On conformally invariant subsets of the planar Brownian curve

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of the planar Brownian curve

Vincent Beffara

To cite this version:

Vincent Beffara. On conformally invariant subsets of the planar Brownian curve. Annales de

l’Institut Henri Poincaré (B) Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2003, 39

(5), pp.793 - 821. �10.1016/S0246-0203(03)00030-X�. �hal-01693163�

arXiv:math/0105192v1 [math.PR] 23 May 2001

the Planar Brownian Curve

Vinent BEFFARA

Département de Mathématiques

Université Paris-Sud

F-91405ORSAY Cedex, Frane

Vinent.Beffaramath.u-psud. fr

Abstrat

Wedeneandstudyafamilyofgeneralizednon-intersetionexponentsforplanarBrow-

nianmotionsthatisindexedbysubsetsoftheomplexplane: Foreah

A ⊂ C

^,wedenean

exponent

ξ(A)

^{thatdesribesthedeayofertain}non-intersetionprobabilities. Toeahof theseexponents,weassoiateaonformallyinvariantsubsetoftheplanarBrownianpath,of

Hausdordimension

2 − ξ(A)

^{. Aonsequeneofthisandontinuityof}

ξ(A)

^asafuntionof

A

^{isthealmost sureexisteneofpivoting pointsofanysuientlysmallangle onaplanar}

Brownianpath.

Résumé

Nousdénissons et étudions unefamilled'exposantsde non-intersetiongénéralisés en-

tre mouvements browniens plans, indexée par lesparties du plan omplexe : pour haque

A ⊂ C

^{nous dénissonsunexposant}

ξ(A)

^{dérivantladéroissanede} probabilitésde non- intersetion.Àhaundees exposantsest assoiéeunepartiedelatrajetoirebrownienne

qui est invariante sous l'ation des transformations onformes et qui a une dimension de

Hausdor égale à

2 − ξ(A)

^{. Une onséquene de e résultat et de la ontinuité de}

ξ(A)

omme fontion de

A

^{est l'existene presque sûre de points pivotants de tout angle assez}

petit surunetrajetoirebrownienneplane.

Contents

Introdution 2

1 Generalized intersetion exponents 4

1.1 Denition of the exponents . . . 4

1.2 Strong approximation . . . 5

2 Properties of the funtion

A 7→ ξ(A)

¹¹ 3 Hausdor dimension of the orresponding subsets of the path 13 3.1 Conformally invariant subsetsof the Brownian path . . . 13

3.2 Seondmoments . . . 15

3.3 Hausdor dimensions . . . 15

3.4 Remark about ritialases . . . 20

4 Bounds and onjetures on the exponent funtion 21

4.1 Knownexat values of

ξ

^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21}

4.2 An upperboundfor the exponent . . . 22

4.3 Conjetured andexperimentalvalues . . . 25

5 Appendix 26

5.1 Sub-additivity. . . 26

5.2 Extremaldistane . . . 26

5.3 Sometopologial tools . . . 27

Introdution

It hasbeen onjeturedfor more than twenty years by theoretialphysiiststhat onformalin-

varianeplaysanimportantrole tounderstandthebehaviourofritialtwo-dimensional models

of statistialphysis. They justifyby amathematially non-rigorous argument involving renor-

malization ideas that inthe saling limit these models behave in a onformally invariant way;

they have been able to lassify them via a real-valued parameter orresponding to the entral

harge of the assoiated Virasoro algebra, and to predit the exat value of ritial exponents

thatdesribe thebehaviourofthese systems. Dierent models (forinstane,self-avoidingwalks

and perolation) withthe sameentral hargehave thesame exponents.

Reently, Shramm [19℄ introdued new mathematial objets that give insight into these

onjetures. Thesearerandomset-valued inreasing proesses

(K _t ) _t>0

^{that healledStohasti}

LöwnerEvolutionproesses. Foreahpositivenumber

κ

^{,thereexistsonesuhproessofparam-}

eter

κ

^{, inshort}

SLE _κ

^{. He proved thatfor various models,if they have a onformally invariant}

salinglimit,thenitanbeinterpretedintermsofoneofthe

SLE _κ

^{'s(theparameter}

κ

^isrelated

to the entral harge of the model). One an then interpret the onjetures from theoretial

physiists interms ofproperties ofthis proess.

Inpartiular,Lawler,ShrammandWerner [13,14 ℄showedthatforonespeivalueofthe

parameter

κ

^(namely

κ = 6

^)whih onjeturallyorrespondedto thesalinglimitofperolation luster interfaes, the

SLE ₆

^{has the remarkable restrition property that enables to relate its}

ritial exponents to the so-alled intersetion exponents between planar Brownian motions.

This lead [13, 14,15 , 12 ℄ to the derivation of the exat value of theexponents between planar

Brownian paths. Furthermore, it turned out [23 ℄ that in fat, the outer boundary of a planar

Brownian urve hasexatlythesame lawthan that of an

SLE ₆

^{. Inother words, thegeometry}

of ritialtwo-dimensional perolation lusters intheir saling limit should be exatlythat ofa

planar Brownian outer frontier.

In a very reent paper Smirnov [20℄ showed that ritial site perolation in the triangular

lattieisonformallyinvariantinthesalinglimitsothatthegeometryofritialtwo-dimensional

perolation lusters boundariesintheir salinglimitisidentialthatofaplanarBrownianouter

frontier.

Before all these reent developments, geometri properties of planar Brownian paths had

alreadybeensubjetofnumerousstudies(seee.g.[18℄forreferenes). Inpartiular,theHausdor

dimension of various subsets ofthe planar Brownian urvedened ingeometri termshadbeen

determined. For instane,Evans[4℄showedthatthe Hausdordimension ofthesetoftwo-sided

one points of angle

θ

^(i.e.points

B _t

^{suh thatboth}

B _[0,t]

^and

B _[t,1]

^{are ontained inthe same}

one of angle

θ

^{with endpoint at}

B t

^{) is}

2 − 2π/θ

^{. In a series of papers (see [9℄ for a review),}

LawlerprovedthatthedimensionofvariousimportantsubsetsoftheplanarBrownianurve an

be related to Brownian intersetionexponents. In partiular [8℄, he showed that thedimension

oftheset

C

^{of utpoints (i.e.points}

B t

^suhthat

B _[0,1] \ { B t }

^{isnot onneted) is}

2 − ξ

^where

ξ

^{istheBrownian} intersetionexponent dened by

p R = P(B _[0,T ^{1 1}

R ] ∩ B _[0,T ^{2 2}

R ] = ∅) = R ^{− ξ+o(1)}

⁽¹⁾

(for independent Brownian paths

B ¹

^and

B ²

^starting respetively from

1

^and

− 1

^,

T _R ¹

^and

T _R ²

standing for theirrespetive hittingtimes oftheirle

C (0, R)

^).

In order to derive suh results and in partiular the more diult lower bound

d > 2 − ξ

^,

the strategy is rst to rene the estimate (1) into

p R ≍ R ^{− ξ}

^{(we shall use this notation to}

denotetheexisteneoftwo positive onstants

c

^and

c ^′

^suhthat

cR ^{− ξ} 6 p _R 6 c ^′ R ^{− ξ}

^{), toderive}

seond-momentestimatesandtousethesefatstoonstrutarandommeasureofnite

r

^-energy

supportedon

C

^{, for all}

r < 2 − ξ

^{. The} determination of the value of theritial exponents via

SLE ₆

^{[13, 14℄ then implies that the dimension of}

C

^is

3/4

^{. Similarly, in [7℄ the Hausdor}

dimensionoftheouterfrontierofaBrownianpathanbeinterpretedintermsofanotherritial

exponent,andthedeterminationofthisexponentusing

SLE 6

^{thenimplied(see[16 ℄forareview)}

thatthis dimensionis

4/3

^{asonjeturedby} Mandelbrot.

Inthe present paper, we dene andstudy a family of generalizationsof theBrownian inter-

setion exponent

ξ

parameterized bysubsets of the omplex plane. For eah

A ⊂ C

^{, we dene}

anexponent

ξ(A)

^{asfollows. Let}

B ¹

^and

B ²

^betwoindependent planarBrownianpathsstarting fromuniformlydistributedpointson theunitirle : then

ξ(A)

^{isdened by}

p R (A) = P (B _[0,T ^{1 1}

R ] ∩ A.B _[0,T ^{2 2}

R ] = ∅) = R ^{− ξ(A)+o(1)}

⁽²⁾

(withthenotation

E ₁ .E ₂ = { xy : x ∈ E ₁ , y ∈ E ₂ }

^{). Note that thease}

A = { 1 }

^orrespondsto

the usualintersetionexponent. InSetion 1,we rst showthatfor a wide lassofsets

A

p _R (A) ≍ R ^{− ξ(A)} .

⁽³⁾

In Setion 2, we study regularity properties of the mapping

A 7→ ξ(A)

^{. In partiular, we}

proveuniform ontinuity (withrespetto theHausdormetri) onertainfamiliesof sets. One

important tool for this result is the fatthat the onstants impliit in (3) an in fatbe taken

uniform overthese familiesof sets.

InSetion 3,weassoiate toeah set

A

^{a subset}

E A

^{oftheplanarBrownian urve denedin}

geometri terms:

E A = { B _t : ∃ ε > 0, (B _[t − ε,t] − B _t ) ∩ A.(B _(t,t+ε] − B _t ) = ∅} .

Using the strong approximation and ontinuity of the mapping

A 7→ ξ(A)

^{, we then show that}

the Hausdor dimension of this subset of theplanar Brownian urve is almost surely

2 − ξ(A)

(and

0

^inase

ξ(A) > 2

^{). For example,when}

A = { e ^iθ , 0 6 θ 6 α }

^,theorrespondingsubset

C _α

of the Brownian urve isthe set of (loal) pivoting points, i.e.pointsaround whih one half of

thepath anrotate of any anglesmallerthan

α

^without intersetingtheother half.

When

A ⊂ A ^′

^{, then}

E A ^′ ⊂ E A

^{. In partiular, when}

A

^ontains

1

^{, then}

E A

^{is a subset of}

the set of (loal) ut points, and therefore theshape of thepath ina neighbourhood of suh a

point is the same as the Brownian frontier in the neighbourhood of a ut-point. This shows in

partiular that (atleast some of) the exponents

ξ(A)

^{desribe also the Hausdor dimension of}

sets ofexeptional points ofthesaling limit ofritialperolation lusters.

In Setion 4, we derive some bounds on the exponents

ξ(A)

^{for small sets}

A

^{in the same}

spirit as theupperbounds for disonnetion exponents derived in [22 ℄. In partiular, for small

α

^{, we show that the exponent}

ξ(C α )

^{is stritly smaller than}

2

^{, whih implies the existene of}

pivotingpoints(ofsmallangle)ontheplanarBrownianurve. Wethenbrieypresentresultsof

simulations thatsuggestthatthere existpivoting pointsof angle upto an anglelose to

3π/4

^.

Atually, itis easy to dene other generalized exponentsin a similarfashion, bystudying

non-intersetionpropertiesbetweenBrownianmotionsandsomeoftheirimagesunderisometries

and salings, i.e. one an view

A

^{as a subset of the linear group. Also, one an onsider non-}

intersetionpropertiesbetween

B

^anditsimage

f (B)

^{byaonformalmap. Itiseasytoseeusing}

thefuntion

z 7→ z ²

^{that theexponent desribing the} non-intersetionbetween

B

^and

− B

^{is in}

fat twie thedisonnetion exponent. The methods of thepresent paper an then be adapted

to suh situations.

Similarly, one ould also extend the denitions to higher dimensions (the ases

d > 4

^an

also be interesting if the set

A

^{is suiently large), but onformal invariane an not be used}

anymore, sothat some ofthetoolsthat we useinthe present paper do not apply.

Aknowledgments

IthankWendelinWernerforsuggestingmetolookforpivotingpointsontheplanarBrownian

urve andfor neverrefusinghelp andadvie.

Notations

Throughout this paper, we will use the following notations for the asymptoti behaviour of

positive funtions(and sequenes, withthesame meaning):

• f ∼ g

^if

lim

t →∞

f (t)

g(t) = 1

^and

f

^and

g

^{aresaid to be} equivalent;

• f ≈ g

^if

log f ∼ log g

^,i.e.if

lim

t →∞

log f (t)

log g(t) = 1

f

^and

g

^arethenlogarithmiallyequivalent;

• f ≍ g

^if

f /g

^{is bounded and bounded by below, i.e. if there exist two positive nite}

onstants

c

^and

C

^{suh that for all}

t

^,

cg(t) 6 f (t) 6 Cg(t)

^{whih we all strong}

approximation of

f

^by

g

^.

1 Generalized intersetion exponents

1.1 Denition of the exponents

Proposition and Denition :

Let

A

^{be a non-empty subset of the omplex plane and}

B ¹

^,

B ²

^{be two} independent Brownian paths starting uniformlyon theunit irle

C (0, 1)

^{; dene the hittingtime}

T _R ⁱ

^of

C (0, R)

^by

B ⁱ

^{and let}

τ _n ⁱ = T _exp(n) ⁱ

^,

E _n = E _n (A) = { B _[0,τ ¹ 1

n ] ∩ AB _[0,τ ² 2

n ] = ∅ } , q _n (A) = P (E _n )

^and

p _R (A) = P (E _{log R} ).

Then,assumingtheexisteneofpositiveonstants

c

^and

C

^suhthat

p _R (A) > cR ^{− C}

^,there

existsareal number

ξ(A)

^{suhthat, when}

R → ∞

^,

p _R (A) ≈ R ^{− ξ(A)} .

//

^{This is a standard} sub-multipliativity argument. If

B

^{is a Brownian path}

startingon

C (0, 1)

^{withany law}

µ

^{,thenthelawof}

B _{τ 1 (B)}

^ontheirle

C (0, e)

^hasa

density (relative to the Lebesgue measure)bounded and bounded awayfrom zero

byuniversalonstants (i.e.independently of

µ

^{). Combining this remark withthe}

Markovpropertyat thehittingtimes oftheirle ofradius

e ⁿ

^showsthat:

∀ m, n > 1 q _m+n 6 cq _n q _m − 1 .

Hene the family

(cq _n − 1 )

^is sub-multipliative and using Proposition 7 we have

q _n ≈ e ^{− ξn}

^,with

ξ ∈ (0, ∞ )

^{,aswell asa lowerbound}

q _n > c ^{− 2} e ^{− ξ(n+1)}

^.

//

Remarks: Forsomehoiesof

A

^{thereisaneasygeometri} interpretationoftheevent

E _n (A)

^:

ξ( { 1 } )

^isthelassialintersetionexponent;if

A = (0, ∞ )

^,the

E n (A)

^{istheeventthatthepaths}

stayindierent wedges.

If

A

^{is suh thatno lower bound}

p _R (A) > cR ^{− C}

^{holds, we let}

ξ(A) = ∞

^{. However, inmost}

oftheresultspresentedhere,wewill restritourselvesto alassof sets

A

^{forwhih itiseasyto}

derivesuhlower bounds:

Denition :

A non-empty subset

A

^{of the omplex plane is said to be nie if it is ontained in the}

intersetion of an annulus

{ r < | z | < R }

^(with

0 < r < R < ∞

^{) with a wedge of angle}

stritly lessthan

2π

^{and vertexat}

0

^.

Indeed, let

A

^{be suh a set and let}

α < 2π

^{be the angle of a wedge ontaining}

A

^:

B ¹

^and

AB ²

^{will not interset provided eah path remains ina well-hosen wedgeof angle}

(2π − α)/2

^,

andthenit isstandard to derive the following bound:

p _R (A) > cR ^{− 4π/(2π − α)} .

⁽⁴⁾

The fat that

A

^{be ontained in an annulus will be needed in the following proof. The only}

usualasewhere thisdoesnot holdiswhen

A

^{is awedgeitself;butinthisasea diretstudy is}

possible,basedonthe derivationofone exponents in[4 ℄ andtheexatvalueof

ξ

^isthenknown

(f.next setionfor details).

We will often onsider the ase where

A

^{is a subset of the unit irle. For suh sets,}

A

^is

nieifand only if

A ¯ ∂U

^{(it isinfateasy to prove thatfor}

A ⊂ ∂U

^,

ξ(A) = ∞

^{ifandonly if}

A ¯ = ∂U

^).

1.2 Strong approximation

Thiswhole subsetion will be dediated to the renement of

p _R ≈ R ^{− ξ}

^into

p _R ≍ R ^{− ξ}

^{. This is}

not anedotial, sine thisstrong approximation will be neededon several oasionslater.

Theorem 1 :

For every nie

A

^,

p _R (A) ≍ R ^{− ξ(A)}

^{,i.e. there exist positive onstants}

c(A) < C(A)

^suh

that

cR ^{− ξ(A)} 6 p _R (A) 6 CR ^{− ξ(A)} .

Moreover,theonstants

c(A)

^and

C(A)

^{an betakenuniformlyonaolletion}

A

^ofsubsets

of theplane, provided the elementsof

A

^{are ontained inthesame nieset.}

//

^{Note that sine}

_{A ∈ A}

^{is nie, the exponents}

_ξ(A)

^{exists and is uniformly}

bounded(for

A ∈ A

^{). The}subadditivityargumentshowed that

q _n > ce ^{− ξ(A).(n+1)}

^,

whihimpliesreadilythelowerboundinthetheorem. Itismorediultto derive

the upper bound. By Proposition 7, it will be suient to nd a nite onstant

c − (A)

^{(thatan bebounded uniformlyfor}

A ∈ A

^{) suh that}

∀ n, n ^′ q _n+n ′ > c − q _n q _n ′ .

⁽⁵⁾

In order to make the proof more readable, it is arried out here for a xed

A

^;

howeverit is easy to see that, at eah step, theonstantsan be taken uniformly

forall

A

^{ontained insome xednie set}

A ₀

^{. Moreover,weshallrstassumethat}

A 0

^{isa subset ofthe unitirle: We briey indiate at theend of theproof what}

arethe fewmodiations neededto adapt it to thegeneral ase.

ThebasimethodisadaptedfromLawler'sprooffornon-intersetionexponents

in[10℄,withsometehnialsimpliationsmadepossibleusingtheabseneofthe

λ

exponent. Themainideaistoobtain aweakindependenebetween thebehaviour

of the paths before and after they reah radius

e ⁿ

^{. The rst step is an estimate}

onerning the probability that the paths are well separated when they reah

radius

e ⁿ

^{(morepreisely,thattheyremainintwoseparatedwedgesbetweenradius}

e ^{n − 1}

^{and radius}

e ⁿ

^):

Lemma (Tehnial) :

Let

η > 0

^and

α < 2π − η

^suhthat

A

^{isontained inawedgeof angleless}

than

α

^{. Dene}

W _α = n

re ^iθ : r > 0, | θ | < α 2

o , δ _n = e ^{− n} [d(B _τ ¹ 1

n , AB _[0,τ ² 2

n ] ) ∨ d(AB _τ ² 2 n , B _[0,τ ¹ 1

n ] )]

^{andthefollowing events:}

U _n ¹ = n B _[0,τ ¹ 1

n ] ∩ {| z | > e ^{n − 1} } ⊂ − W _2π − α − η

o , U _n ² = n

AB _[0,τ ² 2

n ] ∩ {| z | > e ^{n − 1} } ⊂ W _α o ,

and

U _n = U _n ¹ ∩ U _n ²

^{. Then:}

∃ c, β > 0 ∀ ε > 0 ∀ r ∈ 3

2 , 3

P (E _n+r , U _n+r | E _n , δ _n > ε) > cε ^β .

///

^{ThisisandiretonsequeneoflassialestimatesonerningBrownianmotion}

inwedges; the value of

β

^{is not important, sonot muh are isneeded in nding}

thelowerbound. Notethattheexisteneof

α

^requiresthat

A

^benie.

///

If

F n

^{stands for the}

σ

^{-eld generated by both paths up to radius}

e ⁿ

^{(so that}

for instane

E _n

^isin

F n

^{),we now provethatpathsonditionednot to interset up}

to radius

e ⁿ⁺²

^{have a good hane to be well separated at this radius, uniformly}

withrespetto their behaviourup to radius

e ⁿ

^:

Lemma (End-separation) :

There exists

c > 0

^{suh that, for every}

n > 0

^:

P (U n+2 | E n+2 , F n ) > c

(i.e. the essential lower bound of

P (U _n+2 | E _n+2 , F n )

^{, as an}

F n

-measurable funtion,isnot lessthan

c

^).

///

^{The tehnial lemma states that} start-separation ours ifthe starting points aresuiently far fromeah other; morepreisely,we havefor all

ε > 0

^:

P(U _n+2 | E _n+2 , F n , δ _n > ε) > cε ^β .

⁽⁶⁾

Hene, what is to be proved is that two paths onditioned not to interset have

a positive probability to be far from eah other after a relatively short time. To

prove this fat,one hasto useonditioning on thevalueof

δ _n

^.

Fix

k > 0

^{, and assumethat}

2 ^{− (k+1)} 6 δ _n < 2 ^{− k}

^{; let}

τ _k

^{be thesmallest}

r

^suh

thatone ofthefollowing happens: either

δ _n+r > 2 ^{− k}

^,or

E _n+r

^{doesnot hold. It is}

easy to usesalingto prove thatfor some

λ > 0

^,

P (τ _k > 2 ^{− k} ) 6 2 ^{− λ} ,

meaningthatwithpositiveprobability(independentof

k

^and

n

^{)thepathsseparate}

or meet before reahing radius

e ^{n+2 −k}

^{. Hene by the strong Markov property,}

applying this

k ²

^{timesleads to}

P (τ _k > k ² 2 ^{− k} ) 6 2 ^{− λk 2} .

⁽⁷⁾

The tehnial lemmastates that

P (E _n+2 | δ _n > 2 ^{− (k+1)} ) > c2 ^{− βk}

^{: ombining both}

estimates thenleads to

P (τ _k > k ² 2 ^{− k} | E n+2 , δ n > 2 ^{− (k+1)} ) 6 c2 ^{βk − λk 2} .

⁽⁸⁾

Consider now a generi starting onguration at radius

e ⁿ

^{, satisfying}

E n

^and

hene

δ _n > 0

^{. Fixalso}

k ₀ > 0

^{and introdue theradii}

τ _k

^(for

k ₀ 6 k < ∞

^)dened

by

τ _k = Inf { r : δ n+r > 2 ^{− k} }

(sothat

τ _k = 0

^aslongas

2 ^{− k} 6 δ

^{). Equation (8)an be rewritten(using thefat}

thatthe tehnial lemmais valid for all

r > 3/2

^{) as}

P (τ _k − τ _k+1 > k ² 2 ^{− k} | E _n+2 , τ _k+1 6 1

2 ) 6 c2 ^{βk − λk 2} .

Fix

k ₀

^suhthat

∞

X

k=k 0

k ² 2 ^{− k} < 1

2 ,

andsum thisestimate for

k ₀ 6 k < ∞

^{: this leadsto}

P ( ∀ k > k ₀ , τ _k − τ _k+1 6 k ² 2 ^{− k} | E _n+2 ) > 1 − c

∞

X

k=k 0

2 ^{βk − λk 2} .

Inpartiular, if

k ₀

^{istakenlarge enough, this} probability isgreater than

1/2

^,and

we obtain

P(τ _k 0 6 1

2 | E _n+2 ) > 1 2 .

Itis thensuient to ombine thisand Equation(6) to get

P (U _n+2 | E _n+2 ) > c2 ^{− βk 0} > 0,

andisanbeseenthattheobtainedonstantdoesnotdependontheonguration

at radius

e ⁿ

^provided

E _n

^issatised.

///

Therstonsequene oftheend-separationlemmais

P (E _n , U _n ) ≍ q _n

^{;but itis}

easy to see, using estimates on Brownian motion inwedges again and the strong

Markovproperty,that

P(E n+1 | E n , U n ) > c > 0

(with

c

independent of

n

^{), and ombining both estimates leads to}

q _n+1 > cq _n

^,

i.e.

q _n+1 ≍ q _n

^{. Now if}

q ¯ _n

^{stands for the upper bound for the} non-intersetion probabilities,namely

¯

q _n = ^∧ Sup

B ¹ ₀ ,B ² ₀ ∈U

P(E _n | B ₀ ¹ , B ₀ ² ),

thepreviousremark onerningthe lawof

W _{τ 1 (W )}

^{anbe usedtoprove that}

q ¯ _n 6 cq _n − 1

^{: hene,}

¯ q _n ≍ q _n .

Nowthatwe knowthatpaths onditionednot to intersethave agoodhane

to exit a disk at a large distane from eah other, what remains to be proven is

that paths starting from distant points on

C (0, e ⁿ )

^{remain well separated for a}

suiently long time and beome (in a sense to be speied later) independent

fromtheir behaviour beforeradius

e ⁿ

^.

Lemma (Start-separation) :

Let

α

^and

η

^{be asin thetehnial lemma,}

η ^′ = η/2

^and

α ^′ = (2π + α)/2

^;

introdue

J _n ¹ = n B _[0,τ ¹ 1

n ] ∩ B (0, 2) ⊂ − W _2π − α ^′ − η ^′ \ B (0, 1 − η ^′ ) o , J _n ² = n

AB _[0,τ ² 2

n ] ∩ B (0, 2) ⊂ W _α ′ \ B (0, 1 − η ^′ ) o ,

and

E ˜ _n = E _n ∩ J _n ¹ ∩ J _n ²

^{. Dene}

q ˜ _n

^as

˜

q n (x, y) = P( ˜ E n | B ₀ ¹ = x, B ₀ ² = y).

Then there exists

c > 0

^{suh that, for all}

n > 2

^{and uniformly on all pairs}

(x, y)

^satisfying

U ₀

^{(i.e.suh that}

U ₀

^holdswhen

B ₀ ¹ = x

^and

B ₀ ² = y

^):

˜

q _n (x, y) > cq _n .

///

^{Introdue the following} (forbidden)sets:

K ¹ = B (0, e) \ − W _2π − α ^′ − η ^′

∪ B (0, 1 − η ^′ );

K ² = ( B (0, e) \ W _α ′ ) ∪ B (0, 1 − η ^′ ).

For all

n

^wehave

J _n ¹ = { B _[0,τ ¹ 1

n ] ∩ K ¹ = ∅ }

^and

J _n ² = { AB ² _[0,τ 2

n ] ∩ K ² = ∅ }

^{. Forthe}

rest of the proof we shall x

n

^{, and ondition thepaths by their starting points;}

introdue thefollowing stopping times (forpositive values of

k

^):

T ₀ ¹ = Inf { t > 0 : B _[0,t] ¹ ∩ C (0, 3) 6 = ∅ } , S _k ¹ = Inf { t > T _k ¹ − 1 : B _[T ¹ 1

k − 1 ,t] ∩ K ¹ 6 = ∅ } , T _k ¹ = Inf { t > S _k ¹ : B _[S ¹ 1

k ,t] ∩ C (0, 3) 6 = ∅ } ,

and

S _k ²

^,

T _k ²

^{similarly, replaing allourrenes of}

B ¹

^by

AB ²

^and

K ¹

^by

K ²

^{. We}

shall also use the notation

N ⁱ

^{for the number of rossings by}

B ¹

^(resp.

AB ²

⁾

between

K ⁱ

^and

C (0, 3)

^{,dened as}

N ⁱ = Max { k : S _k ⁱ < τ _n ⁱ } .

Withthose notations,

J _n ⁱ = J ₁ ⁱ ∩ { N _n ⁱ = 0 }

^anda.s.

N ⁱ < ∞

^{. Moreover, uniformly}

on the startingpointsonsidered here(satisfying theondition

U 0

^{),we have}

J ₁ ⁱ >

c > 0

^{bythe tehnial lemma, where}

c

^{dependsonly on}

η

^.

First,wesplittheevent

E n

^{aordingtothevalueof,say,}

N ²

^{: wewrite}

P (E n ) = P ∞

k=0 P (E _n , N ² = k)

^{. By the Beurling estimate, on}

{ N > k }

^{, the} probability that

B _[0,τ ¹ 1

n ]

^and

AB _[S ² 2

k ,T _k ² ]

^{do not interset is bounded by some universal onstant}

λ < 1

^{(whih an even be hosen} independent of

A

^), independently of

B ¹

^and

the two remaining partsof

B ²

^{. By the strong Markov propertyat time}

T _k ²

^,when

N ² = k

^the probability that

AB ²

^after

T _k ²

^{does not interset}

B ¹

^{is bounded by}

P (B ¹ ∩ AB _[T ² 2

0 ,τ _n ² ] = ∅, N ² = 0)

^{(i.e.thepath after}

T _k ²

^when

N ² = k

^isthesameas

the entirepath when

N ² = 0

^{). Introduing thosetwoestimateinthesumleadsto}

P (E _n ) 6

∞

X

k=0

λ ^k P (E _n , N ² = 0) = 1

1 − λ P (E _n , N ² = 0).

Doing this deomposition again aordingto

N ¹

^{(with the same onstant}

λ < 1

⁾

we thenobtain

P (E _n ) 6 1

(1 − λ) ² P (E _n , N ¹ = N ² = 0),

i.e.

P(N ¹ = N ² = 0 | E _n ) > (1 − λ) ² > 0

^{. This, and the previous remark that}

P (J _n ⁱ | N ⁱ = 0)

^{isboundedbybelowbyaonstantprovidedthatthestartingpoints}

satisfy

U 0

^,gives:

P ( ˜ E _n | B ₀ ¹ = x, B ₀ ² = y) > cP (E _n | B ₀ ¹ = x, B ₀ ² = y).

⁽⁹⁾

Conditioning on

B ²

^{shows thatthemap}

f : x 7→ P (E _n | B ₀ ¹ = x, B ₀ ² = 1)

⁽¹⁰⁾

isharmonianddoesnotvanishontheomplement of

A

^{. Moreover,itssupremum}

ontheunitirleisequal to

q ¯ n

^{bydenition: ApplyingtheHarnakpriniple then}

proves that

f

^{is bounded by below by}

cq _n

^{on the set of}

x

^satisfying

U 0

^{, whih}

ompletes the proof.

///

Anotherestimateanbeobtainedusingtheverysameproof: Onlykeepingthe

onditionsinvolvingdisksand relaxing those involvingwedges, we obtain

P B _[0,τ ¹ 1

n ] ∩ B(0, 1 − η) = ∅, AB _[0,τ ² 2

n ] ∩ B (0, 1 − η) = ∅

B ₀ ¹ , B ₀ ² , E _n

> c > 0,

⁽¹¹⁾

where

c

^{does not depend on the initial positions}

B ₀ ¹

^and

B ₀ ²

^{, nor on}

n

^{(it learly}

depends on

η

^{, though, and a loser look at the proof shows that we an ensure}

c > η ^β

^as

η → 0

^{,for some}

β > 0

^{). Thisestimate will be needed inthederivation}

ofHausdordimensions, f.Setion 3.

We know have all theneeded estimates to derive the lower bound in thesub-

additivityondition,andhenetheonlusionofthetheorem. Taketwopathswith

independent starting points uniformly distributed on theunit irle and killed at

radius

e ^m+n

^{, onditioned not to interset between radii}

1

^and

e ⁿ

^{. This happens}

withprobability

q _n

^{. Withlarge} probability(i.e. witha positiveprobability, inde- pendentof

m

^and

n

^{)thepathsuptoradius}

e ⁿ

^{endupwellseparated inthesense}

of the end-separation lemma. In partiular, the points where they reah radius

e ⁿ

^{, after suitable resaling, satisfy the hypothesis of the} start-separation lemma:

Hene with probability greater that

cq _m

^{, the paths between radii}

e ⁿ

^and

e ^m+n

remainseparatedupto radius

e ⁿ⁺¹

^{,donotreahradius}

(1 − η)e ⁿ

^anymoreanddo

not intersetup to radius

e ^m+n

^{. Underthoseonditions, itiseasy to seethat the}

paths do not meet at all. So

q _m+n > cq _m q _n

^{for some positive}

c

^{, and we get the}

onlusion.

Someadaptations areneededif

A

^{isinluded inanannulus, say}

{ r < | z | < R }

with

r < 1 < R

^{. First, replae all ourrenes of}

e

^by

e ₀

^{, with}

e ₀

^{hosen larger}

than

10R/r

^{,andin the}start-separationlemma, replae

B (0, 1 − η)

^by

B (0, r/2R)

inthe denitionof the

J _n

^{. As long as}

r

^and

R

^{are xed, this hanges nothing to}

theproof,exeptthat the onstantswe obtainwill then depend on

R/r

^whih

itselfisbounded provided

A

^{remainsa subset ofsome xednie set.}

A more serious problemarises iftheomplement of

A

^{is not onneted, sine}

the natural domain of the funtion

f

^{(as dened by Equation (10)) is itself not}

onneted. However, sine

A

^{is nie, its omplement has exatly one unbounded}

omponent, and it is easy to see that if

x

^{is not in this omponent then}

f(x)

vanishes for

n > 1

^{. Hene, nothing hanges (as far as} non-intersetion properties areonerned)when

A

^{isreplaedbytheomplement oftheinniteomponentof}

itsomplement (i.e.when llingtheholes in

A

^).

//

In fat, a stronger result an be derived: If the starting points

B ₀ ¹

^and

B ₀ ²

^{are xed, then}

P (E n | B ₀ ¹ , B ₀ ² )

^{is equivalent to}

ce ^{− nξ(A)}

^{, where}

c

^{is a funtion of}

B ₀ ¹

^and

B ₀ ²

^satisfying

c 6

c 0 d(B ₀ ¹ , AB ₀ ² ) ^β

^{. This estimate is related to a strong onvergene result on the law of paths}

onditioned by

B ¹ ∩ AB ² = ∅

^{. However, proving this result would be muh more involved}

(f.[17℄ for the proofinthease

A = { 1 }

^).

2 Properties of the funtion

A 7→ ξ(A)

We rstlista fewsimple properties ofthefuntion

A 7→ ξ(A)

^{. For}

p ∈ Z

^and

A ⊂ C

^,introdue

A ^p = { z ^p , z ∈ A }

^andlet

A ^∗ = { z, z ¯ ∈ A }

^.

Proposition 1 :

Is these statements, all setsare assumedto be non-emptybut do not need to be nie:

(i).

ξ

^is non-dereasing: if

A ⊂ A ^′

^then

ξ(A) 6 ξ(A ^′ )

^;

(ii).

ξ

^is homogeneous: if

λ ∈ C ^∗

^then

ξ(λA) = ξ(A)

^;

(iii).

ξ

^{is symmetri:}

ξ(A ^{− 1} ) = ξ(A ^∗ ) = ξ(A);

(iv).

ξ

^{hasthe following property: if}

n > 1

^then

ξ [

e ^2ikπ/n A

= nξ(A ⁿ ).

//

^{(i): Thisisa trivialonsequene of}

p _R (A) > p _R (A ^′ )

^.

(ii): Applying the saling property with fator

| λ |

^to

B ²

^{proves that one an}

suppose

| λ | = 1

^{; in whih ase we have}

p R (A) = p R (λA)

^{(beause the starting}

pointsareuniformlydistributed ontheunit irle).

(iii): Simplyexhange

B ¹

^and

B ²

^for

A ^{− 1}

^{,andsaythattheomplex onjugate}

of aBrownian path isstill aBrownian path to get

A ^∗

^.

(iv): This is a onsequene of the analytiity of the mapping

z 7→ z ⁿ

^(hene

the fat that

((W _t ) ⁿ )

^{is a Brownian path if}

W

^{is one) together with the remark}

thattheexisteneof

s

^,

t > 0

^and

z ∈ A ⁿ

^with

(B _s ¹ ) ⁿ = z(B _t ² ) ⁿ

^{isequivalent tothe}

existeneof

z ^′

ⁱⁿ

S

e ^2ikπ/n A

^with

B _t ¹ = z ^′ B _t ²

^{notethatthemappingalsohasan}

inuene on

R

^{,henethefator}

n

^.

//

We now turn our attention toward regularity properties of the funtion

A 7→ ξ(A)

^the

following resultbeingakeysteptowardsthederivation ofdimensionsinthenextsetion. Intro-

due theHausdordistane between ompat subsetsof theplane (f.Setion 5 for details). It

willbeonvenient hereto deneneighbourhoods by

V _r (A) = { xe ^z , x ∈ A, | z | < r }

^insteadofthe

usual

A + B (0, r)

^{leadingto the logarithmi Hausdordistane. The}(logarithmi)Hausdor topologyis the metritopology derivedfrom thisdistane.

Proposition 2 :

ξ

^{is ontinuous on the olletion of nie sets, endowed withthe} logarithmial Hausdor topology. For anynieset

A ₀

^,

ξ

^{is uniformlyontinuousin}

{ A : A ⊂ A ₀ }

^.

//

^{Theproofreliesontheuniformityofthestrong}approximationinTheorem1: x anie set

A ₀

^{andassumeall setsonsideredherearesubsetsof}

A ₀

^{. Theonstants}

c

^,

c −

^and

c +

^{appearing duringthe proof mayonly depend on}

A 0

^.

First,x

R > 1

^{andonditionalleventsby}

B _[0,T ² 2

R+1 ]

^{i.e.xtheseondpath.}

For all

A ⊂ A ₀

^,let

d _R (A) = d(B ¹ _[0,T 1

R ] , AB _[0,T ² 2 R ] ) ;

for all

ε > 0

^{introdue the stopping time}

S _ε = Inf { t : d(B ¹ _t , AB _[0,T ² 2

R ] ) < ε } .

Notethat

{ d R (A) < ε } = { S ε < T _R ¹ }

^{. Onthis event, thestrong Markovproperty}

showsthat

B _S ¹ _{ε + ·}

^{isa Brownian path starting}

ε

^-loseto

AB ²

^{. By Beurling'stheo-}

rem,the probabilitythattheydo notmeet beforeradius

R + 1

^{issmallerthan the}

orrespondingprobabilityfor a path nearahalfline; hene,

P(B _[S ¹

ε ,T _R+1 ¹ ] ∩ AB _[0,T ² 2

R+1 ] = ∅ | d _R (A) < ε) 6 √ ε,

sothat, onsidering the wholepath,

P (E _R+1 | d _R (A) < ε) 6 √

ε

^{. Apply theBayes}

formula:

P (d _R (A) < ε | E _R+1 ) = P (d _R (A) < ε)

P (E _R+1 ) P (E _R+1 | d _R (A) < ε);

sineweknowthat

P (E R+1 ) > c − (R + 1) ^{− ξ(A)}

^with

ξ(A) 6 ξ(A 0 )

^{wenallyobtain}

P(d _R (A) < ε | E _R+1 ) 6 cR ^{ξ(A 0 )} √

ε.

Fromnowon,weshallassumethat

ε

^{issuientlysmall tomaketheobtained}

bound smallerthat

1

^{. Taking the omplement leads to}

P(d _R (A) > ε | E _R+1 ) > 1 − cR ^{ξ(A 0 )} √

ε.

Now, remark that when

d _R (A) > ε

^and

d _H (A, A ^′ ) < ε/R

^{, we have}

B _[0,T ¹ 1 R ] ∩ A ^′ B _[0,T ² 2

R ] = ∅

^{: fromthis and the previous equationfollowsthat, aslongas}

A

^and

A ^′

^{remain subsetsof}

A 0

^,

d _H (A, A ^′ ) < ε

R ⇒ p _R (A ^′ ) >

1 − cR ^{ξ(A 0 )} √ ε

p _R+1 (A).

We an apply the estimates on

p _R

^{we derived in Theorem 1 i.e.}

p _R (A) ≍ p _R+1 (A) ≍ R ^{− ξ(A)}

^{: stillfor}

d _H (A, A ^′ ) < ε/R

^and

A

^,

A ^′

^inside

A ₀

^weget

c ₊ R ^{− ξ(A ′ )} >

1 − cR ^{ξ(A 0 )} √ ε

c − R ^{− ξ(A)} ,

andtakingthe logarithm ofeah sideof the inequality leadsto

log c ₊ − ξ(A ^′ ) log R > log c − + log

1 − cR ^{ξ(A 0 )} √ ε

− ξ(A) log R,

heneaftersuitable transformations:

ξ(A ^′ ) 6 ξ(A) + c

log R − log 1 − cR ^{ξ(A 0 )} √ ε

log R .

⁽¹²⁾

Fix

η > 0

^{,and hoose}

R

^suhthat

c/ log R < η/2

^{. Itisthenpossibleto take}

ε

suiently small sothat

| log(1 − cR ^{ξ(A 0 )} √

ε) | < (η log R)/2

^;for

d _H (A, A ^′ ) < ε/R

we then have

ξ(A ^′ ) 6 ξ(A) + η

^{, hene by symmetry}

| ξ(A ^′ ) − ξ(A) | 6 η

^{. This}

proves that

ξ

^{is uniformly ontinuous on}

P c (A 0 )

^{, for all}

A 0

^{, heneontinuous on}

the family ofnie sets.

//

Remark 1: Equation (12) allows the derivation of an expliit modulus of ontinuity for

ξ

inside

A 0

^{,ofthe form}

| ξ(A ^′ ) − ξ(A) | 6 C(A 0 )

| log d _H (A, A ^′ ) |

(take

R = d ^{− 1/2ξ(A 0 )}

^{). But sine}

C(A 0 )

^{is not known, this does not provide numerial bounds}

for

ξ

^.

Remark2: Insideanieset,theusualandlogarithmiHausdortopologiesareequivalent,so

theintrodutionofexponentialneighbourhoods inProposition2anseemartiial;however,it

leadsto onstants thatdonot varywhen

A

^{is multipliedbysome onstant (asin}Proposition1, point (ii)), hene uniform ontinuity holds on the olletion of nie sets ontained in a xed

wedgeand insomeannulus

{ r < | z | < cr }

^{for xed}

c

^{whihis wrongfor theusual Hausdor}

topology,asaonsequene ofthe homogeneityof

ξ

^{applied for small}

| λ |

^.

Note that uniform ontinuity annot hold on the family of nie sets ontained in a given

annulussine

ξ

^{wouldthenbebounded(byaompaityargument),whihitisnot: theexponent}

assoiated to airle isinnite.

3 Hausdor dimension of the orresponding subsets of the path

3.1 Conformally invariant subsets of the Brownian path

It is well-known that the Brownian path is invariant in law under onformal transformations;

inthis setion, we study subsetsof theBrownian urve thatarealso invariant underonformal

maps. Arst example istheset of so-alled Brownian ut-points, i.e.points

B _t

^suhthat

B _[0,t)

and

B _(t,1]

^{aredisjoint;thesepointsformasetofHausdordimension}

2 − ξ( { 1 } ) = 3/4

^{. Relatedto}

thoseareloal ut-points,i.e.pointssuhthatthereexists

ε > 0

^satisfying

B _[t − ε,t) ∩ B _(t,t+ε] = ∅

thedimensionisthesame asforglobalut-points. Other examplesaregivenbyLawlerin[9℄:

inpartiularthe setofpioneer points (suhthat

B _t

^{liesonthefrontieroftheinniteomponent}

ofthe omplement of

B _[0,t]

^{), relatedto the} disonnexionexponent

η ₁

^{;frontier points (points of}

theboundaryoftheinniteomponentoftheomplementof

B _[0,1]

^{),relatedtothe}disonnetion exponent for two paths inthe plane. Another exeptional subset of thepath is theset of one

points (suh that

B _[0,t]

^{is ontained in a one of endpoint}

B _t

^{), related to the one exponents}

(studiedin[18 ℄ forexample).

We will usethe exponent introdued inthe previous setions to desribe a family of exep-

tionalsets, indexed by a subset

A

^{of the omplex plane, having dimension}

2 − ξ(A)

^{, and that}

areinvariant under onformaltransformations, as follows. Fixa Brownian path

B _[0,1]

^{, asubset}

A

^{of the omplex plane,and introdue thefollowing times for all}

t ∈ (0, 1)

^and

r > 0

^:

T r (t) = Inf { s > t : | B s − B t | = r } , S r (t) = Sup { s < t : | B s − B t | = r } .

Denition :

If

0 < ε < R

^and

t ∈ (0, 1)

^,let

Z _t ^[ε,R] (B) =

B _s − B _t B _s ^′ − B t

: s ∈ [T _ε (t), T _R (t)], s ^′ ∈ [S _R (t), S _ε (t)]

;

and introdue

E _A ^[ε,R] = { B _t : Z _t ^[ε,R] ∩ A = ∅ }

^{. Then, letting}

ε

^{go to}

0

^:

Z _t ^R = [ ↑

ε>0

Z _t ^[ε,R] , Z _t = \ ↓

R>0

Z _t ^R , Z ˜ _t = \ ↓

R>0

Z _t ^R ;

dene

E A ^R

^,

E A

^and

E ˜ A

^aordingly.

Weshallalsousethenotation

T A = { t : B t ∈ E A }

^{,for thesetof}

A

-exeptional times,and

T ˜ A = { t : B _t ∈ E ˜ A }

^{,for thesetof}

A

^{-strongly exeptional times.}

Note that, sine

0

^{is polar for planar Brownian motion,}

Z

^{is well-dened for almost any}

t

^.

For

A = { 1 }

^,

E A

^{isthe set ofloal ut-points; more generally,}

t

^{is in}

E A

^{if,and only if,for some}

ε > 0

^{,we have}

(B _(t,t+ε] − B _t ) ∩ A.(B _[t − ε,t) − B _t ) = ∅,

so the setup looks similar to the denition of the exponent

ξ(A)

^{. It is easy to see that for all}

xed

t > 0

^,a.s.

Z _t = C ^∗

^and

Z ˜ _t = C

^,sothatfor

A 6 = ∅

^,

P (t ∈ T A ) = 0

^{,leading to}

E(µ( T A )) = 0

i.e.

µ( T A ) = 0

^{almost surely henethe term exeptional points.}

Theset

E A

^of

A

-exeptional points is generallynot onformallyinvariant. However, itis the ase forstrongly exeptional points:

Proposition 3 :

Let

Φ

^{be a onformal map on a} neighbourhood

Ω

^of

0

^{, with}

Φ(0) = 0

^{, and let}

B ^Ω

^be

B

^{stopped at its rst hitting of}

∂Ω

^{. By onformal invariane of planar Brownian motion,}

Φ(B ^Ω )

^{isaBrownian path stopped at its rsthitting of}

∂Φ(Ω)

^{. Moreover, we have}

E ˜ A (Φ(B ^Ω )) = Φ( ˜ E A (B ^Ω )).

//

^{We prove that}

_Z ^˜

^{isinvariant. Itis suient to prove thefollowing harateri-}

zation:

z ∈ Z ˜ _t (B) ⇐⇒ ∃ (s _n ) ↓ 0, (s ^′ _n ) ↓ 0 : B _{t+s n} − B _t B _t − s ^′ _n − B _t → z,

asonformalmapsonservethe limitsofsuhquotients. Suhasequene iseasily

onstrutedusing thevery denitionof

Z ˜

^.

//

Note that nothing in the preeding uses the fat that

B

^{be a Brownian path, exept for}

the remark about

P (t ∈ T A )

^{. The remaining of the present setion is dediated to deriving}

the Hausdor dimension of

E A

^and

E ˜ A

^{. It will be more onvenient to work in the time set, so}

introdue

T _A ^[ε,R] = { t ∈ [0, 1] : B _[t ¹ − R,t − ε] ∩ A.B ² _[t+ε,t+R] = ∅} .

ThesalingpropertyofBrownianmotionanthenbeusedto show, asin[8,lemmas3.143.16℄,

thatTheorem 1implies the following:

P (t ∈ T A [ε, R]) ≍ ε R

ξ(A)/2

.

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