditioned to have n faes (or n verties) and under a ritiality
assumption, onvergesindistributiononeresaledbyn1/4 toasaledversionofthediameter
of theBrowniansnake. Convergeneresultsfortheso-alledproleofmapsare alsoprovided.
The onvergeneof resaled bipartitemaps to theBrownian map, in the sense introduedby
Markert & Mokkadem, is also shown. The proofs of these results rely on a new invariane
priniplefortwo-typespatialGalton-Watsontrees.
Key Words: Randomplanar maps, labeled mobiles, invariane priniple, spatial Galton-Watson
trees,Brownian snake,Brownian map
M.S.C. Code: 60F17,60J80, 05C30
∗
CNRS, LaBRI, Université Bordeaux 1, 351 ours de la Libération, 33405 Talene edex, Frane,
markertlabri.fr
†
CNRS, Équipe Probabilités, Statistique etModélisation, Bât.425, Université Paris-Sud, 91405 Orsay, Frane
Gregory.Miermontmath.u-psud.fr
Contents
1 Introdution,motivationsand main results 3
1.1 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Boltzmann laws on planarmaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Snakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Mainresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Two illustratingexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5.1 2κ-angulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5.2 qi=βi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Commentsand organization of thepaper. . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Pushing maps to two-type trees 11 2.1 Planar spatial trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Two-type spatial GWtrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 The Bouttier-di Franeso-Guitter bijetion andits onsequenes . . . . . . . . . . . 14
3 An invariane priniple for spatial GW trees 18 3.1 The invariane priniple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Computation ofthe saling onstants assoiatedwithrandom maps . . . . . . . . . . 20
3.3 Proof ofTheorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Convergene of the height proess 23 4.1 GWforests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Controlling the height and numberof omponentsof forests . . . . . . . . . . . . . . 24
4.3 Anestral deomposition ofa GWforest . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4 An estimatefor the size ofGWtrees . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.5 The `onvergene of types'lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.6 Convergene ofthe height proess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5 Convergene of the label proess 36 5.1 Controlling the branhinginonditioned trees . . . . . . . . . . . . . . . . . . . . . . 36
5.2 A boundon the Hölder normof theheight proess . . . . . . . . . . . . . . . . . . . 39
5.3 Tightness of thelabelproess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.4 Finite-dimensional onvergene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6 Convergene tothe Brownianmap 49
1 Introdution, motivations and main results
1.1 Motivation
An embedded graph G is an embedding of a onneted graph in the 2-dimensional sphere S2, in whih edges do not interset exept possibly at their endpoints (the verties). A fae of G is a
onneted omponent of S2\ G. Faes are homeomorphi to open disks, and the degree of a given faeis the numberof edges that are inluded inthe losure of this fae, with theonvention that
ut-edgesareountedtwie,where ut-edgesarethoseedges whoseremovaldisonnetsthegraph.
Ifthe graphis the vertex-graph withonly one vertexand no edges, we adopt theonvention that
itbounds one fae withdegree 0. The degree of a vertexis thenumber of edges adjaent to that
vertex,whereself-loopsareountedtwie,aording totheusualgraph-theoretidenition. Unlike
faes,itdepends onlyon the underlying graph rather than itsembedding inS2.
Wesaythattwo embedded graphsareequivalent ifthereexistsanorientation-preserving home-
omorphism of S2 that maps the rst embedding to the seond. Equivalene lasses of embedded graphs are alled planar maps, and their set is denoted by M0. When onsidering a planar map
m∈ M0,we will slightlyimproperly speakofits verties,edges, faes andtheir respetive degrees
(weshouldrsttakeanelementofthelassmtobeompletelyaurate). WeletS(m), A(m), F(m)
bethesets ofverties,edges andfaesofm. Thedegree ofanelement u∈S(m) or f ∈F(m) will
be denotedbydeg(u), resp. deg(f). We denotethe lassof thevertex-graphby†.
If u, v are verties in a planar map m ∈ M0, and e1, . . . , en are oriented edges, we say that e1, . . . , en is a path from u to v of length n ifthe soure of e1 is u, the target of en is v, and the
target ofei is the soure ofei+1 for all 1≤i≤n−1. Thegraph distane assoiated witha planar
mapm∈ M0 is the funtion dm :S(m)×S(m) → Z+ dened by letting dm(u, v) be the leastn
suh that there exists a path of length n leading from u to v. This an be interpreted by saying thatwe turnminto a metrispae, byendowing edgeswithlengthsall equal to 1.
Planarmapshavebeenofpartiularinteresttophysiistsinthelastdeadeastheyan beon-
sideredasdisretizedversionsofsurfaes. Inordertogiveamathematialgroundto the`stohasti
quantization of 2-dimensional gravity', inwhih an integral with respet to an ill-dened `uniform measure'onRiemanniansurfaesisinvolved,apossibleattemptistoreplaetheintegral byanite
sum over distint `disrete geometries', whose role is performed by planar maps [3℄. Informally,it
isbelieved that
• Arandom maphoseninsomelassofplanarmapswith`size'n(e.g.aquadrangulation with
nfaes,i.e.amapwhose nfaesareallofdegree4),whoseedge-lengthsareproperlyresaled, should onverge indistribution asn→ ∞to a limitingrandom `surfae',and
• The limitingrandom surfae shouldnot depend,upto sale fators, ondetails ofthelassof
maps whih israndomly sampled.
The seond property is alled universality. A similar situation is well-known to probabilists: the
roleof a`Lebesguemeasure onpaths' is performedbyBrownian motion, whih isthesaling limit
ofdisretized randompaths (random walks)whose step distributions havea nite variane.
In a pioneering work,Chassaing &Shaeer [8℄ made a very substantial progress inanswering
the rst question, by establishing that the largest distane to the root in a uniform rooted quad-
rangulation with n faes (see denition below) divided by n1/4 onverges in distribution to some
Brownian snake withlifetimeproess the normalized Brownian exursion). Byusing an invariane
priniple for disrete labeled trees satisfying a positivity onstraint, Le Gall [15℄ has given an al-
ternative proof ofthe results of[8℄. Thisinvolvesa newrandom objet, alledtheBrownian snake
onditioned to bepositive, thatwas introdued inLe Gall and Weill[16 ℄. Markert& Mokkadem
[19℄ gave a desription of quadrangulations by gluing two trees, and showed that these trees on-
verge when suitably normalized as n goes to ∞. They introdued the notion of Brownian map,
and showed that under a ertain topology, resaled quadrangulations onverge in distribution to
the Brownian map. All these results have been obtained by using bijetive methods whih take
their soure in the work of Shaeer [22℄, and whih allow to study random quadrangulations in
termsofertainlabeledtrees. Theniefeatureofthis methodisthatthelabels allowtokeep trak
of geodesi distanes to a referene vertex inthe map, sothat some geometri information on the
mapsispresent inthe assoiated labeledtrees.
On theother hand, theseond question has not been addressed up to now in a purely proba-
bilistiform, and in theontext of saling limitsof planarmaps. Angel[4℄ and Angel& Shramm
[5℄giveevidenethatthelarge-salepropertiesoflargeplanarmapsshouldnotdependontheloal
details of the map (like the degree of faes), but these remarks hold in theontext of loal limits
of random maps, where all edges have a length xed to 1 as the numberof faes of the map goes
to innity (this is an `innite volume limit'), rather than in the ontext of saling limits, where
edge-lengths tend to 0 as the number of faes goes to innity (so that the total `volume' is kept
nite). In a reent artile, Bouttier, di Franeso and Guitter [6℄ have given a generalization of
Shaeer's bijetion to general planarmaps. Theyobtain identities for thegeneratingseries of the
most general family of (weighted) planar maps, and infer a number of lues for the universality
of the `pure 2Dgravity' model, e.g. by omputing ertain saling exponents with a ombinatorial
approah.
Their bijetion suggests a path to prove invariane priniples (the probabilisti word for 'uni-
versality') forrandom maps. Thepresent workexplores this pathinthease ofbipartite maps,by
rstgiving aprobabilisti interpretation oftheidentitiesof [6 ℄.
1.2 Boltzmann laws on planar maps
Aplanar mapis saidto bebipartite ifall itsfaes have evendegree. Inthis paper, we will onlybe
onerned withbipartite maps, notie† isbipartite withour onvention.
Every edge of a map an be given two orientations. A bipartite rooted planar map is a pair
(m, e) where m isa bipartite mapand e isa distinguished oriented edgeof m. The basi objets
thatareonsideredinthisartilearebipartiteplanarmapswhiharerootedandpointed,i.e.triples
(m, e,r) where(m, e) isabipartite rootedplanar mapand risa vertexof m. We letMbetheset
ofrooted,pointed,bipartite planarmaps. The map† annotberootedandan be pointedonly at
its unique vertex, but is still onsidered as an element of M. By abuse of notation, we will often
denoteageneri element ofMbymwithout referringto (e,r) whenitis freeofambiguity.
By the bipartite nature of elements of M, we have |dm(r, u)−dm(r, v)| = 1 whenever u, v ∈ S(m) are neighbors. Therefore, if (m, e,r) ∈ M \ {†}, we have either dm(r, e+) > dm(r, e−) or dm(r, e+) < dm(r, e−), where e− and e+ arethe soure and the target of the oriented edge e. We
let
M+={(m, e,r) ∈ M:dm(r, e+)> dm(r, e−)} ∪ {†}.
Allprobability distributions on mapsin thispaperare goingto bedened ontheset M+. Notie
non-orientededgehasbeen distinguished.
Letq= (qi, i≥1) beasequeneofnon-negative weightssuhthatqi>0forat leastonei >1.
By onvention, let q0 = 1. Consider the σ-nite measure Wq on M+ that assigns to eah map
m∈ M+ a weight qi perfaeof degree 2i:
Wq(m) = Y
f∈F(m)
qdeg(f)/2, (1)
with the onvention Wq(†) = q0 = 1. This multipliative form is reminisent of the measures assoiated withthe so-alled simply generated trees, whih areof theform w(t) =Q
u∈tqct(u) for
any tree t, where ct(u) is the number of hildren of a vertex u in t, and where (qi, i ≥ 0) is a
sequeneofnon-negative numbers [1 , p. 27-28℄.
Let Zq=Wq(M+) be the`partition funtion' ofq. Notie that Zq∈(1,∞] sine Wq(†) = 1.
If Zq < ∞, we say that q is admissible, and introdue the Boltzmann distribution on M+ with
suseptibility qby letting
Pq= Wq Zq
.
For k≥1,letN(k) = 2k−1k−1
. For anyweight sequene q(notneessarilyadmissible) dene
fq(x) =X
k≥0
xkN(k+ 1)qk+1 ∈[0,∞], x≥0.
Thefuntion fq : [0,∞)→ [0,∞]is a ompletely positive powerseries, i.e. itsderivatives of every orderarenon-negative,and sine(qi, i >1) isnot identiallyzero, fq isstritly positive on(0,∞),
andstritlyinreasingontheinterval[0, Rq],whereRqistheradiusofonvergeneoffq. Moreover,
fq onverges to ∞ asx→ ∞,and themonotone onvergene theorementailsthat thefuntion fq
isontinuous from [0, Rq] to [0,∞]. At Rq,two distint behaviors arepossible: fq(Rq) an either
benite,sothatfqjumpsto+∞totherightofRq,orinnite,inwhihasefqisontinuousfrom
[0,∞]to [0,∞]. In thesequel,weunderstand thatfq′(Rq)∈(0,∞]standsfor theleft-derivative of
fqat Rq (whenRq>0).
Consider the equation
fq(x) = 1−1/x , x >0. (2)
Sine x 7→ 1−x−1 is non-positive on (0,1] and fq is inniteon (Rq,∞], a solution of (2)always
belongs to (1, Rq]. Sine x 7→ 1−x−1 is stritly onave on (0,+∞), with derivative x 7→ x−2,
andfqisonvex, stritlyinreasing andontinuouson [0, Rq],wean lassifytheongurations of solutionsfor (2 ) bythefollowing four exlusive ases:
1. there areno solutions
2. thereareexatlytwosolutionsz1< z2 in(1, Rq],inwhihasefq′(z1)< z−21 andfq′(z2)> z2−2
3. there isexatlyone solution z1 in(1, Rq]withfq′(z1)< z1−2
4. there isexatlyone solution z in(1, Rq]withfq′(z) =z−2.
AswillbeshowninSet. 2.3,theadmissibility ofqan beformulated intermsoffqasfollows.
Proposition 1 The weight sequene q is admissible if and only if Equation (2) has at least one
solution. In this ase, Zq is the solutionof (2)that satises Zq2fq′(Zq)≤1.
