Discrete uniformizing metrics on distributional limits of sphere packings

1.2 MB·

by  James R. Lee

Discrete uniformizing metrics
on distributional limits of sphere packings
JamesR.Lee
UniversityofWashington
7
1
0 Abstract
2
Suppose that {G } is a sequence of finite graphs such that each G is the tangency graph
n n n
a of a sphere packing in Rd. Let ρn be a uniformly random vertex of Gn and suppose that
J (G,ρ)isthedistributionallimitof{(G ,ρ )}inthesenseofBenjaminiandSchramm. Thenthe
n n
5 conformalgrowthexponentof(G,ρ)isatmostd. Inotherwords,thereexistsaunimodular
2
“unitvolume”weightingofthegraphmetricon(G,ρ)suchthatthevolumegrowthofballsis

] asymptoticallypolynomialwithexponentd. Thisgeneralizestolimitsofgraphsthatcanbe
G
“quasisymmetrically”packedinanAhlforsd-regularmetricmeasurespace.
M
Inparticular,aresultoftheauthorimpliesthatundermomentconditionsonthedegreeof
. therootρ, thealmostsurespectraldimensionofGisatmostd. Asanotherapplication, we
h
t showthatthespectrumoffinitegraphssphere-packedinRdisdominatedbyavariantofthe
a
d-dimensionalWeyllaw.
m
[

1 1 Introduction
v
7
2 Thetheoryofrandomplanargraphshasbeenanactiveareaofstudyinthelasttwentyyears(see,
2 e.g.,[Ben10]),inspiredpartiallybytheconnectiontotwo-dimensionalquantumgravity[ADJ97].
7
0 AsnotedbyBenjaminiandCurien[BC11],ananalogoustheoryinhigherdimensionshasproved
1. elusive,inpartbasedonthedifficultyofenumerationforhigher-dimenisonalsimplicialcomplexes
0 (see[BZ11]andthereferencestherein).
7
1 To address this discrepancy, the authors of [BC11] explored the extension of analytic and
: probabilisticmethodsbasedonpotentialtheory. AgraphGissaidtobesphere-packedinRd ifGis
v
i thetangencygraphofacollectionofinterior-disjointspheresinRd. BenjaminiandCurienproved
X
thatifafamilyoffinitegraphscanbesphere-packedinRd withspheresofboundedaspectratio(so
r
a
thattheratiooftheradiioftangentspheresisO(1)),thenadistributionallimitofsuchgraphsis
d-parabolic.
Roughly speaking, d-parabolicity mans that the Ld extremal length from a fixed vertex to
∞ is infinite, where the Ld extremal length is a natural analog Cannon’s vertex extremal length
[Can94] (the case d = 2); see also [Duf62] and Section 1.3. It is well-known that the special case
of2-parabolicitycarriesstrongprobabilisticsignificance;forinstance,forgraphswithuniformly
boundeddegrees,2-parabolicityisequivalenttorecurrenceoftherandomwalk(see[Duf62,DS84]).
Unfortunately,ford > 2,thetheoryofLd extremallengthissomewhatmoreunwieldy,andisnot
knowntoyieldsuchcontrolontherandomwalk.
Inthiswork,westudyarelatednotionthatonemightrefertoasthe“extremalgrowthrate.”
Forgraphsthatcanbesphere-packedinRd,weshowthatitispossibletoconstructmetricsthat
1
uniformize their underlying geometry so that the counting measure has d-dimensional volume
growth. Employingtheresultsof[Lee17a],onedoesobtainsubstantialprobabilisticconsequences,
including d-dimensional lower bounds on the diagonal heat kernel (see Theorem 1.4 below).
Moreover,ourresultsholdinconsiderablegenerality;theyrequirenoassumptiononthedegreeor
aspectratioofthepacking,andtheyextendtographsthatcanbe“quasisymmetrically”packedin
anAhlforsd-regularmetricmeasurespace.
ThemethodofproofisbasedpartiallyonacelebratedlemmaofBenjaminiandSchramm[BS01].
Theyshowthatif{G }isasequenceoffiniteplanartriangulationswithuniformlyboundeddegrees
n
and{G }convergestoadistributionallimit(G,ρ),thenalmostsurelyanycirclepackingofGhasat
n
mostoneaccumulationpointintheplane. Ananalogousresultholdsforgraphssphere-packedin
Rd whend > 2[BC11].
Wearguethat,inaquantativesense,aslongastheaccumulationpointsremainseparated,one
canconstructamulti-scalereweightingofthespheresinthepacking,endowingthegraphwitha
metricthatreflectsitsd-dimensionalstructurewithrespecttotheunderlyingcountingmeasure.
ThisiscarriedoutinSection2.
InSection3, wediscussbrieflyaconjecturethatwecallthe“systolicminorproblem”based
on analogies with systolic geometry. This problem arises partly out of an attempt to connect
theconstructionofSection2tothemoretopologicalmethodsof[BLR10,KLPT11,Mat15,Lee16]
as explored in [Lee17a, Lee17b], and to extend the isoperimetric theorem of [BP11] to higher
dimensions.
1.1 Discreteconformalmetricsonsphere-packedgraphs
Consider a locally finite, connected graph G. A conformal metric on G is a map ω : V(G) → R+.
This endows G with a graph distance as follows: Give to every edge {u,v} ∈ E(G) a length
lenω({u,v}) := 12(ω(u)+ω(v)). This prescribes to every path γ = {v0,v1,v2,...} in G the induced
length
(cid:88)
lenω(γ) = lenω({vk,vk+1}).
k(cid:62)0
Nowforu,v ∈ V(G),onedefinesthepathmetricdistω(u,v)astheinfimumofthelengthsofallu-v
pathsinG. Denotetheclosedball
Bω(x,R) = (cid:8)y ∈ V(G) : distω(x,y) (cid:54) R(cid:9) .
We can now state a special case of our main technical theorem; the connection to distributional
limitsandrandomwalksisdiscussedsubsequently.
Theorem1.1. Foreveryd (cid:62) 1,thereisaconstantC = C(d)suchthatthefollowingholds. IfG = (V,E)isa
finitegraphthatcanbesphere-packedinRd,thenthereisaconformalmetricω : V → R+ thatsatisfies
1 (cid:88)
ω(x)d = 1,
|V|
x∈V
andsuchthat
max |Bω(x,R)| (cid:54) CRd(logR)2 ∀R (cid:62) 1.
x∈V(G)
2
1.2 Conformalgrowthexponents
If(G,ρ)israndomrootedgraph,thenaconformalmetricon(G,ρ)isarandomtriple(G(cid:48),ω,ρ(cid:48))with
ω : V(G) → R+ suchthat(G,ρ)and(G(cid:48),ρ(cid:48))havethesamelaw. Wesaythattheconformalweightis
(cid:104) (cid:105)
normalizedifE ω(ρ)2 = 1. Onethinksofsuchametricω : V(G) → R+ asdeformingthegeometry
oftheunderlyinggraphsubjecttoaboundonthetotal“area.” Asshownin[Lee17a],normalized
conformal metrics with nice geometric properties form a powerful tool in understanding the
structureof(G,ρ).
In the present work, we consider unimodular random graphs (see Section 1.6); such graphs
ariseasdistributionallimitsoffiniterandomrootedgraphs{(G ,ρ )}whereρ ∈ V(G )ischosen
n n n n
uniformlyatrandom. Wewillconsideronlyunimodularconformalmetricsωon(G,ρ);inother
words,thesettingwhere(G,ω,ρ)isunimodularasamarkedgraph.
Conformal growth exponents. Consider a unimodular random graph (G,ρ). In [Lee17a], we
definedtheupperandlowerconformalgrowthexponentsof(G,ρ),respectively,by
dim (G,ρ) := inflimsup log(cid:107)#Bω(ρ,R)(cid:107)L∞ , (1.1)
cg
ω R→∞ logR
dim (G,ρ) := infliminf log(cid:107)#Bω(ρ,R)(cid:107)L∞ , (1.2)
cg
ω R→∞ logR
where the infimum is over all normalized unimodular conformal metrics on (G,ρ), and we use
(cid:107)X(cid:107)L∞ todenotetheessentialsupremumofarandomvariableX,and#Stodenotethecardinalityof
asetS.
Whendim (G,ρ) = dim (G,ρ),definetheconformalgrowthexponentby
cg cg
dim (G,ρ) := dim (G,ρ) = dim (G,ρ).
cg cg cg
Notethatthequantitiesdim ,dim ,dim arefunctionsofthelawof(G,ρ);theyarenotdefined
cg cg cg
on(fixed)rootedgraphs.
NotethattheconformalgrowthexponentbearsaphilosophicalresemblancetoPansu’snotion
ofconformaldimension[Pan89]. WerefertoPansu’srecentwork[Pan16]whichexploresindetailthe
relationshipbetweenspherepackingsandthetheoryoflarge-scaleconformalmaps.
Lq conformal growth rate. Let us define a generalization: If (G,ω,ρ) is a unimodular random
conformalgraph,wedenote
(cid:107)ω(cid:107)Lq := (cid:0)Eω(ρ)q(cid:1)1/q .
SaythatωisLq-normalizedif(cid:107)ω(cid:107)Lq = 1.
DefinetheanalogousLq quantities: dimq ,dimq ,dimq wherenowtheinfimain(1.1)and(1.2)
cg cg cg
areoverallLq-normalizedconformalmetricson(G,ρ). Observethat,bymonotonicityofLq norms,
wehave
q (cid:54) q(cid:48) =⇒ dimq (G,ρ) (cid:54) dimq(cid:48) (G,ρ).
cg cg
⇒
The next theorem constitutes the main new result presented here. We use to denote
convergenceinthedistributionalsense;seeSection1.6.
Theorem1.2. Foranyd (cid:62) 1,thefollowingholds. If{G }arefinitegraphsthatcanbesphere-packedinRd
n
and{Gn} ⇒ (G,ρ),thenthereisanLd-normalizedunimodularconformalmetricω : V(G) → R+ suchthat
forallR (cid:62) 1,
(cid:107)#Bω(ρ,R)(cid:107)L∞ (cid:54) O(Rd(logR)2). (1.3)
Inparticular,dim (G,ρ) (cid:54) d.
cg
3
For d = 1, the theorem is uninteresting; the trivial weight ω ≡ 1 suffices. For d (cid:62) 2, the last
assertionfollowsfromdim (G,ρ) = dim2 (G,ρ) (cid:54) dimd (G,ρ).
cg cg cg
Weremarkthatsome(logR)O(1) factorisnecessaryevenforthecased = 2(planargraphs);see
[Lee17a,§2]. Infact,weproveTheorem1.2insomewhatgreatergenerality: Forgraphsthatare
“quasisymmetrically”packedinanAhlforsd-regularmetricmeasurespaceusingbodiesthatare
appropriately“round”(seeSection2fortheprecisedefinitions).
A primary motivation for Theorem 1.2 is that such metrics can be used to obtain estimates
on the almost sure spectral dimension of G. For a locally finite, connected graph G, denote the
discrete-timeheatkernel
pG(x,y) := P[X = y | X = x],
T T 0
where{X }isthestandardrandomwalkonG. WerecallthespectraldimensionofG:
n
−2logpG (x,x)
dim (G) := lim 2n ,
sp
n→∞ logn
wheneverthelimitexists. Ifthelimitdoesexist,thenitisthesameforallx ∈ V(G).
Say that a real-valued random variable X has negligible tails if its tails decay faster than any
inversepolynomial:
logn
= ,
lim 0 (1.4)
n→∞ |logP[|X| > n]|
wherewetakelog(0) = −∞intheprecedingdefinition(inthecasethatXisessentiallybounded).
Thenexttheoremisfrom[Lee17a];itassertsthatifdim (G,ρ) (cid:54) d,thenalmostsurelyGadmits
cg
d-dimensionallowerboundsonthediagonalheatkernel:
pG (ρ,ρ) (cid:62) n−d/2+o(1) as n → ∞.
2n
Theorem1.3. Supposethat(G,ρ)isaunimodularrandomgraphsuchthatdeg (ρ)hasnegligibletails.
G
Thenalmostsurely:
−2logpG (x,x)
limsup 2n (cid:54) dim (G,ρ).
cg
n→∞ logn
Inparticular,ifthereisanumberdsuchthatalmostsurelydim (G) = d,thend (cid:54) dim (G,ρ).
sp cg
Incertainsituations,onecangivestrongerestimates. Indeed,whentheconformalgrowthrate
hasonlyapolylogarithmiccorrectionasin(1.3),oneobtainsstrongerresults(see[Lee17a,§4.2]).
Theorem1.4. Suppose(G,ρ)isthedistributionallimitoffinitegraphsthatcanbesphere-packedinRd,
andthatdeg (ρ)hasexponentialtails. Thenthereisaconstantc (cid:62) 1suchthatfornsufficientlylarge,
G
(cid:34) n−d/2 (cid:35) 1
P pG (ρ,ρ) (cid:62) (cid:62) 1− .
2n (logn)c logn
1.3 Gaugedconformalgrowthandd-parabolicity
ConsideralocallyfiniteconnectedgraphG = (V,E). LetΓ denoteacollectionofsimplepathsinG.
G
The(cid:96) -vertexextremallengthofΓ isdefinedby
d G
γ
lenω( )
VEL (Γ ) := sup inf ,
d G ω γ∈ΓG (cid:107)ω(cid:107)(cid:96)d(V)
4
(cid:16)(cid:80) (cid:17)1/d
wheretheinfimumisoverallconformalmetricsonG,and(cid:107)ω(cid:107)(cid:96)d(V) = v∈Vω(v)d .
Fixavertexv ∈ V andletΓ (v )denotethesetofinfinitesimplepathsinGemenatingfrom
0 G 0
v . OnesaysthatGisd-parabolicifVEL (Γ (v )) = ∞(see[HS95,BS13]). Onecancheckthatthis
0 d G 0
definitiondoesnotdependonthechoiceofv ∈ V.
0
Consider a sequence {G } of finite graphs with uniformly bounded degrees. Furthermore,
n
suppose that each G is sphere-packed in Rd and {G } ⇒ (G,ρ). There are examples where
n n
G is almost surely 2-parabolic, but dimd (G,ρ) (cid:62) dim (G,ρ) = ∞, and other examples where
cg cg
dim (G,ρ) = d (cid:62) 2butGisalmostsurelynotd-parabolic;seeSection2.2.
cg
Gaugedgrowth. Ontheotherhand,thereisacommonstrengtheningoftheconditions. Saythat
(G,ρ)has(C,R,d)-growthifthereisanLd-normalizedconformalmetricω : V(G) → R+ suchthat
(cid:107)#Bω(ρ,R)(cid:107)L∞ (cid:54) CRd. (1.5)
Saythat(G,ρ)hasgaugedd-dimensionalconformalgrowthifthereisaconstantC (cid:62) 1suchthat(G,ρ)
has(C,R,d)-growthforallR (cid:62) 0. Asequence{(G ,ρ )}hasuniformgaugedd-dimensionalconformal
n n
growthifthereisaconstantC (cid:62) 1suchthat(G ,ρ )has(C,R,d)-growthforallR (cid:62) 0andn (cid:62) 1.
n n
Itisstraightforwardtoseethatif(G,ρ)hasgaugedd-dimensionalgrowth,thendimd (G,ρ) (cid:54) d:
cg
Foreachk (cid:62) 0,letω denoteanLd-normalizedconformalmetricon(G,ρ)satisfying(1.5)anddefine
k
ωˆ := π62 (cid:88) ωk2dk1/d .
k(cid:62)0
Establishingd-parabolicityissomewhatmoreinvolved;thed = 2caseofthefollowingtheoremis
[Lee17a,Thm. 2.1].
Theorem 1.5. For every d (cid:62) 2, the following holds. If (G,ρ) is a unimodular random graph such that
deg (ρ)isessentiallyboundedand(G,ρ)hasgaugedd-dimensionalconformalgrowth,thenGisalmost
G
surelyd-parabolic.
InordertoestablishTheorem1.2,weprovethefollowingstrongerstatement;seeCorollary2.2.
Theorem1.6. Supposethat{G }isasequenceoffinitegraphsandthateachG admitsasphere-packingin
n n
Rd. If{G } ⇒ (G,ρ),then(G,ρ)hasgaugedd-dimensionalconformalgrowth.
n
1.4 TheLaplacianspectrumoffinitegraphs
LetG = (V,E)denoteafiniteconnectedgraphandletn = |V|. Let{1−λ (G) : k = 0,1,...,n−1}be
k
theeigenvaluesoftherandomwalkoperatoronG,where
0 = λ0(G) (cid:54) λ1(G) (cid:54) ··· (cid:54) λn−1(G).
Definealso
(cid:88)
∆ (k) := max deg (x),
G S⊆V:|S|(cid:54)k G
x∈S
wheredeg (x)denotesthedegreeofavertexx ∈ V.
G
In[KLPT11],itisshownthatthereisaconstantC > 0suchthatifGisaplanargraph,thenfor
allk = 1,...,n−1,
k
λ (cid:54) C∆ (1) ,
k G
n
5
where ∆ (1) is the maximum degree in G. The estimate for k = 2 is due to Spielman and Teng
G
[ST07]. In[Lee17a],theauthorimprovesthisboundto
∆ (k)
λ (cid:54) C G .
k
n
While the utility of this improvement is not immediately apparent, the correct quantitative
dependenceisessentialtoaspectralargumentprovingthattheuniforminfiniteplanartriangulation
isalmostsurelyrecurrent[Lee17a];thisfactwasfirstestablishedbyGurel-GurevichandNachmias
ff
[GN13]usinge ectiveresistanceestimates.
InSection2.3,weuseTheorem1.1toproveananalogousstatementforgraphssphere-packed
inRd.
Theorem1.7. Foreveryd (cid:62) 2,thereisaconstantc suchthatthefollowingholds. IfGisann-vertexgraph
d
thatcanbesphere-packedinRd,thenfork = 1,2,...,n−1,
∆ (k) (cid:18) n(cid:19)2(cid:32)k(cid:33)2/d
λ (G) (cid:54) c G log .
k d
k k n
Notethat∆ (k)/kistheaveragedegreeofthekverticesoflargestdegreeinG. Uptothefactor
G
of(log(n/k))2,thisboundistightforad-dimensionalbox{1,2,...,n1/d}d consideredasasubgraph
of the integer lattice Zd. Whether the (log(n/k))2 factor can be removed from the bound is an
interestingopenquestion.
1.5 Preliminaries
R = ,∞ Z = Z∩R
Weusethenotations + [0 )and + +.
Allgraphsappearinginthispaperareundirectedandlocallyfiniteandwithoutloopsormultiple
edges. IfGissuchagraph,weuseV(G)andE(G)todenotethevertexandedgesetofG,respectively.
If S ⊆ V(G), we use G[S] for the induced subgraph on S. For A,B ⊆ V(G), we write E (A,B) for
G
the set of edges with one endpoint in A and the other in B. We write dist for the unweighted
G
pathmetriconV(G),andB (x,r) = {y ∈ V(G) : dist (x,y) (cid:54) r}todenotetheclosedr-ballaround
G G
x ∈ V(G). Alsoletdeg (x)denotethedegreeofavertexx ∈ V(G),andd (G) = sup deg (x).
G max x∈V(G) G
Write G (cid:27) G to denote that G and G are isomorphic as graphs. If (G ,ρ ) and (G ,ρ ) are
1 2 1 2 1 1 2 2
rootedgraphs,wewrite(G1,ρ1) (cid:27)ρ (G2,ρ2)todenotetheexistenceofarootedisomorphism.
1.6 Unimodularrandomgraphsanddistributionallimits
We begin with a discussion of unimodular random graphs and distributional limits. One may
ff
consult the extensive reference of Aldous and Lyons [AL07]. The paper [BS01] o ers a concise
introductiontodistributionallimitsoffiniteplanargraphs. Webrieflyreviewsomerelevantpoints.
G G
Let denotethesetofisomorphismclassesofconnected,locallyfinitegraphs;let • denotethe
setofrootedisomorphismclassesofrooted,connected,locallyfinitegraphs. DefineametriconG• as
follows: (cid:0)(G ,ρ ),(G ,ρ )(cid:1) = 1/(1+α),where
loc 1 1 2 2
(cid:110) (cid:111)
α = sup r > 0 : BG1(ρ1,r) (cid:27)ρ BG2(ρ2,r) .
G , {µ },µ G {µ } ⇒ µ
( • loc)isaseparable,completemetricspace. Forprobabilitymeasures n on •,write n
µ µ
when convergesweaklyto withrespectto .
n loc
6
TheMass-TransportPrinciple. LetG•• denotethesetofdoubly-rootedisomorphismclassesof
doubly-rooted,connected,locallyfinitegraphs. AprobabilitymeasureµonG• isunimodularifit
obeysthefollowingMass-TransportPrinciple: ForallBorel-measurableF : G•• → [0,∞],
(cid:90) (cid:90)
(cid:88) (cid:88)
F(G,ρ,x)dµ((G,ρ)) = F(G,x,ρ)dµ((G,ρ)). (1.6)
x∈V(G) x∈V(G)
If(G,ρ)isarandomrootedgraphwithlawµ,andµisunimodular,wesaythat(G,ρ)isaunimodular
randomgraph;wewilloftenomittheword“random”andsimplyreferto(G,ρ)asaunimodular
randomgraph.
Distributionallimitsoffinitegraphs. AsobservedbyBenjaminiandSchramm[BS01],unimod-
ularrandomgraphscanbeobtainedfromlimitsoffinitegraphs. Considerasequence{G } ⊆ Gof
n
finitegraphs,andletρ denoteauniformlyrandomelementofV(G ). Then{(G ,ρ )}isasequence
n n n n
G
of •-valuedrandomvariables,andonehasthefollowing.
Lemma1.8. If{(G ,ρ )} ⇒ (G,ρ),then(G,ρ)isunimodular.
n n
If{(G ,ρ )} ⇒ (G,ρ),wesaythat(G,ρ)isthedistributionallimitofthesequence{(G ,ρ )}. When
n n n n
{G }isasequenceoffinitegraphs,wewrite{G } ⇒ (G,ρ)for{(G ,ρ )} ⇒ (G,ρ)whereρ ∈ V(G )
n n n n n n
ischosenuniformlyatrandom.
Unimodularrandomconformalgraphs. Aconformalgraphisapair(G,ω),whereGisaconnected,
locally finite graph and ω : V(G) → R+. Let G∗ and G∗• denote the collections of isomorphism
classesofconformalgraphsandconformalrootedgraphs,respectively. AsinSection1.6,onecan
defineametriconG∗• asfollows: ∗loc(cid:0)(G1,ω1,ρ1),(G2,ω2,ρ2)(cid:1) = 1/(α+1),where
(cid:26) (cid:16) (cid:17) 1(cid:27)
α = sup r > 0 : BG1(ρ1,r) (cid:27)ρ BG2(ρ2,r)and ω1|BG1(ρ1,r),ω2|BG2(ρ2,r) (cid:54) r ,
wherefortwoweightsω1 : V(H1) → R+andω2 : V(H2) → R+onrooted-isomorphicgraphs(H1,ρ1)
and(H ,ρ ),wewrite
2 2 (cid:13) (cid:13)
(ω1,ω2) := ψ:V(Hi1n)→fV(H2)(cid:13)(cid:13)ω2◦ψ−ω1(cid:13)(cid:13)(cid:96)∞ ,
andtheinfimumisoverallgraphisomorphismsfromH toH satisfyingψ(ρ ) = ρ .
1 2 1 2
{µ } µ G∗ {µ } ⇒ µ
If n and are probability measures on •, we abuse notation and write n to
∗
denote weak convergence with respect to . One defines unimodularity of a random rooted
loc
conformalgraph(G,ω,ρ)analogouslyto(1.6): ItshouldnowholdthatforallBorel-measurable
F : G∗•• → [0,∞],
(cid:90) (cid:90)
(cid:88) (cid:88)
F(G,ω,ρ,x)dµ((G,ω,ρ)) = F(G,ω,x,ρ)dµ((G,ω,ρ)).
x∈V(G) x∈V(G)
Indeed,suchdecoratedgraphsareaspecialcaseofthemarkednetworksconsideredin[AL07],and
againitholdsthateverydistributionallimitoffiniteunimodularrandomconformalgraphsisa
unimodularrandomconformalgraph.
Supposethat(G,ρ)isaunimodularrandomgraph. Aconformalweighton(G,ρ)isaunimodular
conformalgraph(G(cid:48),ω,ρ(cid:48))suchthat(G,ρ)and(G(cid:48),ρ(cid:48))havethesamelaw. Wewillspeaksimplyof
a“conformalmetricωon(G,ρ).” Onlysuchunimodularmetricsareconsideredinthiswork.
7
1.6.1 Conformalgrowthratesunderdistributionallimits
In order to establish our main result, we need to pass from a sequence of conformal metrics on
finitegraphstoaconformalmetriconthedistributionallimit.
Theorem1.9. Considerd,q (cid:62) 1. Suppose{(G ,ρ )}isasequenceoffiniteunimodularrandomgraphsand
n n
{(Gn,ρn)} ⇒ (G,ρ). Ifthereisafunctionh : R+ → R+ suchthath(R) (cid:54) Ro(1) asR → ∞,andasequenceof
Lq-normalizedunimodularrandomconformalgraphs{(G ,ω ,ρ )}satisfying
n n n
(cid:107)Bωn(ρn,R)(cid:107)L∞ (cid:54) Rdh(R), (1.7)
thendimq (G,ρ) (cid:54) d.
cg
Iftheunimodularrandomgraphs{(G ,ρ )}haveuniformgaugedd-dimensionalgrowth,then(G,ρ)has
n n
gaugedd-dimensionalgrowth.
Theprecedingtheoremisestablishedbythenexttwolemmas.
Lemma1.10. Considerasequence{(G ,ω ,ρ )}ofunimodularrandomconformalgraphssatisfyingthe
n n n
followingconditions:
1. Foreveryn (cid:62) 1,E(cid:2)ω (ρ )(cid:3) (cid:54) 1.
n n
2. Thereisasequence{bk : k ∈ N}suchthatforeveryk (cid:62) 1,limsupn→∞(cid:107)#BGn(ρn,k)(cid:107)L∞ (cid:54) bk.
Then{(G ,ω ,ρ )}hasasubsequentialweaklimit.
n n n
Proof. Letusfirstpasstoasubsequence{(Gn,ωn,ρn)}satisfying(cid:107)#BGn(ρn,k)(cid:107)L∞ (cid:54) bk fork (cid:54) n.
Fork (cid:62) 1andaconformalgraph(G,ω)andx,y ∈ V(G),definetheflow
Fk(G,ω,x,y) = ω(x)1{distG(x,y)(cid:54)k}.
EmployingtheMass-TransportPrinciple,fork (cid:54) nwehave
   
E (cid:88) ωn(x) = E (cid:88) Fk(Gn,ωn,x,ρn)
x∈BGn(ρn,k) x∈V(Gn)
 
= E (cid:88) Fk(Gn,ωn,ρn,x)
x∈V(Gn)
(cid:54) b E(cid:2)ω (ρ )(cid:3) = b .
k n n k
λ >
Inparticular,forany 0,
(cid:34) (cid:35)
1
P max ω(x) > λk2b (cid:54) .
x∈BGn(ρn,k) k λk2bk
Denotetheevent
(cid:40) (cid:41)
E (λ) := (G ,ω ,ρ ) : max ω(x) (cid:54) λk2b ∀k = 1,2,...,n .
n n n n k
x∈BGn(ρn,k)
Aunionboundyields
P[E (λ)] (cid:62) 1−O(1/λ). (1.8)
n
8
Letµ (λ)denotethelawof(G ,ω ,ρ )conditionedonE (λ). Then{µ (λ)}admitsasubsequential
n n n n n n
weaklimitbecausethespaceofboundedweightsonballsofradiuskiscompact(recallingthatsuch
ballsalmostsurelyhavesizeatmostb byassumption). Thereforebydiagonalization,thesequence
k
{µ (n)} admits a subsequential weak limit µ. Let {n } be a subsequence for which {µ (2nj)} ⇒ µ.
n j nj
Then(G ,ω ,ρ ) ⇒ µaswellsinceP[E (n )] → 1by(1.8). (cid:3)
nj nj nj nj j
Theorem 1.9 now follows from the following lemma by taking {ω } to be the Lq-normalized
n
conformalweightsthatverify(1.7)for{(G ,ρ )}.
n n
Lemma1.11. Suppose{(G ,ρ )}isasequenceoffiniteunimodularrandomgraphssuchthat{(G ,ρ )} ⇒
n n n n
(G,ρ). Thenanysequenceofunimodularrandomconformalgraphs{(G ,ω ,ρ )}satisfyingE[ω (ρ)] (cid:54) 1,
n n n n
∗
hasasubsequentialweaklimitinthemetric .
loc
Proof. Sincethereareonlycountablymanyrootedisomorphismclassesofrootedballsofradiusr,
wehave
lim P[#B (ρ,r) > b] = 0. (1.9)
G
b→∞
Inparticular,thereexistsasequence{bj,(cid:96)}suchthat
lim P[#BG(ρ,j) (cid:54) bj,(cid:96) ∀j (cid:62) 1] = 1. (1.10)
(cid:96)→∞
(cid:110) (cid:111)
Let(G(cid:96)n,ω(cid:96)n,ρ(cid:96)n)havethelawof(Gn,ωn,ρn)conditionedontheevent #BGn(ρn,j) (cid:54) bj,(cid:96) ∀j (cid:62) 1 (for
(cid:96) ffi ffi
su cientlylargethatthiseventhasnon-zeroprobability). From(1.10),itsu cestoprovethat
{(G(cid:96),ω(cid:96),ρ(cid:96))} has a weak subsequential limit for all (cid:96) sufficiently large, and this is precisely the
n n n
(cid:3)
contentofLemma1.10.
1.7 Systemsofdyadiccubes
Considerapseudo-metricspace(X,d)(i.e.,weallowforthepossibilitythatd(x,y) = 0whenx (cid:44) y).
Throughout the paper, we will deal only with complete, separable, pseudo-metric spaces. For
x ∈ XandtwosubsetsS,T ⊆ X,weusethenotationsd(S,T) = infx∈S,y∈Td(x,y)andd(x,S) = d({x},S).
Definediam(S,d) = sup d(x,y)andforR (cid:62) 0,definetheclosedballs
x,y∈S
B(X,d)(x,R) = {y ∈ X : d(x,y) (cid:54) R}.
Weomitthesubscript(X,d)iftheunderlyingmetricspaceisclearfromcontext. Wesaythat(X,d)
isdoublingifthereisaconstantDsuchthateveryclosedballinXcanbecoveredbyDclosedballs
D D
ofhalftheradius,andwelet (X,d) denotetheinfimal forwhichthisholds.
IfµisameasureontheBorelσ-algebraofX,wereferto(X,d,µ)asametricmeasurespace. Sucha
spaceissaidtobeAhlforsβ-regularifthereareconstantsc ,c > 0suchthat
1 2
c Rβ (cid:54) µ(B(x,R)) (cid:54) c Rβ ∀x ∈ X,R ∈ [0,diam(X)].
1 2
Itwilloccasionallybeconvenienttorecordtheconstantsc ,c ,inwhichcasewesaythat(X,d,µ)is
1 2
(c ,c ,β)-regular. Werecallthefollowingelementaryfact:
1 2
Fact1.12. If(X,d,µ)isAhlforsβ-regularforsomeβ > 0,then(X,d)isdoubling,andD(X,d) (cid:54) Cforsome
constantC = C(c ,c ,β)dependingonlyonc ,c ,β.
1 2 1 2
9
Wewillemployanappropriatesystemofhierarchicaldyadicpartitionsofadoublingmetric
space(X,d)alongthelinesof[Chr90]and[Dav91]. Deterministicandrandomizedconstructionsof
thistypeareabasictoolinharmonicanalysisandmetricgeometry(see,e.g.,[LN05]and[HK12]).
Forourpurposes,itwillbeeasiesttouseaconstructionfrom[HK12]whichwesummarizehere.
Considerametricspace(X,d). Abi-infinitesequenceP = {P : n ∈ Z}ofpartitionsofXissaidtobe
n
ahierarchicalsystemifPn isarefinementofPn+1 foralln ∈ Z. WesaythatPis∆-adicif
S ∈ Pn =⇒ diam(X,d)(S) (cid:54) ∆n ∀n ∈ Z.
Theorem 1.13 ([HK12]). Suppose (X,d) is a doubling metric space. Then there are numbers k,(cid:96),∆ (cid:62) 2
that depend only on D(X,d) such that the following holds. There is a collection {P(1),...,P(k)} of ∆-adic
hierarchicalsystemssuchthatforeverysubsetS ⊆ Xwithdiam(X,d)(S) (cid:54) ∆m,thereisaset
(cid:91)k
Q ∈ P(i)
m+(cid:96)
i=1
suchthatS ⊆ Q.
2 Discrete conformal metrics on d-dimensional graphs
Wefirstintroduceamoregeneralsettingforgraphs“packed”ind-dimensionalspaces.
QSpackings. Considera(complete,separable)metricmeasurespace(X,dist,µ). ABorelsetS ⊆ X
issaidtobeγ-roundifthefollowingholds: ForeveryballBinXwhosecenterliesinS¯ (theclosure
ofS)andforwhichS (cid:42) B,wehaveµ(S∩B) (cid:62) γ·µ(B). Forinstance,ballsinRd are2−d-round.
Consider additionally a collection F of Borel sets in X. For ε > 0, and an at most countable
0
subcollectionF ⊆ F0 ofpairwise-disjointsets,definetheε-QStangencygraphGε[F]asthegraph
withvertexsetF andwithanedgebetweendistinctsetsS,T ∈ F whenever
dist(S,T) (cid:54) ε·min{diam(S),diam(T)}.
Say that a locally-finite graph G is ε-packable in (X,dist,µ) by F0 if Gε[F] is isomorphic to G for
somesubsetF ⊆ F . SaythatGis(ε,γ)-packablein(X,dist,µ)ifitisε-packablebyγ-roundsubsets
0
of X. Say that G is QS-packable in (X,dist,µ) if it is (ε,γ)-packable for some ε (cid:62) 0,γ > 0. For
instance,thetangencygraphofasphere-packinginRd is0-packablebyopenEuclideanballs,and
(0,2−d)-packableinRd.
ε
Wenowpresentthemaintheoremofthissection. Despitetheuseofthesymbol ,oneshould
ε
observeitholdsevenfor large. Fortheremainderofthissection,weusethedefinition
d∗ := max(d,2).
Theorem2.1. Foreveryd (cid:62) 1,c ,c > 0,ε (cid:62) 0,γ > 0,thereisaconstantCsuchthatthefollowingholds.
1 2
SupposeG = (V,E)isafinitegraphthatis(ε,γ)-packedina(c ,c ,d)-regularspace(X,dist,µ). Thenfor
1 2
everyR (cid:62) 0,thereisaconformalweightω : V → R+ thatsatisfies
1 (cid:88)
ω(x)d∗ = 1, (2.1)
|V|
x∈V
andsuchthat
max |Bω(x,R)| (cid:54) CRd∗. (2.2)
x∈V(G)
10