Local convergence of random graph colorings

0.3 MB·English

by  Amin Coja-Oghlan

LOCAL CONVERGENCE OF RANDOM GRAPH COLORINGS
AMINCOJA-OGHLAN∗,CHARILAOSEFTHYMIOU∗∗ANDNORJAAFARI
5
1 ABSTRACT. LetG=G(n,m)bearandomgraphwhoseaveragedegreed=2m/nisbelowthek-colorabilitythreshold.
0 Ifwesampleak-coloringσofGuniformlyatrandom,whatcanwesayaboutthecorrelationsbetweenthecolorsassigned
2 tovertices thatarefarapart? According toaprediction fromstatistical physics, foraveragedegrees below theso-called
n condensationthresholddk,cond,thecolorsassignedtofarawayvertices areasymptotically independent [Krzakalaetal.:
Proc.NationalAcademyofSciences2007].Weprovethisconjectureforkexceedingacertainconstantk0.Moregenerally,
a weinvestigatethejointdistributionofthek-coloringsthatσinduceslocallyonthebounded-depthneighborhoodsofany
J
fixednumberofvertices.Inaddition,wepointoutanimplicationonthereconstructionproblem.
6
2 MathematicsSubjectClassification:05C80(primary),05C15(secondary)

]
O 1. INTRODUCTION ANDRESULTS
C Let G = G(n,m) denote the random graph on the vertex set [n] = 1,...,n with precisely m edges. Unless
. specified otherwise, we assume that m = m(n) = dn/2 for a fixed{number}d > 0. As usual, G(n,m) has a
h property “withhighprobability”(“w.h.p.”)iflim ⌈ P[⌉G(n,m) ]=1.
at A n→∞ ∈A
m 1.1. Backgroundandmotivation. GoingbacktotheseminalpaperofErdo˝sandRe´nyi[20]thatfoundedthetheory
ofrandomgraphs, the problemofcoloringG(n,m)remainsoneofthe longest-standingchallengesin probabilistic
[
combinatorics. Overthepasthalf-century,effortshavebeendevotedtodeterminingthelikelyvalueofthechromatic

1 number χ(G(n,m)) [4, 11, 26, 28] and its concentration [6, 27, 34] as well as to algorithmic problems such as
v
constructingorsamplingcoloringsoftherandomgraph[3,15,16,17,22,23].
1
0 Atantalisingfeatureoftherandomgraphcoloringproblemistheinterplaybetweenlocalandglobaleffects.Locally
3 aroundalmostanyvertextherandomgraphisbipartitew.h.p.Infact,foranyfixedaveragedegreed> 0andforany
6 fixedω the depth-ω neighborhoodof allbuto(n) verticesis justa tree w.h.p.Yetgloballythe chromaticnumberof
0
the random graph may be large. Indeed, for any number k 3 of colors there exists a sharp threshold sequence
1. d = d (n)suchthatforanyfixedε > 0,G(n,m)is≥k-colorablew.h.p.if2m/n< d (n) ε,whereas
k−col k−col k−col
0 therandomgraphsfailstobek-colorablew.h.p.if2m/n > d (n)+ε[1]. Whilstthethresholdsd− arenot
k−col k−col
5
knownprecisely,therearecloseupperandlowerbounds.Thebestcurrentonesread
1
: d =(2k 1)lnk 2ln2+δ liminfd (n) limsupd (n) (2k 1)lnk 1+ε , (1.1)
v k,cond k k−col k−col k
− − ≤ n→∞ ≤ n→∞ ≤ − −
i
X wherelim δ =lim ε =0[4,13,14]. Tobeprecise,thelowerboundin(1.1)isformallydefinedas
k→∞ k k→∞ k
r
a d =inf d>0:limsupE[Z (G(n,m))1/n]<k(1 1/k)d/2 . (1.2)
k,cond k
−
(cid:26) n→∞ (cid:27)
This number, called the condensation threshold due to a connection with statistical physics [24], can be computed
preciselyforkexceedingacertainconstantk [8]. Anasymptoticexpansionyieldstheexpressionin(1.1).
0
Thecontrastbetweenlocaland globaleffectswas famouslypointedoutby Erdo˝s, who producedG(n,m) as an
exampleofa graphthatsimultaneouslyhasa highchromaticnumberanda highgirth[19]. Thepresentpaperaims
atamorepreciseunderstandingofthiscollusionbetweenshort-rangeandlong-rangeeffects. Forinstance,doglobal
effectsentail“invisible”constraintson the coloringsofthe localneighborhoodsso thatcertain “local”coloringsdo
notextendtoacoloringoftheentiregraph?Andwhatcorrelationsdotypicallyexistbetweenthecolorsofverticesat
alargedistance?
Date:January27,2015.
∗TheresearchleadingtotheseresultshasreceivedfundingfromtheEuropeanResearchCouncilundertheEuropeanUnion’sSeventhFrame-
workProgramme(FP/2007-2013)/ERCGrantAgreementn.278857–PTCC.
∗∗ResearchissupportedbyARCGaTech.
1
A naturalway of formalising these questionsis as follows. Let k 3 be a number of colors, fix some number
ω > 0andassumethatd < d sothatG = G(n,m)isk-colorab≥lew.h.p.Moreover,pickavertexv andfixa
k,cond 0
k-coloringσ ofitsdepth-ωneighborhood.Howmanywaysaretheretoextendσ toak-coloringoftheentiregraph,
0 0
andhowdoesthisnumberdependonσ ? Additionally,ifwepickavertexv thatis“faraway”fromv andifwepick
0 1 0
anotherk-coloringσ of the depth-ω neighborhoodof v , is there a k-coloringσ of G that simultaneouslyextends
1 1
bothσ andσ ? Ifso,howmanysuchσexist,andhowdoesthisdependonσ ,σ ?
0 1 0 1
The main result of this paper (Theorem1.1 below) providesa very neat and accurate answer to these questions.
Itshowsthatw.h.p.all“local”k-coloringsσ extendtoasymptoticallythesamenumberofk-coloringsoftheentire
0
graph. Letuswrite (G)forthesetofallk-coloringsofagraphGandletZ (G) = (G) bethenumberofk-
k k k
S |S |
colorings.Moreover,let∂ω(G,v )bethedepth-ωneighborhoodofavertexv inG(i.e.,thesubgraphofGobtained
0 0
bydeletingallverticesatdistancegreaterthanωfromv ). Thenw.h.p.anyk-coloringσ of∂ω(G,v )has
0 0 0
(1+o(1))Z (G)
k
Z (∂ω(G,v ))
k 0
extensionstoak-coloringofG. Moreover,ifwepickanothervertexv atrandomandfixsomek-coloringσ ofthe
1 1
depth-ωneighborhoodofv ,thenw.h.p.thenumberofjointextensionsofσ ,σ is
1 0 1
(1+o(1))Z (G)
k
.
Z (∂ω(G,v ))Z (∂ω(G,v ))
k 0 k 1
Inotherwords,ifwechooseak-coloringσuniformlyatrandom,thenthedistributionofthek-coloringthatσinduces
on the subgraph ∂ω(G,v ) ∂ω(G,v ), which is a forest w.h.p., is asymptotically uniform. The same statement
0 1
∪
extendstoanyfixednumberv ,...,v ofvertices.
0 l
1.2. Results. The appropriate formalism for describing the limiting behavior of the local structure of the random
graphis the conceptof localweak convergence[5, 9]. The concreteinstalmentof the formalismthatwe employis
reminiscentofthatusedin[10,32].(Corollary1.2belowprovidesastatementthatisequivalenttothemainresultbut
thatavoidstheformalismoflocalweakconvergence.)
Let G be the set of all locally finite connected graphs whose vertex set is a countable subset of R. Further, let
G be the set of all triples (G,v ,σ) such that G G, σ : V(G) [k] is a k-coloring of G and v V(G) is
k 0 0
∈ → ∈
a distinguished vertex that we call the root. We refer to (G,v ,σ) as a rooted k-colored graph. If (G′,v′,σ′) is
0 0
anotherrootedk-coloredgraph,wecall(G,v ,σ)and(G′,v′,σ′)isomorphic((G,v ,σ)=(G′,v′,σ′))ifthereisan
0 0 0 ∼ 0
isomorphismϕ : G G′ suchthatϕ(v ) = ϕ(v′),σ = σ′ ϕandsuchthatforanyv,w V(G)suchthatv < w
→ 0 0 ◦ ∈
we haveϕ(v) < ϕ(w). Thus,ϕ preservestheroot,the coloringandtheorderofthe vertices(whicharereals). Let
[G,v ,σ]betheisomorphismclassof(G,v ,σ)andlet bethesetofallisomorphismclassesofrootedk-colored
0 0 k
G
graphs.
Foranintegerω 0andΓ welet∂ωΓdenotetheisomorphismclassoftherootedk-coloredgraphobtained
k
≥ ∈G
fromΓbydeletingallverticeswhosedistancefromtherootexceedsω. ThenanyΓ,ω 0giverisetoafunction
≥
0,1 , Γ′ 1 ∂ωΓ′ =∂ωΓ . (1.3)
k
G →{ } 7→ { }
Weendow withthecoarsesttopologythatmakesallofthesefunctionscontinuous.Further,forl 1weequip l
Gk ≥ Gk
withthecorrespondingproducttopology.Additionally,theset ( l)ofprobabilitymeasureson l carriestheweak
P Gk Gk
topology,asdoestheset 2( l)ofallprobabilitymeasureson ( l). Thespaces l, ( l), 2( l)arePolish[5].
P Gk P Gk Gk P Gk P Gk
ForΓ wedenotebyδ ( )theDiracmeasurethatputsmassoneonΓ.
k Γ k
Let∈GGbeafinitek-colorab∈lePgrGaphwhosevertexsetV(G)iscontainedinRandletv ,...,v V(G). Thenwe
1 l
∈
candefinea probabilitymeasureon l asfollows. LettingG v denotetheconnectedcomponentofv V(G)and
Gk k ∈
σ vtherestrictionofσ :V(G) [k]toG v,wedefine
k → k
l
1
λ(G,v ,...,v)= δ ( l). (1.4)
1 l Z (G) [Gkvi,vi,σkvi] ∈P Gk
k
σ∈XSk(G)Oi=1
Theidea is thatλ capturesthejointempiricaldistributionof coloringsinducedbya randomcoloringof G
G,v1,...,vl
“locally”inthevicinityofthe“roots”v ,...,v . Further,let
1 l
1
λl = E[δ G χ(G(n,m)) k] 2( l).
n,m,k nl λ( (n,m),v1,...,vl)| ≤ ∈P Gk
v1,..X.,vl∈[n]
2
Thismeasurecapturesthetypicaldistributionofthelocalcoloringsinarandomgraphwithlrandomlychosenroots.
Wearegoingtodeterminethelimitofλl asn .
Tocharacterisethislimit,letT∗(d)bne,ma,(kpossib→lyi∞nfinite)randomGalton-Watsontreerootedatavertexv∗ with
0
offspringdistributionPo(d). WeembedT∗(d)intoRbyindependentlymappingeachvertextoauniformlyrandom
point in [0,1]; with probabilityone, all vertices get mapped to distinct points. Let T(d) G signify the resulting
randomtree and let v denote its root. For a numberω > 0 we let ∂ωT(d) denotethe (fi∈nite) rootedtree obtained
0
fromT(d)byremovingallverticesatadistancegreaterthanωfromv . Moreover,forl 1letT1(d),...,Tl(d)be
0
lindependentcopiesofT(d)andset ≥
ϑld,k[ω]=E(cid:20)δNi∈[l]λ(cid:16)∂ωTi(d)(cid:17)(cid:21)∈P2(Gkl), where (1.5)
1
λ(cid:0)∂ωTi(d)(cid:1)= Zk(∂ωTi(d))σ∈Sk(X∂ωTi(d))δ[∂ωTi(d),v0,σ] ∈P(Gkl) (cf.(1.4)).
Thesequence(ϑl [ω]) converges(seeAppendixA)andwelet
d,k ω≥1
ϑl = lim ϑl [ω].
d,k d,k
ω→∞
Combinatorially,ϑl correspondsto samplingl copiesofthe Galton-Watsontree T(d) independently. Thesetrees
d,k
are colored by assigning a random color to each of the l roots independently and proceeding down each tree by
independentlychoosingacolorforeachvertexfromthek 1colorsleftunoccupiedbytheparent.
−
Theorem1.1. Thereisanumberk >0suchthatforallk k ,d<d ,l >0wehavelim λl =ϑl .
0 ≥ 0 k,cond n→∞ n,m,k d,k
Fixnumbersω 1, l 1, choosea randomgraphG = G(n,m) forsomelargeenoughn andchoosevertices
v ,...,v uniform≥lyandi≥ndependentlyatrandom. Thenthedepth-ω neighborhoods∂ω(G,v ),...,∂ω(G,v )are
1 l 1 l
pairwisedisjointandtheunion = ∂ω(G,v ) ∂ω(G,v )isaforestw.h.p.Moreover,thedistancebetween
1 l
anytwotreesin isΩ(lnn)w.Fh.p.GiventhatG∪is··k·-∪colorable,letσbearandomk-coloringofG. Thenσinduces
F
a k-coloring of the forest . Theorem 1.1 implies that w.h.p. the distribution of the induced coloring is at a total
F
variationdistanceo(1)fromtheuniformdistributiononthesetofallk-coloringsof . Formally,letuswriteµ for
k,G
F
theprobabilitydistributionon[k]V(G)definedby
µ (σ)=1 σ (G) Z (G)−1 (σ [k]V(G)),
k,G k k
{ ∈S } ∈
i.e.,theuniformdistributiononthesetofk-coloringsofthegraphG. Moreover,forU V(G)letµ denotethe
k,G|U
⊂
projectionofµ onto[k]U,i.e.,
k,G
µ (σ )=µ σ [k]V : u U :σ(u)=σ (u) (σ [k]U).
k,G|U 0 k,G 0 0
∈ ∀ ∈ ∈
IfH isasubgraphofG,thenweju(cid:0)s(cid:8)twriteµ insteadofµ .(cid:9)L(cid:1)et denotethetotalvariationnorm.
k,G|H k,G|V(H) k·kTV
Corollary1.2. Thereisaconstantk >0suchthatforanyk k ,d<d ,l 1,ω 0wehave
0 0 k,cond
≥ ≥ ≥
1
lim E µ G G G µ G G =0.
n→∞nl k, |∂ω( ,v1)∪···∪∂ω( ,vl)− k,∂ω( ,v1)∪···∪∂ω( ,vl) TV
v1,..X.,vl∈[n] (cid:13) (cid:13)
(cid:13) (cid:13)
Sincew.h.p.thepairwisedistance(cid:13)oflrandomlychosenverticesv ,...,v inGisΩ(lnn)(cid:13),weobservethatw.h.p.
1 l
µ G G = µ G .
k,∂ω( ,v1)∪···∪∂ω( ,vl) k,∂ω( ,vi)
iO∈[l]
With very little work it can be verified that Corollary 1.2 is actually equivalent to Theorem 1.1. Setting ω = 0 in
Corollary1.2yieldsthefollowingstatement,whichisofinterestinitsownright.
Corollary1.3. Thereisanumberk >0suchthatforallk k ,d<d andanyintegerl >0wehave
0 0 k,cond
≥
1
lim E µ G µ G =0. (1.6)
n→∞nl (cid:13) k, |{v1,...,vl}− k, |{vi}(cid:13)
v1,..X.,vl∈[n] (cid:13)(cid:13) iO∈[l] (cid:13)(cid:13)TV
(cid:13) (cid:13)
(cid:13) 3 (cid:13)
(cid:13) (cid:13)
Bythesymmetryofthecolors,µ G isjusttheuniformdistributionon[k] foreveryvertexv. Hence,Corol-
k, |{v}
lary1.3statesthatford<d w.h.p.intherandomgraphGforrandomlychosenverticesv ,...,v thefollowing
k,cond 1 l
istrue:ifwechooseak-coloringσofGatrandom,then(σ(v ),...,σ(v )) [k]l isasymptoticallyuniformlydis-
1 l
∈
tributed. PriorresultsofMontanariandGershenfeld[21]andofMontanari,RestrepoandTetali[33]implythat(1.6)
holdsford<2(k 1)ln(k 1),aboutanadditivelnkbelowd .
k,cond
− −
The above results and their proofs are inspired by ideas from statistical physics. More specifically, physicists
have developeda non-rigorousbut analytic technique, the so-called “cavity method”[29], which has led to various
conjecturesontherandomgraphcoloringproblem. Theseincludea predictionastotheprecisevalueofd for
k,cond
anyk 3[37]aswellasaconjectureasto theprecisevalueofthek-colorabilitythresholdd [25]. While the
k−col
≥
latterformulaiscomplicated,asymptoticallyweexpectthatd = (2k 1)lnk 1+ε ,wherelim ε =
k−col k k→∞ k
− −
0. According to this conjecture, the upper bound in (1.1) is asymptotically tight and d is strictly greater than
k−col
d . Furthermore,accordingto the physicsconsiderations(1.6) holdsfor anyk 3 and anyd < d [24].
k,cond k,cond
≥
Corollary1.3verifiesthisconjecturefork k . Bycontrast,accordingtothephysicspredictions,(1.6)doesnothold
0
≥
ford < d < d . As(1.6)isthespecialcaseofω = 0ofTheorem1.1(resp.Corollary1.2),theconjecture
k,cond k−col
implies that neither of these extendto d > d . In other words, the physics picture suggeststhat Theorem1.1,
k,cond
Corollary1.2andCorollary1.3areoptimal,exceptthattheassumptionk k canpossiblybereplacedbyk 3.
0
≥ ≥
1.3. Anapplication. Supposewedrawak-coloringσ ofG atrandom. Ofcourse,thecolorsthatσ assignsto the
neighborsofavertexvandthecolorofvarecorrelated(theymustbedistinct).Moregenerally,itseemsreasonableto
expectthatforanyfixed“radius”ωthecolorsassignedtotheverticesatdistanceωfromvandthecolorofvitselfwill
typicallybecorrelated. Butwill these correlationspersistasω ? Thisis the“reconstructionproblem”,which
→ ∞
hasreceivedconsiderableattentioninthecontextofrandomconstraintsatisfactionproblemsingeneralandinrandom
graphcoloringinparticular[24,33,35]. ToillustratetheuseofTheorem1.1wewillshowhowitreadilyimpliesthe
resultonthereconstructionproblemforrandomgraphcoloringfrom[33].
Toformallystatetheproblem,assumethatGisafinitek-colorablegraph. Forv V(G)andasubset =
∈ ∅ 6 R ⊂
(G)letµ ( )betheprobabilitydistributionon[k]definedby
k k,G|v
S ·|U
1
µ (i )= 1 σ(v)=i ,
k,G|v
|R { }
|R|σ∈R
X
i.e.,thedistributionofthecolorofvinarandomcoloringσ . Forv V(G),ω 1andσ (G)let
0 k
∈R ∈ ≥ ∈S
(v,ω,σ )= σ (G): u V(G) ∂ω−1(G,v):σ(u)=σ (u) .
k,G 0 k 0
R ∈S ∀ ∈ \
Thus, k,G(v,ω,σ0)containsallk-co(cid:8)loringsthatcoincidewith σ0 onverticeswhosedistance(cid:9)fromv isatleastω.
R
Moreover,let
1 1 1
bias (v,ω,σ )= µ (i (v,ω,σ )) , bias (v,ω)= bias (v,ω,σ ).
k,G 0 2 k,G|v |Rk,G 0 − k k,G Z (G) k,G 0
iX∈[k](cid:12)(cid:12) (cid:12)(cid:12) k σ0∈XSk(G)
Clearly, for symmetry reaso(cid:12)(cid:12)ns, if we draw a k-coloring σ(cid:12)(cid:12) k(G) uniformly at random, then σ(v) is uniformly
distributed over [k]. What bias (v,ω,σ ) measures is ho∈w mSuch conditioningon the event σ (v,ω,σ )
k,G 0 k,G 0
∈ R
biasesthe colorofv. Accordingly,bias (v,ω) measuresthebiasinducedbya random“boundarycondition”σ .
k,G 0
Wesaythatnon-reconstructionoccursinG(n,m)if
1
lim lim E[bias G (v,ω)]=0.
ω→∞n→∞n k, (n,m)
vX∈[n]
Otherwise,reconstructionoccurs.Analogously,recallingthatT(d)istheGalton-Watsontreerootedatv ,wesaythat
0
treenon-reconstructionoccursatdiflimω→∞E[biask,∂ωT(d)(v0,ω)]=0.Otherwise,treereconstructionoccurs.
Corollary1.4. Thereisanumberk >0suchthatforallk k andd<d thefollowingistrue.
0 0 k,cond
≥
ReconstructionoccursinG(n,m) treereconstructionoccursatd. (1.7)
⇔
Montanari,RestrepoandTetali[33]proved(1.7)ford<2(k 1)ln(k 1),aboutanadditivelnkbelowd .
k,cond
− −
Thisgapcouldbepluggedbyinvokingrecentresultsonthegeometryofthesetofk-colorings[7,13,31]. However,
weshallseethatCorollary1.4isactuallyanimmediateconsequenceofTheorem1.1.
4
The point of Corollary 1.4 is that it reducesthe reconstructionproblemon a combinatoriallyextremely intricate
object, namely the random graph G(n,m), to the same problem on a much simpler structure, namely the Galton-
Watson tree T(d). Thatsaid, the reconstructionproblemonT(d) is farfromtrivial. The bestcurrentboundsshow
that there exists a sequence (δ ) 0 such that non-reconstruction holds in T(d) if d < (1 δ )klnk while
k k k
→ −
reconstructionoccursifd>(1+δ )klnk[18].
k
1.4. Techniquesandoutline. Noneoftheargumentsinthepresentpaperareparticularlydifficult. Itisratherthata
combinationofseveralrelativelysimpleingredientsprovesremarkablypowerful. Thestartingpointoftheproofisa
recentresult[7]ontheconcentrationofthenumberZ (G(n,m))ofk-coloringsofG(n,m). Thisresultentailsavery
k
preciseconnectionbetweenafairlysimpleprobabilitydistribution,theso-called“plantedmodel”,andtheexperiment
of sampling a random coloring of a random graph, thereby extending the “planting trick” from [2]. However, this
planting argumentis not powerful enough to establish Theorem 1.1 (cf. also the discussion in [10]). Therefore, in
the presentpaperthe keyidea isto use theinformationaboutZ (G(n,m)) tointroducean enhancedvariantof the
k
planting trick. More specifically, in Section 3 we will establish a connection between the experiment of sampling
a random pair of coloringsof G(n,m) and another, much simpler probability distribution that we call the planted
replicamodel.Weexpectthatthisideawillfindfutureuses.
Apart from the concentration of Z (G(n,m)), this connection also hinges on a study of the “overlap” of two
k
randomlychosencoloringsofG(n,m). Theoverlapwasstudiedinpriorworkonreconstruction[21,33]inthecase
thatd < 2(k 1)ln(k 1)basedonconsiderationsfromthesecondmomentargumentofAchlioptasandNaor[4]
− −
thatgavethe bestlowerboundonthe k-colorabilitythresholdatthe time. To extendthe studyofthe overlapto the
wholeranged (0,d ), we crucially harnessinsightsfromthe improvedsecondmomentargumentfrom[14]
k,cond
∈
andtherigorousderivationofthecondensationthreshold[8].
AswewillseeinSection4,thestudyoftheplantedreplicamodelallowsustodrawconclusionsastothetypical
“local” structure of pairs of random colorings of G(n,m). To turn these insights into a proof of Theorem 1.1, in
Section5we extendanelegantargumentfrom[21], whichwasusedtheretoestablishtheasymptoticindependence
of the colors assigned to a bounded number of randomly chosen individualvertices (reminiscent of (1.6)) for d <
2(k 1)ln(k 1).
− −
ThebottomlineisthatthestrategybehindtheproofofTheorem1.1israthergeneric. Itprobablyextendstoother
problemsofa similarnature. A naturalclasstothinkofarethebinaryproblemsstudiedin[33]. Anothercandidate
mightbethehardcoremodel,whichwasstudiedin[10]byasomewhatdifferentapproach.
2. PRELIMINARIES
2.1. Notation. Forafiniteorcountableset wedenoteby ( )thesetofallprobabilitydistributionson ,which
X P X X
weidentifywiththesetofallmapsp : [0,1]suchthat p(x) = 1. Furthermore,ifN > 0isaninteger,
X → x∈X
then ( ) is the set of all p ( ) such that Np(x) is an integer for every x . With the conventionthat
N
P X ∈ P X P ∈ X
0ln0=0,wedenotetheentropyofp ( )by
∈P X
H(p)= p(x)lnp(x).
−
x∈X
X
LetGbe a k-colorablegraph. By σk,G,σk,G,σk,G,... (G) we denoteindependentuniformsamplesfrom
(G). WhereG,kareapparentfromthecont1ext,w2eomitth∈esSukperscript.Moreover,ifX : (G) R,wewrite
k k
S S →
1
X(σ) = X(σ).
h iG,k Z (G)
k
σ∈XSk(G)
Moregenerally,ifX : (G)l R,then
k
S →
1
X(σ ,...,σ ) = X(σ ,...,σ).
h 1 l iG,k Z (G)l 1 l
k
σ1,...,Xσl∈Sk(G)
WeomitthesubscriptGand/orkwhereitisapparentfromthecontext.
Thus,thesymbol referstotheaverageoverrandomlychosenk-coloringsofafixedgraphG. Bycontrast,
h·iG,k
thestandardnotationE[ ],P[ ]willbeusedtoindictatethattheexpectation/probabilityistakenoverthechoiceof
· ·
therandomgraphG(n,m). Unlessspecifiedotherwise,weusethestandardO-notationtorefertothelimitn .
→ ∞
Throughoutthepaper,wetacitlyassumethatnissufficientlylargeforourvariousestimatestohold.
5
By a rootedgraphwe mean a graphGtogetherwith a distinguishedvertexv, the root. The vertexset is always
assumedtobeasubsetofR. Ifω 0isaninteger,then∂ω(G,v)signifiesthesubgraphofGobtainedbyremoving
≥
allverticesatdistancegreaterthanωfromv(includingthoseverticesofGthatarenotreachablefromv),rootedatv.
Anisomorphismbetweentworootedgraphs(G,v),(G′,v′)isanisomorphismG G′oftheunderlyinggraphsthat
→
mapsvtov′ andthatpreservestheorderofthevertices(whichiswhyweinsistthattheybereals).
2.2. Thefirstmoment. ThepresentworkbuildsuponresultsonthefirsttwomomentsofZ (G(n,m)).
k
Lemma2.1. Foranyd>0,E[Z (G)]=Θ(kn(1 1/k)m).
k
−
AlthoughLemma2.1isfolklore,webrieflycommentonhowtheexpressioncomesabout.Forσ :[n] [k]let
→
k σ−1(i)
(σ)= | | (2.1)
F 2
i=1(cid:18) (cid:19)
X
bethenumberofedgesofthecompletegraphthataremonochromaticunderσ. Then
n (σ) n
P[σ (G)]= 2 −F 2 . (2.2)
k
∈S m m
(cid:18)(cid:0) (cid:1) (cid:19)(cid:30)(cid:18)(cid:0) (cid:1)(cid:19)
By convexity, we have (σ) 1 n for all σ. In combination with (2.2) and the linearity of expectation, this
implies that E[Z (G(n,Fm))] =≥ Ok(k2n(1 1/k)m). Conversely, there are Ω(kn) maps σ : [n] [k] such that
k
n/k σ−1(i) √nforalli,and(cid:0) (cid:1)(σ)/−n =1/k+O(1/n)forallsuchσ. ThisimpliesE[Z (G→)]=Ω(kn(1
1/k)m−).|Thefol|lo≤wingresultshowsthFatZ (G2)istightlyconcentratedaboutitsexpectationfordk<d . −
(cid:12) (cid:12) k(cid:0) (cid:1) k,cond
(cid:12) (cid:12)
Theorem2.2([7]). Thereisk >0suchthatforallk k andalld<d wehave
0 0 k,cond
≥
lim lim P[ lnZ (G) lnE[Z (G)] ω]=1.
k k
ω→∞n→∞ | − |≤
Forα = (α ,...,α ) ([k]) we letZ (G) bethe numberof k-coloringsσ of G suchthat σ−1(i) = α n
1 k n α i
∈ P | |
for all i [k]. Conversely, for a map σ : [n] [k] let α(σ) = n−1(σ−1(i)) ([k]). Additionally, let
i∈[k] n
∈ → ∈ P
α¯ =k−11=(1/k,...,1/k).
Lemma2.3([7,Lemma3.1]). Letϕ(α)=H(α)+ dln 1 α 2 . Then
2 −k k2
E[Z (G)]=O(exp(nϕ(α))) (cid:16) (cid:17) uniformlyforallα ([k]),
α n
∈P
E[Z (G)]=Θ(n(1−k)/2)exp(nϕ(α)) uniformlyforallα ([k])suchthat α α¯ k−3.
α ∈Pn k − k2 ≤
2.3. Thesecondmoment. Definetheoverlapofσ,τ :[n] [k]asthek kmatrixρ(σ,τ)withentries
→ ×
1
ρ (σ,τ)= σ−1(i) τ−1(j) .
ij
n ∩
Thenthenumberofedgesofthecompletegraphthatare(cid:12)monochromaticu(cid:12)ndereitherσorτ equals
(cid:12) (cid:12)
nρ (σ,τ)
ij
(σ,τ)= (σ)+ (τ) .
F F F − 2
i,Xj∈[k](cid:18) (cid:19)
Fori [k]letρ signifytheithrowofthematrixρ,andforj [k]letρ denotethejthcolumn. Anelementary
i· ·j
∈ ∈
applicationofinclusion/exclusionyields(cf.[7,Fact5.4])
m
(n)−F(σ,τ)
P[σ,τ (G)]= 2 m =O 1 ( ρ (σ,τ) 2+ ρ (σ,τ) 2)+ ρ(σ,τ) 2 . (2.3)
∈Sk (n)  − k i· k2 k ·i k2 k k2 
(cid:0) m2 (cid:1) iX∈[k]
  
We canview ρ(σ,τ) asa(cid:0)distr(cid:1)ibutionon[k] [k], i.e., ρ(σ,τ) ([k]2). Letρ¯be theuniformdistributionon
n
× ∈ P
[k]2. Moreover,forρ ([k]2)letZ⊗(G)bethenumberofpairsσ ,σ (G)withoverlapρ. Finally,let
∈Pn ρ 1 2 ∈Sk
(ω)= ρ ([k]2): i [k]: ρ α¯ , ρ α¯ ω/n , and (2.4)
Rn,k ∈Pn ∀ ∈ k i· − k2 k ·i− k2 ≤
n d p o
f(ρ)=H(ρ)+ ln(1 2/k+ ρ 2). (2.5)
2 − k k2
6
Lemma2.4([4]). Assumethatω =ω(n) butω =o(n). Forallk 3,d>0wehave
→∞ ≥
E[Z⊗(G)]=O(n(1−k2)/2)exp(nf(ρ)) uniformlyforallρ (ω)s.t. ρ ρ¯ k−3,
ρ ∈Rn,k k − k∞ ≤
E[Z⊗(G)]=O(exp(nf(ρ))) uniformlyforallρ (ω).
ρ ∈Rn,k
Moreover,ifd<2(k 1)ln(k 1),thenforanyη >0thereexistsδ >0suchthat
− −
f(ρ)<f(ρ¯) δ forallρ (ω)suchthat ρ ρ¯ >η. (2.6)
− ∈Rn,k k − k2
Thebound(2.6)appliesford<2(k 1)ln(k 1),aboutlnkbelowd .Tobridgethegap,letκ=1 ln20k/k
k,cond
− − −
andcallρ ([k]2)separableifkρ (0.51,κ)foralli,j [k]. Moreover,σ (G)isseparableifρ(σ,τ)is
n ij k
separablef∈orPallτ (G). Otherwise6∈,wecallσ inseparable∈. Further,ρiss-sta∈blSeiftherearepreciselysentries
k
∈ S
suchthatkρ κ.
ij
≥
Lemma2.5([14]). Thereisk suchthatforallk >k andall2(k 1)ln(k 1) d 2klnkthefollowingistrue.
0 0
− − ≤ ≤
(1) LetZ˜ (G)= σ (G):σisinseparable . ThenE[Z˜ (G)] exp( Ω(n))E[Z (G)].
k k k k
|{ ∈S }| ≤ −
(2) Let1 s k 1. Thenf(ρ)<f(ρ¯) Ω(1)uniformlyforalls-stableρ.
≤ ≤ − −
(3) Foranyη >0thereisδ >0suchthatsup f(ρ):ρis0-stableand ρ ρ¯ >η <f(ρ¯) δ.
{ k − k2 } −
Lemma2.5omitsthek-stablecase. Todealwithit,weintroduce
(G,σ)= τ (G):ρ(σ,τ)isk-stable . (2.7)
k
C { ∈S }
Lemma2.6([8]). Thereexistk andω =ω(n) suchthatforallk k ,2(k 1)ln(k 1) d<d we
0 0 k,cond
→∞ ≥ − − ≤
have
nl→im∞P h|C(G,σ)|iG,k ≤ω−1E[Zk(G)] =1.
h i
2.4. Atailbound. Finally,weneedthefollowinginequality.
Lemma 2.7 ([36]). Let X ,...,X be independent random variables with values in a finite set Λ. Assume that
1 N
f :ΛN Risafunction,thatΓ ΛN isaneventandthatc,c′ >0arenumberssuchthatthefollowingistrue.
→ ⊂
Ifx,x′ ΛN aresuchthatthereisk [N]suchthatx =x′ foralli=k,then
∈ ∈ i i 6
c ifx Γ, (2.8)
f(x) f(x′) ∈
| − |≤ c′ ifx Γ.
(cid:26) 6∈
Thenforanyγ (0,1]andanyt>0wehave
∈
t2 2N
P[f(X ,...,X ) E[f(X ,...,X )] >t] 2exp + P[(X ,...,X ) Γ].
| 1 N − 1 N | ≤ −2N(c+γ(c′ c))2 γ 1 N 6∈
(cid:18) − (cid:19)
3. THE PLANTEDREPLICA MODEL
Throughoutthissectionweassumethatk k forsomelargeenoughconstantk andthatd<d .
0 0 k,cond
≥
InthissectionweintroducethekeytoolfortheproofofTheorem1.1,theplantedreplicamodel.Thisistheprobability
distributionπpr ontriples(G,σ ,σ )suchthatGisagraphon[n]withmedgesandσ ,σ (G)inducedby
n,m,k 1 2 1 2 ∈Sk
thefollowingexperiment.
PR1: Sampletwomapsσˆ ,σˆ :[n] [k]independentlyanduniformlyatrandomsubjecttotheconditionthat
1 2
(σˆ ,σˆ ) n m. →
F 1 2 ≤ 2 −
PR2: ChooseagraphGˆ on[n]withpreciselymedgesuniformlyatrandom,subjecttotheconditionthatboth
(cid:0) (cid:1)
σˆ ,σˆ areproperk-colorings.
1 2
Wedefine
πpr (G,σ ,σ )=P (Gˆ,σˆ ,σˆ )=(G,σ ,σ ) .
n,m,k 1 2 1 2 1 2
Clearly,theplantedreplicamodelisquitetamesothatithshouldbeeasytobringtheiknowntechniquesfromthetheory
of randomgraphsto bear. Indeed,the conditioningin PR1 is harmlessbecause E[ (σˆ ,σˆ )] (2/k 1/k2) n
F 1 2 ∼ − 2
whilem = O(n). Hence,bytheChernoffboundwehave (σˆ ,σˆ ) n mw.h.p.Moreover,PR2justmeans
F 1 2 ≤ 2 − (cid:0) (cid:1)
7
(cid:0) (cid:1)
thatwe draw m randomedgesoutof the n (σˆ ,σˆ ) edgesof the complete graphthatare bichromaticunder
bothσˆ ,σˆ . Inparticular,wehavetheexpli2cit−foFrmul1a 2
1 2
(cid:0) (cid:1)
1 n (τ ,τ ) −1
πpr (G,σ ,σ )= 2 −F 1 2 .
n,m,k 1 2 (τ ,τ ) [k]n [k]n : (τ ,τ ) n m m
1 2 ∈ × F 1 2 ≤ 2 − τ1,τ2:[n]→[k],XF(τ1,τ2)≤(n2)−m(cid:18)(cid:0) (cid:1) (cid:19)
(cid:12)(cid:8) (cid:0) (cid:1) (cid:9)(cid:12)
The purpose of the(cid:12) planted replica model is to get a handle on ano(cid:12)ther experiment, which at first glance seems
far less amenable. The random replica model πrr is a probability distribution on triples (G,σ ,σ ) such that
n,m,k 1 2
σ ,σ (G)aswell. Itisinducedbythefollowingexperiment.
1 2 k
∈S
RR1: ChoosearandomgraphG=G(n,m)subjecttotheconditionthatGisk-colorable.
RR2: Sampletwocoloringsσ ,σ ofGuniformlyandindependently.
1 2
Thus,therandomreplicamodelisdefinedbytheformula
n −1
πrr (G,σ ,σ )=P[(G,σ ,σ )=(G,σ ,σ )]= 2 P[χ(G) k]Z (G)2 . (3.1)
n,m,k 1 2 1 2 1 2 m ≤ k
(cid:20)(cid:18)(cid:0) (cid:1)(cid:19) (cid:21)
Sinceweassumethatd<d ,Gisk-colorablew.h.p.Hence,theconditioninginRR1isinnocent.Butthisisfar
k,cond
fromtrueofthe experimentdescribedin RR2. For instance, we havenoideaas tohowonemightimplementRR2
constructivelyfordanywhereneard . Infact, thebestcurrentalgorithmsforfindingasinglek-coloringofG,
k,cond
letalonearandompair,stopworkingfordegreesdaboutafactoroftwobelowd (cf.[2]).
k,cond
Yetthemainresultofthissectionshowsthatford < d ,the“difficult”randomreplicamodelcanbestudied
k,cond
bymeansofthe“simple”plantedreplicamodel.Moreprecisely,recallthatasequence(µ ) ofprobabilitymeasures
n n
iscontiguouswithrespecttoanothersequence(ν ) ifµ ,ν aredefinedonthesamegroundsetforallnandiffor
n n n n
anysequence( ) ofeventssuchthatlim ν ( )=0wehavelim µ ( )=0.
n n n→∞ n n n→∞ n n
A A A
Proposition3.1. Ifd<d ,thenπrr iscontiguouswithrespecttoπpr .
k,cond n,m,k n,m,k
The rest of this section is devoted to the proof of Proposition 3.1. A key step is to study the distribution of the
overlapoftworandomk-coloringsσ ,σ ofG,whosedefinitionwerecallfromSection2.3.
1 2
Lemma3.2. Assumethatd<dk,cond. ThenE[hkρ(σ1,σ2)−ρ¯k2iG]=o(1).
In words, Lemma 3.2 asserts that the expectation over the choice of the random graph G (the outer E) of the
averageℓ -distanceoftheoverlapoftworandomlychosenk-coloringsofGfromρ¯goesto0asn . Toprove
2
→ ∞
this statement the following intermediate step is required; we recall the α( ) notation from Section 2.2. The d <
·
2(k 1)ln(k 1)caseofLemma3.2waspreviouslyprovedin[33]bywayofthesecondmomentanalysisfrom[4].
− −
Asitturnsout,theregime2(k 1)ln(k 1) < d < d requiresasomewhatmoresophisticatedargument. In
k,cond
− −
any case, for the sake of completeness we give a full prove of Lemma 3.2, including the d < 2(k 1)ln(k 1)
− −
(which adds merely three lines to the argument). Similarly, in [33] the followingclaim was established in the case
d<2(k 1)ln(k 1).
− −
Claim3.3. Supposethatd<d andthatω =ω(n)issuchthatlim ω(n)= butω =o(n). Thenw.h.p.
k,cond n→∞
Gissuchthat ∞
1 α(σ) α¯ > ω/n exp( Ω(ω)).
k − k2 G ≤ −
Proof. WecombineTheorem2.2Dwitnhastandard“firstmpomenot”Eestimatesimilartotheproofof[33,Lemma5.4].The
entropyfunctionα ([k]) H(α) = k α lnα isconcaveandattainsitsglobalmaximumatα¯. Infact,
∈ P 7→ − i=1 i i
theHessianofα H(α)satisfiesD2H(α) 2id. Moreover,sinceα α 2 isconvex,α dln(1 α 2)is
7→ (cid:22)P− 7→k k2 7→ 2 −k k2
concaveandattainsisglobalmaximumatα¯ aswell. Hence,lettingϕdenotethefunctionfromLemma2.3,wefind
D2ϕ(α) 2id. Therefore,weobtainfromLemma2.3that
(cid:22)−
O(1) if α α¯ >1/lnn,
E[Z (G)] exp(n(ϕ(α¯) α α¯ 2)) k − k2 (3.2)
α ≤ −k − k2 ·(O(n(1−k)/2) otherwise.
Further,letting
Z′(G)= Z (G)
α
α∈Pn([k]):kXα−α¯k2>√ω/n
8
andtreatingthecasesω ln2nandω ln2nseparetely,weobtainfrom(3.2)that
≤ ≥
E[Z′(G)] exp( Ω(ω))exp(n(ϕ(α¯)). (3.3)
≤ −
SinceLemma2.1showsthatE[Z (G)]=Θ(kn(1 1/k)m)=exp(nϕ(α¯)),(3.3)yieldsE[Z′(G)]=exp( Ω(ω))E[Z (G)].
k k
− −
Hence,byMarkov’sinequality
P[Z′(G) exp( Ω(ω))E[Z (G)]] 1 exp( Ω(ω)). (3.4)
k
≤ − ≥ − −
Finally,since α(σ) α¯ > ω/n =Z′(G)/Z (G)andbecauseZ (G) E[Z ]/ωw.h.p.byTheorem2.2,
k − k2 G k k ≥ k
theassertionfDollowsfrom(3.4).p E (cid:3)
ProofofLemma3.2. Webound
Λ= kρ(σ1,σ2)−ρ¯k2 =Zk(G)2hkρ(σ1,σ2)−ρ¯k2iG
G
σ1,σ2X∈Sk( )
byasumofthreedifferentterms. First,letting,say,ω(n)=lnn,weset
Λ = 1 α(σ ) α¯ > ω/n =Z (G)2 α(σ) α¯ > ω/n .
1 k 1 − k2 k k − k2 G
G
σ1,σ2X∈Sk( ) n p o D p E
Todefinetheothertwo,let ′(G)bethesetofallσ (G)suchthat α(σ) α¯ ω/n. Letη >0beasmall
Sk ∈Sk k − k2 ≤
butn-independentnumberandlet
p
Λ = 1 ρ(σ ,σ ) ρ¯ η ρ(σ ,σ ) ρ¯ , Λ = 1 ρ(σ ,σ ) ρ¯ >η .
2 {k 1 2 − k2 ≤ }k 1 2 − k2 3 {k 1 2 − k2 }
σ1,σ2X∈Sk′(G) σ1,σ2X∈Sk′(G)
Since ρ(σ ,σ ) ρ¯ 2forallσ ,σ ,wehave
k 1 2 − k2 ≤ 1 2
Λ 4(Λ +Λ )+Λ . (3.5)
1 2 3
≤
Hence,weneedtoboundΛ ,Λ ,Λ . WithrespecttoΛ ,Claim3.3impliesthat
1 2 3 1
P Λ exp( Ω(√n))Z (G)2 =1 o(1). (3.6)
1 k
≤ − −
To estimate Λ2, we let f denotethe f(cid:2)unctionfromLemma2.4. Ob(cid:3)servethatDf(ρ¯) = 0, becauseρ¯maximisesthe
entropyandminimisestheℓ -norm.Further,astraightforwardcalculationrevealsthatforanyi,j,i′,j′ [k], (i,j)=
2
∈ 6
(i′,j′),
∂2f(ρ) = 1 + d 2dρ2ij , ∂2f(ρ) = 2dρijρi′j′ .
∂ρ2ij −ρij 1−2/k+kρk22 − (1−2/k+kρk22)2 ∂ρij∂ρi′j′ −(1−2/k+kρk22)2
Consequenctly,choosing,say,η <k−4,ensuresthattheHessiansatisfies
D2f(ρ) 2id forallρsuchthat ρ ρ¯ 2 η. (3.7)
(cid:22)− k − k2 ≤
Therefore,Lemma2.4yields
E[Λ ] ρ ρ¯ E[Z⊗(G)]
2 ≤ k − k2 ρ
ρ∈RXn,k(η)
O(n(1−k2)/2)exp(nf(ρ¯)) ρ ρ¯ exp(n(f(ρ) f(ρ¯)))
≤ k − k2 −
ρ∈RXn,k(η)
O(n(1−k2)/2)exp(nf(ρ¯)) ρ ρ¯ exp( nk−2 ρ ρ¯ 2) [by(3.7)]. (3.8)
≤ k − k2 − k − k
ρ∈RXn,k(η)
Further,sinceρ =1 ρ foranyρ (η),substitutingx=√nρin(3.8)yields
kk − (i,j)6=(k,k) ij ∈Rn,k
E[Λ ] O(n(1P−k2)/2)exp(nf(ρ¯)) kxk2 exp( k−2 x 2)dx=O(n−1/2)exp(nf(ρ¯)). (3.9)
2 ≤ ZRk2−1 √n − k k2
Sincef(ρ¯)=2lnk+dln(1 1/k),Lemma2.1yields
−
exp(nf(ρ¯)) O(E[Z (G)]2). (3.10)
k
≤
9
Therefore,(3.9)entailsthat
E[Λ ] O(n−1/2)E[Z (G)]2. (3.11)
2 k
≤
To bound Λ , we consider two separate cases. The first case is that d 2(k 1)ln(k 1). Then Lemma 2.4
3
≤ − −
and(3.10)yield
E[Λ ] exp(nf(ρ¯) Ω(n)) exp( Ω(n))E[Z (G)]2. (3.12)
3 k
≤ − ≤ −
Thesecondcaseisthat2(k 1)ln(k 1) d<d . Weintroduce
k,cond
− − ≤
Λ = 1 σ failstobeseparable ,
31 1
{ }
σ1,σ2X∈Sk′(G)
Λ = 1 ρ(σ ,σ )iss-stableforsome1 s k ,
32 1 2
{ ≤ ≤ }
σ1,σ2X∈Sk′(G)
Λ = 1 ρ(σ ,σ )is0-stableand ρ(σ ,σ ) ρ¯ >η ,
33 { 1 2 k 1 2 − k2 }
σX1,σ2
Λ = 1 ρ(σ ,σ )isk-stable ,
34 1 2
{ }
σ1,σ2X∈Sk′(G)
sothat
Λ Λ +Λ +Λ +Λ . (3.13)
3 31 32 33 34
≤
BythefirstpartofLemma2.5andMarkov’sinequality,
P[Λ exp( Ω(n))Z (G)E[Z (G)]]=1 o(1). (3.14)
31 k k
≤ − −
Further,combiningLemma2.4withthesecondpartofLemma2.5,weobtain
P[Λ exp(nf(ρ¯) Ω(n))]=1 o(1). (3.15)
32
≤ − −
Addionally,Lemma2.4andthethirdpartofLemma2.5yield
P[Λ exp(nf(ρ¯) Ω(n))]=1 o(1). (3.16)
33
≤ − −
Moreover,Lemma2.6entailsthat
P[Λ exp( Ω(n))Z (G)E[Z (G)]]=1 o(1). (3.17)
34 k k
≤ − −
Finally,combining(3.14)–(3.17)with(3.10)and(3.13)andusingMarkov’sinequalityoncemore,weobtain
P Λ exp( Ω(n))E[Z (G)]2 =1 o(1). (3.18)
3 k
≤ − −
Insummary,combining(3.5),(3.6(cid:2)),(3.11),(3.12)and(3.18)and(cid:3)setting,say,ω =ω(n)=lnlnn,wefindthat
P Λ ω/nE[Z (G)]2 =1 o(1). (3.19)
k
≤ −
Sfrionmce(Λ3.1=9)Z.k(G)2hkρ(σ1,σ2)−ρ¯k2iGh andpasZk(G)≥E[Zki(G)]/ωw.h.p.byTheorem2.2,theassertionfollow(cid:3)s
Lemma3.2putsusinapositiontoproveProposition3.1byextendingtheargumentthatwasusedto“plant”single
k-coloringsin[7,Section2]tothecurrentsettingof“planting”pairsofk-colorings.
ProofofProposition3.1. Assume for contradiction that ( ′ ) is a sequence of events such that for some fixed
An n≥1
numberε>0wehave
lim πpr [ ′ ]=0 while limsupπrr [ ′ ]>2ε. (3.20)
n→∞ n,m,k An n→∞ n,m,k An
Letω(n) = lnln1/πpr [ ′ ]. Thenω = ω(n) . Let bethe setof allpairs(σ ,σ ) ofmaps[n] [k]
n,m,k An → ∞ Bn 1 2 →
suchthat ρ(σ ,σ ) ρ¯ ω/nanddefine
k 1 2 − k2 ≤
p An ={(G,σ1,σ2)∈A′n :(σ1,σ2)∈Bn}.
ThenLemma3.2and(3.20)implythat
lim πpr [ ]=0 while limsupπrr [ ]>ε. (3.21)
n→∞ n,m,k An n→∞ n,m,k An
10