On the precision of the spectral profile

0.12 MB·

by  Gady Kozma

On the precision of the spectral profile
GadyKozma
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TheWeizmannInstituteofScience,RehovotPOB76100,Israel.
0
0 E-mailaddress:
gady.kozmaweizmann.a
.il
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n Abstract. Weexaminethe spectralprofilebound ofGoel, MontenegroandTetali
a fortheL∞mixingtimeofcontinuous-timerandomwalkinreversiblesettings. We
J
find that it is precise up to a loglog factor, and that this loglog factor cannot be
3
improved.

]
R
P
.
h 1. Introduction
t
a
m Ofalltheformulassuggestedintheliteratureasboundsforthemixingtimeof
afinite graph(seee.g.[LK99,MP05,FR07]),possiblythemostpromising, froma
[
geometricpointofview,isthespectralprofileformula. IntroducedbyGoel,Mon-

2 tenegro and Tetali [GMT06], it brings into the realm of finite graphs the idea of
v
Faber-Krahninequalities. AFaber-Krahninequalityisaninequalityrelatingthevol-
2
umeof aset A andthe firsteigenvalueofthe Laplacianwith Dirichletboundary
1
1 conditionsonA—wewillgivealldefinitionsinthediscretesettingsbelow,butfor
0 the history of the topic, mainlyin continuous settings, one should consult [G94],
. [C01, §VIII.6]or [B01], which hasa somewhat differenttake onthis topic and an
9
0 excellenthistoricalsurvey.Thisapproachispromisingbecause,asGrigory’andis-
7 covered [G94], on a general complete manifold it gives sharp estimates on the
0 decay of the heat kernel, even in cases where the manifold does not have polynomial
:
v volumegrowth. The requirementof polynomial growth was essential in previous
i approachestothisproblem,usingSobolev[V85]orNash[CKS87]inequalities.
X
LetusdescribeFaber-Krahninequalitiesinthediscretesettings. Wewillwork
r
a withweighted,undirected,finitegraphs. LetGbesuchagraphandω : G×G →
[0,∞)theweightfunction. Theheatkernelisdefinedby
ω(x,y)
K(x,y):= ω(x):=∑ω(x,z).
ω(x)
z
The heat kernel is a stochastic matrix (i.e. ∑ K(x,y) = 1) and hence describes a
y
Markovchain on G. The symmetry ω(x,y) = ω(y,x) givesthat it isself-adjoint
withrespecttothestationarymeasureπdefinedby
ω(x)
π(x):=
∑ ω(y)
y
2000MathematicsSubjectClassification. 60J27(primary),68W20,58J65.
Keywordsandphrases. L∞mixingtime,L2mixingtime,uniformmixingtime,maxingtime,spectral
profile,Faber-Krahninequality,roughisometry,quasiisometry,coarseisometry,bi-Lipschitzmap.
1
Ontheprecisionofthespectralprofile 2
and therefore the associated Markov chain is reversible. It is important to note
thatthe resultsof [GMT06] arenot restrictedto the reversiblecase, and applyto
any finite Markov chain, but in this paper we will restrict our attention to the
reversiblecase.TheLaplacian,whichisanoperatoronL2(G,ω)isdefinedsimply
as∆:= I−Kandisself-adjointandpositive.
When A ⊂ G is some subset, we will introduce the restricted Laplacian with
Dirichletboundaryconditions
∆f(x) v∈ A
(∆ f)(x)= (1)
A
(0 otherwise.
The smallest eigenvalue for ∆ will be denoted by λ (A). It is easy to see that
A 0
λ (A)mayalsobedefinedas
0
h∆f, fi
λ (A)= inf (2)
0 suppf⊂A ||f||2
2
f6≡0
wherehf,gi = ∑ f(x)g(x)π(x)and||f||p = ∑ f(x)pπ(x). Itissomewhatmore
x p x
eleganttodescribetheresultsof[GMT06]withthefollowingquantityinstead,
h∆f, fi
λ(A) = inf (3)
suppf⊂A ||f||2−||f||2
2 1
f≥0,f6≡const
and we will adhere to this convention. Note that as long as π(A) ≤ 1−ǫ the
quatitiesλ andλarecomparable[GMT06,eq.(1.4)].AFaber-Krahninequalityis
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aninequality of the form λ(A) ≤ Λ(π(A)) for some function Λ, so the minimal
functionΛsatisfyingthisisdefinedby
Λ(r):= inf λ(A).
0<π(A)≤r
Λ(r)isthespectralprofile. Itisdefinedforallr ≥π∗ :=min∅6=A⊂Gπ(A).
∞
Themainresultof[GMT06]isaboundfortheL mixingtimeofthecontinuous-
timerandomwalkintermsofthespectralprofile. Letusgivethenecessarydefi-
nitions. Thecontinuous-timerandomwalkonGisdefinedusing−∆astheinfin-
itesimalgenerator. Explicitly,wedefine
∞ tn
H =e−t∆ =e−t ∑ Kn(x,y)
t
n!
n=0
andthink about H (x,y) asthe probabilitythata particle doing continuous-time
t
random walk on G, starting from x will be at y at time t. Hence we define the
mixingtimeusing
H (x,y)−π(y)
τ∞(ǫ):=inf t >0: sup t ≤ ǫ .
( x,y∈G(cid:12) π(y) (cid:12) )
(cid:12) (cid:12)
Wemaynowstatethemainresultof[GMT(cid:12)06], (cid:12)
(cid:12) (cid:12)
4/ǫ 2dr
τ∞(ǫ)≤ . (4)
Z4π∗ rΛ(r)
Intherestofthediscussionwewillfixǫ = 1 anddenotethelefthandsidebyτ∞
2
andtherighthandsidebyρ.
Ontheprecisionofthespectralprofile 3
1.1. The starting point of this short note was the hope that in fact (4) is precise
in the sense that ρ < Cτ∞ (1). This was motivated by the fact that Faber-Krahn
inequalitiesgivesharpboundsinmanyinterestingmanifolds,andbythefactthat
(4)isinfactsharp underacertainδ-regularitycondition (see[GMT06, §3]). And
infact,thetechniquestheregivequiteeasily(andwithnoregularityassumption),
thefollowing:
Theorem1. ForanyfinitegraphG,
ρ< Cτ∞loglog1/π∗(G).
Unfortunately,itturnsoutthatthiscannotbeimproved. Indeedwehave
Theorem2. Thereexistasequencenk → ∞andgraphsGkofsizenk andπ∗ = n1k such
that
ρ≥ cτ∞loglognk.
The proof oftheorem2 isalsonot difficult—the graphs G willbe (detailsin
k
section2)composedofloglogn piecesH whereeachH correspondstoadistinct
k i i
range of r-s in the integral defining ρ (4). On the other hand, a random walker
starting at H will see only H — when it finally leaves H it will already be too
i i i
mixed to notice any effects from the other H-s. Thus, perhaps the most natural
j
questiontoaskis
Question. IsitpossibletohavethegraphsG transitive?
k
A graph G is transitive if for all x,y ∈ G there exists anautomorphism of the
graphtaking x to y. Itwould be extremelyexcitingif the answertothe question
weretobeno. Alessexciting,butnonethelessverynaturalquestionisasfollows.
Question. Is it possible to have the graphs G unweighted and of uniformly bounded
k
degrees?
Herethe rationalefor the question is geometric. The analogybetween graphs
andmanifoldsworksbestformanifoldswithboundedgeometryandgraphswith
bounded degrees. Hence there is a certain discord in the fact that the examples
constructedintheorem2areweighted. Onewouldbetemptedtosolvetheques-
simple
tionbyconstructingthegraphs(callthemG )randomly,namelyputanedge
k
simple
betweenxandyinG withprobabilityω(x,y)whereωistheweightfunction
k
ofG . However,morecareisneeded—applyingtherecipeabovenaivelywould
k
immidiatelycreatelogarithmictailsthatwoulddominatethemixingtime.
Finally, we remark that in non-reversible settings existing bounds are quite
weak. For example, it is possible to have ρ ≥ c|G|2 while τ∞ ≤ C|G|log|G|. A
carefuldiscussionofthisphenomenoncanbefoundin[MT06,examples5.3-5.5].
1.2. Anotherrelevantsetofproblemsrevolvesaroundthefollowing: isthemix-
ingtimeageometricproperty? Thisisparticularlyinterestingsincemanyresults
inmixinghavebeenachievedusingrepresentationtheory([BD92]isprobablythe
1
HereandbelowweuseCandctodenoteabsolutepositiveconstantsthatmaybedifferentfrom
placetoplace. Cwillbeusedforconstantswhichare“largeenough”andcforconstantswhichare
“smallenough”.Thenotation f ≈gwillstandforcf ≤g≤Cf.
Ontheprecisionofthespectralprofile 4
most famous) or using coupling (e.g. [LRS01]), techniques which are better de-
scribedas“algebraic”ratherthan“geometric”.Tomakethequestionformalletus
definethenotionofaroughisometry.
Definition. LetX andY bemetricspacesandlet f : X → Y beafunctionandlet
K ∈ (0,∞). We say that f is a K-rough isometry if the following two properties
hold:
(1) Foranya,b∈ X,
1
d(a,b)−K ≤d(f(a), f(b))≤Kd(a,b)+K.
K
(2) Foranyy∈Ythereexistsanx ∈ Xsuchthat
d(f(x),y)≤ K.
TousethisforMarkovchainswewillrestrictourselvestothesimplestsettings,
thatofrandomwalkona(unweighted)graphwithboundeddegree. Inthiscase
thegraphhasanaturalmetricstructuregivenbythepathmetric,i.e.thedistance
d(v,w)isdefinedtobethelengthoftheshortestpathbetweenv andw. Andwe
ask: isthemixingtimeinvarianttoroughisometries? Formally:
Conjecture. Let G, H be two graphs with degG,degH ≤ d. Let f : G → H be a
K-roughisometryinthepathmetricsonGand H. Then
τ(G)≤C(K,d)τ(H). (5)
Sincearoughisometryisreversible,thiswouldinfactimplythatτ(G)≈ τ(H).
ItisaninterestingobservationthatallapproximationsforthemixingtimeIam
awareofareroughisometryinvariants.Itiseasytoseethatisoperimetricinequal-
ities are rough-isometry invariants, and hence both the Lovász-Kannan integral
[LK99] and the Fountoulakis-Reed integral [FR07] (which bounds the L1 mixing
time rather than our τ∞, but the conjecture is just as relevant for τ1) are rough-
isometryinvariants. Toseethat,forexample,thespectralgapisaroughisometry
invariantsonehastodefineitusingfunctionalinequalitiesi.e.(3)—notethatthe
spectral gap is exactly λ(G) — and then it becomes easy to check that the spec-
tralgapandthespectralprofilearebothroughisometryinvariants. Thusaprecise
boundofthisstyleforthespectralgapwouldprobablyimplytheconjecture.
Inparticular,combining[GMT06, theorem1.1]withtheorem1givesaweaker
formof(5):
τ∞(G)≤C(K,d)(loglog|G|)τ∞(H). (6)
Thisresult,however,isnotnew. Indeed,τ∞ iscomparabletothebestconstantin
thelogarithmicSobolevinequalityαdefinedby
h∆f, fi
α = inf
Entπ f26=0Entπ f2
inthesensethat
c Cloglog1/π∗
≤ τ∞ ≤ .
α α
See [MT06] for historical background, the definition of the entropy Ent and for
π
the equivalence (theorem 4.13 ibid). It is easy to see that α is a rough isometry
invariancehencethisgivesanotherderivationof(6).
We end this discussion with an observation of Itai Benjamini, that the mixing
time from a given point is not a rough isometry invariant. Thus, for example, the
Ontheprecisionofthespectralprofile 5
mixingtimefromtherootofabinarytreeofheighthis≈h. However,themixing
time from a neighbor of the root is ≈ 2h (see [AF, chapter 5] for both). Since
there is a rough isometry of a tree on itself carrying the root to a neighbor, this
demonstratestheclaim.
I wish to thank László Lovász and Prasad Tetali for many useful discussions.
This material is partially based upon work supported by the National Science
Foundation underagreementDMS-0111298. Anyopinions, findings andconclu-
sions or recommendations expressedinthis materialaremine and donot neces-
sarilyreflecttheviewsoftheNationalScienceFoundations
2. Proofs
Proofoftheorem1. Denote by A a Rayleigh set of measure 2−k i.e. π(A ) ≤ 2−k
k k
and
λ(A )=min{λ(S) : π(S) ≤2−k}.
k
Whereλisfrom(3). Itiseasytoseethat
⌊log21/π∗⌋ 1
ρ ≈ ∑ .
λ(A )
k=1 k
Ontheotherhand,by[GMT06,lemma3.1],foranyk ≥2
−log(π(A )) ck
τ∞ ≥c k ≥ . (7)
λ(A ) λ(A )
k k
As for k = 1, we have 1/λ(A ) ≤ 1/λ(G) but λ(G) is just the spectralgap and
1
hence 1/λ(G) ≤ Cτ∞ [MT06, theorem 4.9]. Hence (7) holds for k = 1 as well.
Therefore
⌊log21/π∗⌋ Cτ
ρ ≤ ∑ ≈τloglog1/π∗. (cid:3)
k
k=1
Proofoftheorem2. We may assume w.l.o.g. that k is sufficiently large. We define
simply
n = (k−⌈logk⌉+1)22k
k
wherelog, here andbelow, isto base2. The graphwill consist of k−⌈logk⌉+1
pieces,whichwedenotebyH ,...,H ,eachwith22k vertices. Wecanalready
⌈logk⌉ k
definetheweightfunctionbetweentheHs: foreveryv ,v indifferentHsweset
l 1 2 l
k2−k
ω(v ,v ) = .
1 2 |G|
Let A beasetofverticesofsize22k−2l and B asetofsize22l. Setwisewedefine
l l
H = A ×B andthendefinetheweightfunctionωasfollows:
l l l
1 2l−k+ 1 k2−k b 6= b
ω((a ,b ),(a ,b ))= |Hl| |G| 1 2
1 1 2 2 (|A1l| + |H1l|2l−k+ |G1|k2−k b1 = b2.
With this definition of ω we would have that ω(v) = 1+2l−k+k2−k for every
v ∈ H. The inhomogeneity of ω issomewhat bothersome sowe modify ω(v,v)
l
Ontheprecisionofthespectralprofile 6
tofixthis,writing
1 1 1
ω(v,v)= (1−2l−k)+ + 2l−k+ k2−k ∀v ∈ H
|A | |H | |G| l
l l
withtheresultbeingthatω(v)=2+k2−kforallv.
With our graph G defined we can start investigating its properties. We first
estimatethespectralprofileρ. BythediscreteinverseCheegerinequality[AM85,
lemma2.1],foranysetS,
π(∂S) ω(∂S)
λ(S) ≤ C =C .
π(S) ω(S)
wherewe consider the weight function ω asa measurewhich is aconstant mul-
tiple of π. We use it for the set A ×{pt} which we confusingly call A˜ . Now,
l l
ω(A˜ ) ≈ |A | = 22k−2l. Therearetwotypesof edgescoming out of A˜ , edgesto
l l l
H andedgestotheother Hs. Thefirsttypehasweight
l i
1 1 1
2l−k+ k2−k = 2l−k(1+o(1))
|H | |G| |H |
l l
where the o notation above and also below means “as k → ∞, uniformly in l ∈
[logk,k]”. so, after summing over all couples (v ,v ), v ∈ A˜ and v ∈ H \
1 2 1 l 2 l
A˜ gives a total contribution ≤ |A |2l−k(1+o(1)). The second type has weight
l l
1 k2−k so aftersumming overall (v ,v ), v ∈ A˜ and v ∈ G\H gives atotal
|G| 1 2 1 l 2 l
contribution≤|A |k2−k. Sincel >logkweget
l
ω(∂A˜ )
λ(A˜ ) ≤C l ≤C2l−k. (8)
l ω(A˜ )
l
Now, π(A˜ ) = |A |/|G| = 1/(k−⌈logk⌉+1)22l. Forbrevitydenoteǫ = 1/(k−
l l
⌈logk⌉+1). Wegetthat,
ǫ/22l−1 dv 1 ǫ/22l−1 dv 1 (8)
≥ ≥ ·c2l ≥ c2k.
Zǫ/22l vΛ(v) Λ(ǫ/22l) Zǫ/22l v λ(A˜l)
Summingweget
8 dv
ρ = ≥c2k(k−⌈logk⌉+1)≥ck2k.
Z4π∗ vΛ(v)
Theproofwillbefinishedonceweshowthatτ ≤C2k.
Let us therefore investigate the random walk on G. It will be convinient to
representit as follows. Assume the walker is at a vertex v ∈ H. We first throw
l
a coin which has probability k/2kω(v) of success. Call the event that this throw
succeededξ andinthiscasechooseoneoftheverticesofGrandomlywithequal
1
probability and move there. If ξ did not occur, throw a second coin which has
1
probability
2l−k
=2l−k−1
ω(v)−k2−k
to succeed. Call the event that this throw succeeded ξ and in this case choose
2
oneoftheverticesofH randomlywithequalprobabilityandmovethere.Finally,
l
Ontheprecisionofthespectralprofile 7
throwacoinwithprobability
1 1
=
ω(v)−k2−k−2l−k 2−2l−k
to succeed (if l = k it always does). Call the event that this throw succeeded ξ
3
andinthiscasechooseoneoftheverticesofthecopyof A containingvrandomly
l
with equal probability and move there. If none of ξ , ξ and ξ succeeded, stay
1 2 3
at v. It is easy to see that this is equivalent to the walk on the graph (infact, we
definedtheweightsonthegraphwiththisrepresentationinmind).
Examinearandomwalkoflength2k+1,startingfromsome v ∈ H. Letw ∈ G
l
andexamineP(R(2k+1) = w)(thatstartingpointwillalwaysbev—wewillnot
remind this fact in the notation). We first note that after an event of type ξ the
1
walkiscompletlymixed. Defineτ tobethefirsttimewhenξ occurred,whichis
1 1
astoppingtime. UsingthestrongMarkovpropertyweget,foreveryt≤2k+1,
1
P({τ =t}∩{R(2k+1) =w}) =P(τ = t)
1 1 |G|
andsummingovertgives
1 1
P({τ ≤2k+1}∩{R(2k+1) =w}) =P(τ ≤2k+1) ≤ . (9)
1 1 |G| |G|
Now,theeventτ >2k+1canbeestimatedsimplyusing
1
2k+1 2k+1
k2−k k2−k
P(τ >2k+1) = 1− ≤ 1− ≤e−2k/3 (10)
1 2+k2−k! 3 !
whichimmediatelygivesalowerboundP(R(2k+1) = w) ≥ (1−o(1))/|G|valid
forallw. Further,ifw 6∈ H then(9)givesanupperbound,sinceonecannotreach
l
from v to w without a ξ event. Hence we will henceforth assume w ∈ H and
1 l
τ >2k+1. Theestimate(10)isnice,butfarfromourgoalof 3 .
1 2|G|
Afteraneventoftypeξ thewalkistotallymixedin H. Thereforeifwedefine
2 l
τ to be the first time when ξ occurred then a similar calculation to the above
2 2
showsthat
P({τ >2k+1}∩{τ ≤2k+1}∩{R(2k+1) =w}) =
1 2
1 (10) 1 1
P(τ >2k+1)P(τ ≤2k+1|τ >2k+1) ≤ e−2k/3 =o .
1 2 1 |H| |H | |G|
l l (cid:18) (cid:19)
With(9)wehave
1
P({min{τ ,τ } ≤2k+1}∩{R(2k+1) =w}) ≤ (1+o(1)). (11)
1 2 |G|
Againwenotetheprobabilitythatτ >2k+1:
2
P(min{τ ,τ }>2k+1) ≤ 1−2l−k−1 2k+1 ≤ e−2l (12)
1 2
(cid:16) (cid:17)
Ontheprecisionofthespectralprofile 8
Forthelastpartweassumew ∈ A˜ wherehere A˜ isthecopyof A containingv.
l l l
Wedefineτ asthefirsttimeξ occurredandget
3 3
P({min{τ ,τ }>2k+1}∩{τ ≤2k+1}∩{R(2k+1)= w}) =
1 2 3
1 (12)
P({min{τ ,τ } >2k+1})P(τ ≤2k+1| min{τ ,τ }>2k+1) ≤
1 2 3 1 2 |A |
l
≤e−2l ·22l−2k = o 1 . (13)
|G |
(cid:18) k (cid:19)
Finally,inthecasethatτ ,τ ,τ >2k+1(sowmustbev)wedefinitelyhave
1 2 3
P(min{τ ,τ ,τ } >2k+1)≤ 1− 1 2k+1 ≤2−2k+1 = o 1 . (14)
1 2 3 2−2l−k |G |
(cid:18) (cid:19) (cid:18) k (cid:19)
Summingup(9),(11),(13)and(14)wefinallyget
1
Pv(R(2k+1) =w) ≤ (1+o(1))
|G|
andhenceforksufficientlylarge,τ ≤2k+1. Thisendsthetheorem. (cid:3)
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