Girsanov’s transformation for SLE(,) processes, intersection ......121 Girsanov s transformation...

ANNALES DE LA FACULTÉ DES SCIENCES DE TOULOUSE

WENDELIN WERNERGirsanov’s transformation for SLE(κ,ρ) processes,intersection exponents and hiding exponentsAnnales de la faculté des sciences de Toulouse 6e série, tome 13,no 1 (2004), p. 121-147<http://www.numdam.org/item?id=AFST_2004_6_13_1_121_0>

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121

Girsanov’s transformation for SLE(03BA,03C1) processes,intersection exponents and hiding exponents(*)

WENDELIN WERNER (1)

Annales de la Faculté des Sciences de Toulouse Vol. XIII, n° 1, 2004

ABSTRACT. - We relate the formulas giving Brownian (and other) in-tersection exponents to the absolute continuity relations between Besselprocess of different dimensions, via the two-parameter family of Schramm-Loewner Evolution processes SLE(03BA, p) introduced in [23]. This allows usalso to compute the value of some new exponents ( "hiding exponents" )related to SLEs, planar Brownian motions and the conjectured scalinglimit of two-dimensional critical systems.

RÉSUMÉ. - Nous faisons le lien entre les formules donnant les valeurs

des exposants d’intersection entre mouvements browniens plans et les rela-tions d’absolue continuité entre processus de Bessel de différentes dimen-

sions, via la famille à deux paramètres de processus de Loewner-SchrammSLE(03BA, p) introduite dans [23]. Ceci permet en particulier de déterminerla valeur de nouveaux exposants critiques pour le mouvement brownienplan et les SLE.

1. Introduction

The value of the intersection exponents between planar Brownian mo-tions has been derived in the series of papers [15, 16, 17, 18] using the re-lationship with the exponents for the Schramm-Loewner Evolution processwith parameter 6 (in short SLE6) that can be computed directly. For in-stance, if B 1, ... , BP denote p independent planar Brownian motions startedfrom p fixed different points on the unit circle, the probability that the ptraces BI [0, t], ... , Bp[0, t] remain disjoint and all stay in the same (fixed)half-plane decays like t-(p/2 as t tends to infinity. The exponent (p is called

(*) Reçu le 11 février 2003, accepté le 4 septembre 2003(1) Laboratoire de Mathématiques, Université Paris-Sud, 91405 Orsay cedex, France.

E-mail: [email protected]

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the half-space intersection exponent between p Brownian motions and it isproved in [15] that (p = p(2p + 1)/3, as conjectured in [7].

Before SLE allowed to determine the value of these exponents, it wasshown in [25] that in order to understand these Brownian exponents, itis convenient to introduce "generalized" Brownian exponents 03BEp(03B11,..., ap)that correspond (in the case where all ai’s are integers) to the decay ofthe probability of non-intersection between p unions of planar Brownianmotions in a half-plane containing respectively 03B11,..., ap paths. For instance

03BEp(1,..., 1) = (p. In particular, one can define the function

and show (this is not mysterious, it is basically a consequence of conformainvariance of planar Brownian motion) that

This, combined with the conjectures by Duplantier-Kwon [7] for (p allowedto predict the value of U and of the generalized exponents 03BE. Duplantier [4]then observed that this type of equation can also be viewed as coming fromthe quantum gravity formalism, which provided yet another way to predictthe exact form of the function U.

In [26], the relation between the Brownian exponents and the expo-nents for self-avoiding walks and critical percolation was pointed out. Moreprecisely, a "universality" argument was presented that showed that all

conformally invariant models that possess a certain locality condition mustbasically have the same exponents i.e. the same function U. This allowed torecover the predictions (see [26] and the references therein) for the criticalexponents for self-avoiding walks or critical percolation from the above-mentioned prediction for U, and conversely to show that the value of theBrownian exponents would follow from the computation of the exponentsfor any other local conformally invariant object. This is the strategy thatwas successfully used in [15] : Show that SLE6 is local, and compute its ex-ponents. The derivation of thé SLE6 exponents (in the half-plane) is in facta computation related to the (real) Bessel flow. This gave the rigorous proofof the fact that indeed U(x) = x + 1/24 - 1/24 as predicted in [25].

In the recent paper [23], the same basic idea was developed in a differentsetting. There, the family of random sets satisfying the so-called confor-mal restriction property is fully described and classified (the correspondingprobability measures are called "restriction measures"). This leads [22] to

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the precise conjecture that SLEs/3 is the scaling limit of the half-plane self-avoiding walk. It also proves [23] that the boundary of planar Brownianmotion, the boundary of the scaling limit of critical percolation cluster in-terfaces (that Smirnov [32] proved to be indeed corresponding to SLE6 ) andthe (conjectured) scaling limit of the self-avoiding walk, do not only have thesame exponents but are in fact the same random object. The family of re-striction measures is parametrized by a positive real parameter a that can beinterpreted as the number of planar Brownian motions that this restrictionmeasure is equivalent to. More precisely, when a is a positive integer, onecan construct the restriction measure with exponent a by considering theunion of a independent Brownian excursions (i.e. in the half-plane, Brown-ian motions started from the origin that are "conditioned" to stay forever inthe upper half-plane). This shows that the half-plane intersection exponents03BE correspond to intersection exponents between restriction measure samples.Note (but this will not be directly relevant here even if it provides one addi-tional motivation, since one would wish to also understand the relation withthe intersection exponents) that the restriction measures are closely relatedto highest-weight representations of some infinite-dimensional Lie algebras(see [10]).

As shown in [23], the restriction measures (more precisely, their outerboundary) can be described via variants of SLEs/3 called SLE(8/3, p) (eachp corresponds to a value of a). As we shall briefly recall in the next section,SLE(8/3, p) is defined as SLE8/3 except that the driving Brownian motionis replaced by a (multiple of) a Bessel process (actually, it is a little bit

more complicated than that). We shall see in the present paper that withthis SLE(03BA, p) approach, the computation of the intersection exponents canbe interpreted as the standard absolute continuity relations between Besselprocesses of différent dimensions (following from Girsanov’s theorem).

This provides the value of various new exponents, some of which describeprobabilities of events that are associated to planar Brownian motions: Forinstance, consider n + m independent Brownian motions in the complexplane that are started from i, and stopped at their first hitting of the line{Q(z) = R}. What is the probability that they all stay in the upper half-plane and that none of the n first Brownian motions contributes to the"right-hand" boundary of the union of the n + m paths restricted to thestrip {Q(z) E [1,R]} (i.e. the n paths are hidden from +~ by the m otherpaths - note that this does not imply non-intersection between the paths)?When R -+ oo, the probability that this happens decays like a negativepower of R and the corresponding exponent is

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(the n+m part is just corresponding to the fact that the n + m paths remainin the upper half-plane). Let us comment that just as for the generalizedintersection exponents, the values of these "hiding" exponents are rationalonly for exceptional values of n, m. For instance, even the exponent form = n == 1 is the irrational number (3 + 7)/2. However, for m = 1 andn = 4, the exponent is 7. These "hiding" exponents do not seem to haveappeared before in the theoretical physics literature.

The fact that such exponents can be determined can seem somewhat

surprising. It is due to the fact that the SLE(8/3, p) approach makes itpossible to separate the information given by the boundary of the randomsets (i.e. the law of the exterior boundary of a union of Brownian paths) fromwhat happens "in the inside" . An example of such facts is the symmetry ofthe Brownian frontier as described in [23].

Last but not least, this description not only provides the values of theintersection exponents, but it also gives directly the law of the paths thatare conditioned not to intersect. Of course, all this is very closely relatedto the computations of the exponents in [15] as principal eigenvalues ofsome differential operators, and to the corresponding eigenfunction (andthe underlying stationary diffusion, for instance the diffusion conditionedto never hit the boundary of the domain), but it is simply formulated interms of these SLE(K, p) processes. This is much less involved than the

corresponding non-intersection conditioning in the Brownian case (see [13,20]).

.

The results are not restricted-to the K = 8/3 case. Hence, one obtains also"hiding/intersection exponents" in the general case. In particular, a non-intersection exponent between p SLE03BA’s (with some Brownian loops addedin a proper way) turns out to be simply p(p - 1)/03BA. These exponents areconjectured to be relevant in the study of two-dimensional critical systemsfrom statistical physics. Recall in particular [29] that SLE03BA for all 03BA E [4,8]are supposed to correspond to the scaling limit of two-dimensional criticalmodels, and that their outer boundaries are conjectured (see [2]) to beclosely related to the SLE(16/03BA, 03C1) processes.

As explained at the end of the paper, it also gives a new and simple inter-pretation of the "quantum gravity function" from [12] predicted by Knizh-nik, Polyakov and Zamolodchikov that has been used in various forms topredict the values of exponents of two-dimensional critical systems by theo-retical physicists (see e.g. [5] and the references therein). Loosely speaking,the "quantum gravity exponent" (conjectured to correspond to the samesystem but on a random lattice) is just the value p that appears in theSLE(03BA, p).

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When writing up this paper, 1 had basically the choice between on theone hand being sloppy at times, but with reasonable heuristic intuition, orgiving precise complete statements and proofs that would hide the intuitionbehind stochastic calculus considerations and technical setups. 1 deliberatelychose the first option, since I believe that the gaps left are reasonable.

Acknowledgements. - I would like to express my deep gratitude to GregLawler and Oded Schramm. This paper can be viewed as an addendum toour joint papers, and it could have been (co)-authored by them as well.I have also benefited from stimulating and enlightening discussions withJulien Dubédat.

2. Background

2.1. (One-sided) restriction

We now recall some facts and notation from [23]. Define the family A ofclosed subsets A of the closed upper half-plane H such that

1. H B A is simply connected.

2. A is bounded and bounded away from the negative half-line.

To each such A, associate the conformal mapping ~A from H B A onto Hsuch that ~A(0) = 0 and ~A(z) ~ z when z ~ oo.

We say that a closed set K C H is left-filled if

2022 K and H B K are both simply connected and unbounded

2022 K ~ R = R_

We say that a random left-filled set satisfies one-sided restriction if forall A E A, the law of K is identical to that of ~A(K) given the event{K ~ A = ~}. It is not very difficult (see [23]) to prove that this implies thatfor some positive constant a, one has for all A,

Conversely (see [23]), for each a > 0, there exists a unique random left-filledset satisfying this identity. Its law is called the one-sided restriction measurewith exponent a. It can be explicitly constructed using the SLE(8/3, p)processes that will be described below.

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Define now the family At just as A except that condition 2. is replacedby the condition that A is bounded and bounded away from 0. This im-mediately leads to the definition of "two-sided" restriction properties (see[23]) that we shall also use in the present paper. These are the random sets(which are no longer left-filled) such that for all A At, the law of K isidentical to that of ~A(K) conditionally on {K n A = ~}. Again, (2.1) hasto hold for all A E At and some fixed a. It is proved in [23] that this canbe realized if and only if 03B1 5/8. Furthermore, the only random simplepath that satisfies the two-sided restriction property is SLE8/3 for whichthe corresponding exponent is a = 5/8.

2.2. Absolute continuity relation between Bessel processes

Suppose that (Xt, t 0) is a Bessel process of dimension d 1 startedfrom x > 0 (see e.g. [28] for more details on the content of this subsection).In other words,

where (Bt, t 0) is a standard one-dimensional Brownian motion startedfrom 0. As customary, we will also use the index v related to the dimensiond by

Recall that X hits the origin if and only if d 2.

Suppose for a moment that d = 2 and that J-l is some non-negative realnumber. Then Itô’s formula shows immediately that

is a local martingale. It is then possible to apply Girsanov’s theorem to un-derstand (for each fixed t > 0) the behavior of X under the new probabilitymeasure Qt defined by

(it is standard to check that in this particular case, the exponential localmartingale exp(p log Xt - 03BC2 (log X~t/2) is a martingale, so that Qt is indeeda probability measure): Under this new probability measure,

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for s E [0, t] is a Brownian motion. In other words, as

it follows that under this new probability measure, (Xs, s E [0, t]) is a Besselprocess of dimension d’ = 2 + 2J-L i.e. of index y.

Note that the probability measures Qt are compatible in the sense thatthe QT probability of any 03C3(Bs, s E [0, t]) measurable set is independent ofT > t. Hence, one can in fact define a probability measure Q that coincideswith Qt on 03C3(Bs, s E [0, t]) for all t. Under this probability measure Q, theprocess (Xs, s 0) is a Bessel process of index ,u.

Conversely, suppose now that d > 2 (i.e. v > 0) and define

Then, under the new probability measure Q, the process X is a two-dimen-sional Bessel process.

Plugging in these two facts together shows that if X is a Bessel processof dimension 2 + 2v 2 started from x > 0, then under the probabilitymeasure Q that is induced by the probability measures Qt defined by

the process X is a Bessel process of index 03BC (instead of v) started from x.As we shall see, this relation between 03BC - v and the exponent y 2 - v2 willbasically be the reason for the particular form of the critical exponents (i.e.the fact that the function U-1 is quadratic) in our two-dimensional context.

Note that (unless il = v), Q is not absolutely continuous with respectto P (the limiting behavior of X when t ~ oo depends on its dimension).Similarly, one can let x go to zero, and interpret heuristically the resultas the relation between Bessel processes of different dimension that arestarted from zero. This is not formally true since dQt/dP is not well-definedanymore (Qt is singular with respect to P because the almost sure behaviorof X at time 0+ depends on its dimension).

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2.3. The SLE(03BA, p) processes

We now recall the definition of the SLE (03BA, 03C1) processes. Suppose that03BA > 0 and p > -2. Let X denote a Bessel process of dimension

that is started fro]

Define 03BAX and

and also

so that

Then, one defines the Loewner chain gt with driving function Wt i.e. for at 0 and z in the closed upper half-plane H,

(as long as gt(z) does not hit Wt). For each t, gt is a conformal mapfrom a domain Ht onto H, where Ht is the set of points z e IHI such thai|gs(z) - Ws| > 0 for s E [0, t]. We call this process SLE(03BA, p). When p = 0this is the (usual) chordal SLE03BA process.

We will for the time being assume that d 2 (so that X does not hi10). This means that

When p = 0, this corresponds to the fact that 03BA 4.

Suppose that Yo = a > 0. Then, for all fixed t > 0, the law of W[0, t] isabsolutely continuous (even if the density may not be bounded, or boundedaway from zero) with respect to that of 03BABt, and therefore the law ofthe Loewner chain up to time t is absolutely continuous with respect tothat of SLE03BA. In particular, it is almost surely generated by a continuouscurve (see [29, 21]). If Yo = 0, then this does not hold directly, but one canapply the same reasoning to the chains gto+t °g-1t0 to deduce that SLE(03BA, p)is generated by a continuous curve, that we shall denote by 03B3. In other

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words, gt is the conformal map from the unbounded connected componentof H B 03B3[0, t] onto H that is normalized at infinity by gt(z) = z + o(1). Thiscurve is simple if and only if Ii 4 ([29]).

The process

should be understood as the left-image gt(0-) of the origin under gt (whenY0 = 0, then the origin can correspond to two prime ends in gt 1 (H) i.e.

the origin has two images corresponding to the limit from the left and fromthe right of the curve). The fact that d 2 ensures that the left-image of 0is never "swallowed" by the SLE(03BA, p) curve, i.e. that the curve never hitsthe negative half-line. On the other hand, the SLE(03BA, p) hits the positivehalf-line if 03BA > 4.

The intuition behind the drift term when 03C1 ~ 0 is the following: It is arepulsion from the origin (more precisely from 0-) if p > 0 or an attractive

force toward the origin if p 0. The fact that d 2 ensures that the

repulsion/attraction is such that the SLE curve never hits the negative half-line : For instance, when 03BA = 6, the repulsion has to be sufficiently strong sothat the SLE does not swallow the origin (i.e. one must have 03C1 1). WhenK = 8/3, the process can be attracted toward zero without swallowing it(all the values 03C1 -2/3 work).

Here since O0 = 0 and 14,’o == a, we say that the SLE(03BA, p) process isstarted from (0, a). Similarly, for any o w, one can define an SLE(03BA, p)started from (o, w) by translating the SLE started from (0, w - o) by o.

Note that the SLE(03BA, p) curve is obtained deterministically (via theLoewner chain) from the process Y (or X).

2.4. Restriction and SLE(03BA, p) processes

In [23], it is proved that the boundary of the sample of a one-sidedrestriction measure of exponent ’TJ is an SLE(8/3, p) process where

(here and in the sequel, we will use the bars to indicate that this is a functionand not a parameter). It is also shown that if one adds (or decorates) anSLE03BA curve with parameter 03BA 8/3 with a Poisson cloud of Brownian loopswith intensity À, where

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(of course, this depends on the actual normalizing factor in the definition ofthe loop soup), and then "left-fills" the obtained set, one obtains a sampleof a (one-sided) restriction measure of exponent

The same argument is generalized in [2], where it is shown that for allr, 8/3, one can decorate the SLE(I1:, p) curve with a Poisson cloud ofBrownian loops of intensity ÀK and obtain a one-sided restriction measuresample with exponent

We refer to [23, 2] for further details.

The Brownian loop decoration procedure can be roughly summarizecas follows: There exists an infinite measure M supported on (unrooted;Brownian loops in the half-plane. A realization of the Brownian loop-soupwith intensity A is a Poisson point process with intensity 03BBM. A sample ojthe loop-soup is therefore an infinite countable collection of Brownian loop!:in the upper half-plane. One decorates a curve with the loop-soup by addingto the curve all the loops of the loop-soup that it intersects. See [27, 23] foimore details. When K = 2, this is also closely related to the fact that SLE2is the scaling limit of loop-erased random walk as proved in [21].

2.5. Conditioned Bessel processes

In this paper, we will interpret the probability measure Q defined il]

Subsection 2.2 in terms of conditioning (i.e. "Q is P conditioned on somEevent" ). This conditioning is singular (it is with respect to an event of zercprobability), so that this interpretation has to be made more precise inorder to be rigorous. It is very similar to the interpretation of the three-dimensional Bessel process as one-dimensional Brownian motion "condi-

tioned to remain positive" . More generally, when d 2, it is well-knownthat the d dimensional Bessel process "conditioned to remain positive" is a4 - d dimensional Bessel process. In order to clarify what we will mean, itis worthwhile to briefly recall these classical facts (see e.g. [28]):

Suppose in the present subsection that X is a d dimensional Besselprocess with dimension d 2, that is started from x > 0. Let P denote itslaw and T its hitting time of the origin (T is a.s. finite because d 2). Itis easy to check that (X2-dmin(t,T),t 0) is a martingale, and it follows that

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Girsanov’s transformation for SLE(r,, p) processes

if one defines for any t > 0, the probability measure Qt by

on the 03C3-field Ft generated by (Xs, s t), one gets a compatible familyof probability measures (just as in Subsection 2.2), which in turn defines aprobability measure Q, such that for any t, Q = Qt on Ft. Clearly, underQ, the process X does never hit the origin (because for each t, X[0, t] doesnot hit the origin Qt almost surely) .

The probability measure Q will be interpreted as the law of the Besselprocess X conditioned never to hit the origin. The weighting in the def-inition of Qt can be decomposed as follows: The 1tT term correspondsto the conditioning by the event that X[0,t] does not hit the origin, andthe (Xt/x)2-d term corresponds to the "renormalized probability" that thefuture of X does not hit the origin.

A way to make this interpretation rigorous goes as follows: Considerfor any R > x, the hitting time TR of R by X. Then, the law of X[0, TR]conditionally on the event TR T is identical to the law of X[0, TR] underthe measure Q (and this holds for each R, so that the corresponding lawsare compatible). Here, there is no weighting in terms of the future of Xafter TR because XTR = R so that all paths contribute the same way.

Recall also (and this is a classical consequence of Girsanov’s theorem)that the law of X under the probability measure Q is that of a 4 - d dimen-sional Bessel process. This fact will be useful later on.

3. Absolute continuity between SLE(r,, p)’s

We are now going to combine the previous considerations. Consider anSLE(03BA, p) with 03BD 0 that is started from (0, a) as before, where a > 0.

Recall that

i.e.,

Define for 03BC 03BD, the probability measure Q induced by the measures Qt asbefore i.e. for all t > 0

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where

Then, under the probability measure Q, the process X is a Bessel process ofdimension 2 + 2J-l (instead of 2 + 2v) started from x = 03B1/03BA. Hence, underthis new probability measure, the Loewner chain gt corresponds to that ofan SLE(K,;5), where

1

Recall that

(this formally follows from the differentiation of (2.2) with respect to z)Therefore,

n

so that

Hence,

where

In order to interpret (3.1), it is convenient to introduce an auxiliary inde-pendent sample K of a one-sided restriction measure of exponent a. Then

where X is a Bessel process of dimension 2+2J1 started from x. Let us stressthat this is an exact identity and not just an asymptotic expansion.

We can let a ~ 0 for fixed t. The previous formula shows readily that

wher(

with Xo = 0 (the density of Xl near 0 behaves like y1+203BC so that thisexpectation is finite). This gives the value 03BC - 03BD for the intersection exponentbetween a one-sided restriction measure with exponent a and the SLE(03BA, p)process.

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Furthermore, as in Subsection 2.5, it is possible to interpret Q as thelaw of this SLE conditioned to never intersect K. For instance for any R,the law of X[0,TR] conditionally on X[0,?R] n K = 0 is Q (when TR isthe hitting time of R by the underlying Bessel process). This conditionedSLE(03BA, p) is therefore an SLE(03BA, 03C1).

To avoid notational confusion and for future reference, let us sum up therelation between the exponents, p’s, a’s etc. Even if for ease, the statementsare loosely stated, they are rigorous when formulated in the way that wehave just described.

e An SLE, conditioned to avoid a sample of a one-sided restrictionmeasure of exponent a is an SLE(03BA, 03C1) where

Conversely, an SLE (03BA, p) can be viewed as an SLE, conditioned notto intersect a one-sided restriction sample of exponent

e An SLE(03BA, p) conditioned to avoid a one-sided restriction sample ofexponent 03B1 is an SLE(03BA, p) where

e The exponent associated to the non-intersection event between an

SLE(03BA, p) and a one-sided restriction sample of exponent Q is

More precisely, if an SLE(03BA, 03C1) is started from a > 0 and runs upto time 1, the probability that it does not intersect the one-sided

restriction sample of exponent a decays like a constant times aU when03B1 ~ 0.

Note that

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which is not surprising: Conditioning the SLE to avoid a restriction sampleof exponent a and then to avoid a restriction sample of exponent 13 is thesame as conditioning to avoid a restriction sample of exponent a + 13.

Let us briefly insist on the following fact: The conditioning is a condition-ing of both the SLE and the restriction sample (i.e. the pair is conditionedso that they do not intersect). In the previous statements, we consider themarginal law of this conditioned SLE, and we do not describe the law of thecouple. However, the restriction property shows easily how to recover thelaw of the restriction sample conditioned not to intersect that SLE (i.e. it

gives the conditional law of the restriction sample given the (conditioned)SLE). More precisely, consider an SLE(03BA, p) as before. Let 0393_ be the con-

nected component of H B -y which has the negative half-line on its boundary.Define also a conformal map 03A8_ from H onto r - that fixes both the ori-

gin and infinity, and let K denote an independent sample of the one-sidedrestriction measure of exponent 03B1(03BA,03C1). Then, the joint law of a restrictionmeasure sample conditioned not to intersect an SLE03BA (in the sense describedabove) is just that of (03A8_ (K), 03B3).

4. Exponents

This implies a variety of results concerning the value of critical expo-nents. To illustrate this, we now briefly describe some of them, leavingdetails and further exponents for the interested reader.

4.1. Hiding exponents between one-sided restriction measures

Let us first focus on the case 03BA = 8/3. In this case, SLE(8/3, p) is itself theright-boundary of a one-sided restriction sample of exponent 1] = ~(8/3, p).Suppose that d 2 i.e. ~ 1/3 and 03C1 -2/3. Then, the intersection expo-nent between this right-boundary and another one-sided restriction sampleof exponent 13 is

or = a(1] hides /3) = 03C3(8/3, p, 03B2)

This can be interpreted as a hiding exponent between one-sided restrictionmeasures of exponents 1] and /3: Consider two independent samples K,7 andK{3 of one-sided restriction measures with respective exponents ~ 1/3 and

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03B2 0. Consider the probability that the right-boundary of KTf U K,3 in thestrip {çs(z) E [1,R]} consists only of points in K~. This probability decayslike R-u as R ~ oo.

In the special case where ~ = 5/8, the right-boundary of the restrictionmeasure sample is the SLE8/3 curve itself. Hence, non-intersection betweenthe right-boundary of K5/8 and K,3 is just non-intersection between K,3 andthe SLE curve, so that the exponent cr in this case is the same as the non-intersection exponent (5/8,03B2) = 03BE(5/8,03B2) - 5/8 - /3 between restrictionmeasures. This gives another way (if one combines the obtained value of03BE(5/8, 0) with the cascade formula (1.1)) to recover the Brownian half-plane exponents that were derived in [15, 17] using computations involvingthe SLE6 processes.

In the very special case where q = /3 = 5/8, one gets a description ofthe right-boundary of the union of two Brownian excursions in terms of oneSLE8/3 conditioned not to intersect another independent one. We will comeback to this is the two-sided case.

When 7/ > 5/8, the hiding exponent cr is smaller than (~, 03B2), which isnot surprising since the corresponding events are larger.

Note again, that for the values of q such that 1 + 24n is a perfect square(for instance 7y = 1 and q = 2 corresponding to one or to the union of twoBrownian excursion), the obtained exponents are simpler.

Let us insist on the fact that this is valid for all 03B2 0, and ~ 1/3(corresponding to the d 2 assumption) but that it does a priori nothold for 1] 1/3; we shall see in the next subsection what to do in thiscase. Recall that 1] = 1/3 corresponds to the scaling limit of conditionedpercolation cluster boundaries (see [23]). The exponent cr is in this specialcase equal to 303B2/2.

There exist various alternative ways to formulate the "hiding events"since restriction measures can be described in terms of Brownian excur-sions or conditioned SLE6’s (see [23]). For example, one way to phrase thisin terms of planar Brownian motions is described at the end of the intro-duction. The proof is a consequence of the previous considerations, and ofthe relation between exponents for Brownian excursions and for Brownianmotions as developed for instance in [20].

Note that the existence of these hiding exponents itself is a non-trivialfact (sub-multiplicativity does not simply hold as it does for non-intersectionevents) .

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4.2. When 77 1/3

As we have just mentioned, the previous expression for the hiding expo-nent (1 is not valid when 77 1/3. One way to circumvent the difficulty isto first condition the boundary of the one-sided restriction sample of expo-nent 17 not to hit the negative half-line (it cannot hide another restrictionmeasure if it hits the negative half-line). Recall (see Subsection 2.5) that aBessel process X of dimension d 2 started from x E [0, 1] hits 1 before0 with probability x2-d (because X 2-d is a local martingale), and that theprocess "conditioned" not to intersect 0 is a Bessel process of dimension4 - d.

Suppose now that p E (-2, -2/3), so that d 2. The probability thatan SLE(8/3, p) started from (0, a) does not intersect the negative half-linebefore its capacity (the Loewner time-parametrization) reaches one, decayslike a constant times a2-d = a-1/2-303C1/4 when a ~ 0. Furthermore, it followsfrom the relation between the d and 4 - d dimensional Bessel processes thatthe conditioned process is an SLE(8/3, p*) where

(see [2] for a similar facts). The corresponding exponents = ~(8/3, p) and~* = ~(8/3, p*) satisfy

Straightforward computations then show that for all /3 > 0 and 7y e (0,1/3],

03C3(~ hides (3)

Hence, one can sum up things by saying that the formula

in fact holds for all ~ > 0 and (3 > 0. Let us note that when ~ ~ 0+, onegets a non-trivial limit:

which is somewhat surprising (one might have guessed at first sight thatthe exponent should blow up when ~ ~ 0+). Indeed, for each fixed largE

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R, the probability that K1/N hides KI (with obvious notation) in the strip{J(z) E [1, R]} is anyway smaller that 1/(N+1). This is due to the fact thata restriction measure of exponent (N + 1)/N can be viewed as the unionof N + 1 independent copies of K1/N, so that the probability that Kl/Nhides all N others is no larger than 1/(N + 1). However, when ~ ~ 0+ (i.e.N ~ oo), even if the probabilities (for fixed R) go to zero, this does notaffect the exponents (only the "multiplicative constants" vanish).

4.3. Iterations

The description of conditioned SLE(03BA, p) as another SLE(03BA, p) allowsto iterate the procedure (this is for instance apparent in the formulationwith the hitting times TR), and to obtain exponents describing the jointbehavior of more than two restriction measures. For instance, in the simplestcase where ’" = 8/3, one gets readily the exponents describing the non-intersection between p SLE8/3’S (these are the exponents corresponding [22]to the non-intersection of self-avoiding walks in a half-plane):

For each positive integer p, consider p independent SLE8/3’S that areconditioned not to intersect (appropriately defined). Define 1]p the exponentof the obtained restriction measure, and define pp such that the right-mostSLE is an SLE(8/3, pp). Clearly, ~p = ~(8/3, pp). Furthermore, for eachp 0,

(where pl = 0). This shows readily that

and

Hence, the exponent describing the probability that p independent chordal

SLE8/3 (up to time 1) started at distance a of each other are mutuallyavoiding is

This result is not new since (in the notation of [ 15, 17]) qp = 03BEp(5/8, ... , 5/8);these exponents also correspond to those conjectured in [8] for self-avoidingwalks (see [22] for the conjectured relation between self-avoiding walks andSLE8/3).

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4.4. Other /1;’8

One can easily generalize the iterative procedure for other 03BA’s. SupposEfor instance that we consider the conditioned measure for p SLE03BA’s foi03BA 8/3 that are conditioned to mutually avoid each other and by thEevent that no Brownian loop in the Brownian loop-soup with intensity ÀKintersects two different paths. The right-most path is then an SLE(03BA, 03C1p)for some pp that a priori depends on 03BA, but it turns out that

If one adds another independent Brownian loop-soup with intensity ÀK tothis right-most path and looks at the obtained right-most boundary, oneobtains a restriction measure with exponent

For K = 2, the exponents correspond to those for loop-erased random walkderived by Kenyon [11] and Fomin [9] (previously conjectured in [3]). Thiis not surprising since loop-erased random walks converge to SLE2 in thescaling limit (see [21]).

One equivalent way to describe the corresponding event goes as followsRun p independent chordal SLE03BA’s 5i,..., Sp started from nearby points(for instance from the points a, 2a, ..., pa) up to time one. Consider independent Brownian loop-soups of intensity ÀK, and define for each j pthe union Sj of the loops in thé j-th soup that intersect Sj. Consider novthe event that for j = 2 up to j = p,

Then, the probability of this event decays like 03B103C3 when a ~ 0, where

In the special case 03BA = 2 that we just mentioned, the relation between SLE2and loop-erased random walks [21] and Wilson’s algorithm [36] gives to thisevent a natural interpretation in terms of uniform spanning trees.

For 03BA 8/3, the previous description does not make much sense (thedensity of the loop-soup is negative), and it raises the interesting problemto find a simple geometric way to interpret the exponent in terms of aphysical model. Conjecturally, the exponents correspond to the (asymptotic)probability of occurrence of the corresponding configuration in a sample ofthe corresponding critical FK-percolation model.

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4.5. The "quantum gravity" function

As the formulas show, 03C1(03BA, 0, a) is in fact the same as the quantum grav-ity function U (this actually also holds for 03BA =1= 8/3), if one compares withthe "KPZ relation" [12] used e.g. in Duplantier [5] (see [6] for a survey andreferences). Hence, the SLE(03BA,03C1) approach does give another interpretationof the "quantum gravity equations," and also permits (using the relationwith restriction measures) to identify precisely what exponents (i.e. whatevents) are given by this quantum gravity formalism. When 03BA ~ 8/3, thiswas not so obvious.

On a rigorous level, since the exponents computed via SLE (for instancein the present paper) are rigorously derived, while the KPZ relation is not,one may view the SLE derivation of the exponents as a derivation of the KPZrelation (modulo the assumption that the critical exponents for statisticalmechanics systems on a random planar graph exist and are universal).

4.6. Négative a’s

In fact (but we prefer to emphasize it in this separate paragraph), theabsolute continuity relation and the derivation of the hiding exponents alsoapply for (some) negative a. In order for the absolute continuity betweenBessel processes to hold, the condition is that both have a dimension notsmaller than 2. In other words, if one starts with an SLE(03BA, p) such that

then, the arguments developed in Section 3 go through except that thereis no interpretation of the weighting as a non-intersection probability (theweighting is here an unbounded function of the path). The constraint thatthe obtained conditioned Bessel process has dimension at least 2 means that

(note that this does not depend on p). Loosely speaking, when et is too

negative, then the SLE is not able to compensate the weighting (so thatQ is still a probability measure). This basically shows that - as one mighthave expected from the formulas - that the hiding exponents make sense onthe interval of values of et for which it can be extended analytically (as afunction of a).

In the special case where 03BA = 8/3, the lower bound on a is -1/24. Inthe special case p = 0, the hiding exponent is the intersection exponent

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Wendelin Werner

03BE(5/8, 03B1) - 5/8 - cx. We have just argued that a to this power describesindeed the asymptotic behavior of the quantity E[g’1(0)03B1] for an SLEs/3started from a as a vanishes, for all values of 03B1 -1/24.

If one then applies the cascade ideas, as developed in [25] say, it is

then simple to see that this for instance enables to deduce that exponents03BE(1,03B1) - 1 - a for instance describe the asymptotic behavior of E[(g’1(0)03B1]when 91 corresponds this time to the conformal map associated to a Brow-nian excursion started from a, up to time 1, when 03B1 ~ 0. Recall that

In particular,

5. The two-sided picture

5.1. The SLE(r,,p) martingales

Before turning our attention to the two-sided picture, let us point outthe following by-product of the description of the SLE(03BA, p)’s as an SLE,conditioned not to intersect a one-sided restriction sample of exponent a (p)It is a simple heuristic explanation to the (useful) martingales associated toSLE(03BA, p) derived and used in [23, 2]. Let us first focus on the 03BA = 8/3 casestudied in [23].

Let 4 E A. Consider the event that the SLE(8/3, p) does avoid A. Let usnow focus on the conditional probability of this event given the path up totime t. This is a function of Wt, Ot and of the image of 4 under gt. Defineas in [23] the conformal map ht from H B gt (A) onto H that is normalizedby ht(z) = z + 0(1) when z ~ oo (this is just a real shift of ~gt(A) as de-fined in the preliminary section). If one views the SLE(8/3, p) as an SLEs/3conditioned to avoid a restriction sample K, the conditional probability canbe decomposed as follows. First, the SLE§/3 started from Wt has to avoidgt(A): This event has probability h’t(Wt)5/8. Second, the restriction samplehas to avoid the set gt(A) as well. This occurs with probability h’t(Ot)03B1.Conditionally on these two events, the image under ht of the SLEs/3 is anSLE8/3 in H started from ht(Wt) and the image of the restriction measuresample is a restriction measure sample in H started from ht(Ot). The "prob-ability" of non-intersection between these two sets is going to be affected

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by the scaling factor given by the non-intersection exponent v - 1/4 = 03C1/03BAi.e.

Hence, the quantity

is a martingale. This is proved analytically in [23].

The same argument can be used for the local martingales associated toSLE(03BA, p)’s for 03BA ~ 8/3 as derived in [2] (with an additional "loop-soupterm" ) .

5.2. The two-sided case

In fact, Alt is a martingale also in the two-sided case. More precisely,suppose that A is the symmetric image with respect to the imaginary line ofa set in A (i.e. it is attached to the negative half-line). We will suppose in thissubsection that 03BA = 8/3 and p > 0. Then, Mt is still a bounded martingale(this is proved in [23]), that is well-defined up to the (possibly infinite) timeT at which the SLE curve hits A. Just as when A E A (see [23]):

2022 If T is finite, then there exists a sequence tn ~ T such thatlimn~~ Mtn = 0.

2022 If T is infinite, then there exists an unbounded sequence tn such that

This is basically due to the fact that gt (A) becomes smaller andsmaller, so that ht becomes closer to the identity.

However, the term h’t(Ot) does not tend to one (when t ~ T = oo), becauseeven if gt(A) becomes smaller, Ot gets closer and closer to gt(A). But sincethe SLE path is transient, the term h(0t) has a (non-trivial) limit whent ~ oo (if T = oo) that can be interpreted as follows:

The SLE(8/3, p) is a simple curve r that separates the upper half-planeinto two connected components r- and 0393+ (defined in such a way that thenegative half-line is on the boundary of F-). We now focus on r -. Let 03A6_

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Wendelin Werner

denote a anti-conformal map (i.e. 03A6_ (z) is analytic) from H onto 0393_ suchthat 03A6_(0) = 0 and 03A6_(~) = oo (i.e. 03A6_(x + iy) = 03A8_(-x + iy) wherE03A8_ is as before). In particular, the image of the positive half-line is thEnegative half-line, and the image of the negative half-line is the curve 03B3Consider a sample K of a one-sided restriction measure of exponent a that

. is independent of the SLE 03B3. Define

Note that the set K consists of 03B3 and of a subset of r -. In particular, its"right-boundary" is 03B3. Since, K is scale-invariant, the actual choice of 03A6_does not change the law of K. Then, almost surely on the event T = oo,

In particular, this implies that almost surely,

Since the martingale is bounded (by one), the optional stopping theorershows that

But, since 03B3 satisfies one-sided restriction (to the right) with exponent rl, itfollows that in fact

for all A E At. In other words, K is a sample of the two-sided restrictionmeasure with exponent 7î.

In the special case where 77 = 2, we see that the restriction measurewith exponent 2 corresponds to two SLEs/3’s conditioned not to intersect.This is closely related to the predictions concerning the scaling limits ofself-avoiding polygons [22].

5.3. Two-sided exponents

Two-sided hiding

This description of the two-sided restriction measures leads naturally tothe following exponent that describe the probability that if one considers twoindependent two-sided restriction measure samples K17 and K(3 of exponents

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77 and /3 (where 77 > 5/8 and 03B2 5/8), then K{3 n {J(z) E [1, R]} c K~decays like R-03C4 as R ~ oo, where

Hence,

03C4(~ hides t

Note in particular that

In particular, in both these cases, the exponent of the conditioned restrictionmeasure is 5.

No cut-points

A by-product of these calculations is the exponent that describes thedecay of the probability that a two-sided restriction measure of exponent17 > 5/8 has no cut-point. More precisely, when ~ E (5/8, 35/24), the prob-ability that a sample K~ of the two-sided restriction sample of exponent ~has no cut-point inside the strip {J(z) E [1, R]} decays like R-03B4(~) whenR ~ oo, where

Furthermore, the conditional law is that of the two-sided restriction measurewith exponent

i.e.

The reason is that the absence of cut-points means that the left-bounda]does not intersect the right-boundary. In the interpretation described abovthis occurs if K does not hit the negative half-line (or more precisely a se,ment on the negative half-line). The exponent that describes the probabilithat a d-dimensional Bessel process does not hit the origin is 2 - d and t]conditioned process is a 4 - d dimensional Bessel process. It follows that tl

restriction exponent of the conditioned set is

where p is chosen so that 03BE(5/8, ~(8/3, 03C1)) = ~.

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When ~ 35/24, the restriction measure sample has a.s. no cut-poiniso that the problem is not relevant. When ~ 5/8, the two-sided restrictio:measure does not exist. When q = 5/8, then K1J is almost surely a simplpath, so that the probability that it has no cut point in an annulus i

0. However, when 1] -t 5/8+, one sees that 8 tends to 2, and that thconditional law "tends" to that of a restriction measure of exponent 21/8that can therefore be viewed as the filling of an SLE8/3 conditioned to havno cut-point! Of course, since SLE8/3 is a.s. a simple curve, this depends a loon the limiting procedure used to define this conditioned object (here: firsreplace SLE8/3 by a restriction measure of exponent 5/8 + 6, then conditioit to have no cut point (in larger and larger annuli), and finally let E tenito zéro).

It is worthwhile stressing the special case where ~ = 1. The exponent is equal to 1 and it is related to Bàlint Virâg’s Brownian beads [34]. It givea description of the restriction measure of parameter "1’ = 2 as the fillin;of one single path. More precisely: "The filling of a Brownian excursio]conditioned to have no cut point has the same law as the filling of the unio]of two Brownian excursions." It raises the question whether this conditione4Brownian excursion has soinething to do with the path that is obtained bconsidering the appropriate SLE, to which one chronologically attacheBrownian loops as in [23] in order to construct a restriction measure sampiof exponent 2.

Note also that the two-sided measure obtained if one conditions K,7 thide K03B2, is the same as the one obtained if one conditions K~’ to hide K03B2This is not surprising: One first conditions K17 to have no cut point, ancthen weights it by the "space" it leaves in its inside.

Mixed two-sided hiding

One can also define exponents associated to "mixed" two-sided hidingConsider the exponent (~, /3) that is associated to the fact that the leftboundary of K,7 UK{3 consists only of points in K,7 while the right-boundarconsists of points in K,3. This time

where as before 03BE(5/8, a) = yy. This leads to

where.

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Radial hiding

All two-sided hiding exponents yield readily the corresponding exponentin the radial setting, using the mapping described for example in [26] andthe disconnection exponents computed in [16, 17] (see also [24]).

For instance, consider n + p independent Brownian motions started fromthe origin and stopped when they hit the unit circle. Consider the event thatthe union of these n + p paths do not disconnect the circle of radius r from1, and if the boundary of the connected component of U B (Bl U ... Bn+p)that contains 1 consists of points of B1 U ... U Bn. Then, the probability ofthis event decays like rP when r ~ 0, where

Note that when n = 2 or n = 1, the hiding exponent is just

(in the notation of [16]), which is not surprising because of the inside/outsidesymmetry of the Brownian frontier pointed out in [23]. The inside/outsidesymmetry of the Brownian frontier also shows that a single Brownian mo-tion started from the origin, "conditioned not to disconnect the origin frominfinity and to have no cut point" also separates the plane into the "inside"I and the "outside" 0 in such a way that (1, 0) and (0, 1) have the samelaw.

When the half-plane exponent is 5, then the radial exponent is 2. Forinstance, p(l hides 1) = 2, so that the corresponding existence problem is"critical" : Are there points BT on the outer boundary of a planar Brownianpath (Bt, t E [0,1]) such that (locally) the outer boundary consists only ofthe future after BT (or only of the past before BT)?

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