An almost sure KPZ relation for SLE and Brownian motion
AAn almost sure KPZ relation for SLE and Brownian motion
Ewain Gwynne
MIT
Nina Holden
MIT
Jason Miller
Cambridge
Abstract
The peanosphere construction of Duplantier, Miller, and Sheffield provides a means of representing a γ -Liouville quantum gravity (LQG) surface, γ ∈ (0 , κ , κ = 16 /γ ∈ (4 , ∞ ), η as a gluing of a pair of trees which are encoded by a correlated two-dimensional Brownian motion Z . We prove a KPZ-type formula which relates the Hausdorff dimension ofany Borel subset A of the range of η which can be defined as a function of η (modulo time parameterization)to the Hausdorff dimension of the corresponding time set η − ( A ). This result serves to reduce the problemof computing the Hausdorff dimension of any set associated with an SLE, CLE, or related processes inthe interior of a domain to the problem of computing the Hausdorff dimension of a certain set associatedwith a Brownian motion. For many natural examples, the associated Brownian motion set is well-known.As corollaries, we obtain new proofs of the Hausdorff dimensions of the SLE κ curve for κ (cid:54) = 4; the doublepoints and cut points of SLE κ for κ >
4; and the intersection of two flow lines of a Gaussian free field. Weobtain the Hausdorff dimension of the set of m -tuple points of space-filling SLE κ for κ > m ≥ m − π/ Contents
The Schramm-Loewner evolution (SLE κ ) [Sch00] and related processes such as SLE κ ( ρ ) [LSW03,SW05,MS16d]and the conformal loop ensembles (CLE κ ) [She09, SW12] have been an active area of research for thepast sixteen years. One line of research in this area has been the confirmation of exponents computednon-rigorously by physicists in the context of discrete models from statistical physics. Many of theseexponents were derived using the so-called KPZ relation [KPZ88], which is a non-rigorous formula whichrelates exponents for statistical physics models on random planar maps to the corresponding exponentsfor the model on a Euclidean lattice, such as Z . Exponents derived in this way are said to be obtainedfrom “quantum gravity methods.” This method of deriving exponents has been very successful because1 a r X i v : . [ m a t h . P R ] O c t he computation of an exponent in many cases boils down to a counting problem which turns out to bemuch easier when the underlying lattice is random (i.e., one considers a random planar map). Perhapsthe most famous example of this type are the so-called Brownian intersection exponents , which give theexponent of the probability that k Brownian motions started on ∂B (cid:15) (0) at distance proportional to (cid:15) fromeach other make it to ∂ D without any of their traces intersecting. These exponents were derived usingquantum gravity methods by Duplantier in [Dup98]. The values of the Brownian intersection exponentswere then verified mathematically in one of the early successes of SLE by Lawler, Schramm, and Wernerin [LSW01a, LSW01b, LSW02]. Following these works, a number of other exponents (hence also Hausdorffdimensions) have been calculated using SLE techniques, many of which were previously predicted in the physicsliterature [ABJ15, AS08, Bef08, GMS17b, JVL12, MSW14, MW17, MWW15, NW11, RS05, SSW09, WW15].Our main result is a rigorous version of the KPZ formula that relates the a.s. Hausdorff dimension of a setassociated with space-filling SLE κ (cid:48) [MS13], κ (cid:48) ∈ (4 , ∞ ), to the a.s. Hausdorff dimension of a certain Brownianmotion set in the context of the so-called peanosphere construction of [DMS14], which we review below. Thisserves to reduce the problem of calculating the Hausdorff dimension of any set associated with SLE κ , SLE κ ( ρ ),or CLE κ for κ (cid:54) = 4 in the interior of a domain to the problem of calculating the Hausdorff dimension of a certain(explicitly described) set associated with a correlated two-dimensional Brownian motion. There are numerousformulations of the KPZ formula in the literature, see e.g. the original physics paper [KPZ88], in addition tomore recent and rigorous formulations in e.g. [Aru15, BGRV14, BJRV13, BS09b, DMS14, DRSV14, DS11, RV11].As explained just above, the KPZ formula is typically applied to compute the Euclidean dimension of fractalsets, after deriving the quantum dimension heuristically or rigorously by quantum gravity techniques. In ourformulation the quantum dimension is explicitly given by the dimension of some Brownian motion set, henceour formula is directly useful for computations.To illustrate the application of our main theorem, we will obtain new proofs of the a.s. Hausdorff dimensionsof several sets, including the SLE curve for κ (cid:54) = 4, the double points of SLE, the cut points of SLE, andthe intersection of two flow lines of a Gaussian free field [She16a, MS16d, MS16e, MS16a, MS13]. We willalso use our theorem to calculate the a.s. Hausdorff dimension of the m -tuple points of space-filling SLE κ (cid:48) , κ (cid:48) ∈ (4 , m − m − κ (cid:48) gasket for κ (cid:48) ∈ (4 , κ . If η is anSLE κ curve for κ ∈ (0 , Y is a deterministic subset of R with Hausdorff dimension d ∈ [0 , H f ( Y ) = 132 κ (cid:16) κ − (cid:112) (4 + κ ) − κd (cid:17) (cid:16)
12 + 3 κ + (cid:112) (4 + κ ) − κd (cid:17) for almost every choice of conformal map f from H to a complementary connected component of η whichsatisfies f ( Y ) ⊂ η . We will now provide a brief review of Liouville quantum gravity and the peanosphere construction which willbe necessary to understand our main result below. Suppose that h is an instance of the Gaussian free field The Brownian intersection exponents were also derived earlier using a different method by Duplantier and Kwon in [DK88]. In order to be consistent with the notation of [MS16d, MS16e, MS16a, MS13], unless explicitly stated otherwise we willassume that κ ∈ (0 ,
4) and κ (cid:48) = 16 /κ ∈ (4 , ∞ ). D and γ ∈ (0 , γ -Liouville quantum gravity (LQG) surface associated with h formally corresponds to the surface with Riemannian metric e γh ( z ) ( dx + dy ) , (1.1)where z = x + iy = ( x, y ) and dx + dy denotes the Euclidean metric on D . This expression does not makeliteral sense because h takes values in the space of distributions and does not take values at points. The areameasure µ h associated with (1.1) has been made sense of using a regularization procedure (see e.g., [DS11]),namely by taking e γh ( z ) dz to be the weak limit as (cid:15) → (cid:15) γ / e γh (cid:15) ( z ) dz , where h (cid:15) ( z ) is the average of h onthe circle ∂B (cid:15) ( z ). One can similarly define a length measure ν h by taking it to be the weak limit as (cid:15) → (cid:15) γ / e γh (cid:15) ( z ) / dz . We refer to µ h (resp. ν h ) as the quantum area (resp. boundary length) measure associatedwith h . Quantum boundary lengths are well-defined for piecewise linear segments [DS11], their conformalimages, and SLE type curves for κ = γ [She16a]. The metric space structure associated with (1.1) has alsobeen recently constructed in [MS15a, MS15b, MS16b, MS16c, MS15c] in the special case that γ = (cid:112) /
3, inwhich case it is isometric to the Brownian map [Mie13, Le 13]. It remains an open question to construct themetric space structure for γ (cid:54) = (cid:112) / γ values arise by consideringdifferent discrete models [DS11]. So far, this conjecture has been proven only in the case of the Tutteembedding of the γ -mated-CRT map (a discretized version of the peanosphere) for γ ∈ (0 ,
2) [GMS17a].However, the convergence of other random planar maps decorated with a statistical physics model to LQGdecorated with SLE/CLE has been proved with respect to the peanosphere topology, which we will describebelow [She16b, DMS14, GMS15, GS17, GS15, MS15c, GKMW16, KMSW15, GHS16, LSW17]. See also theforthcoming work [GM17b] for a strengthening of this topology in the case of FK-weighted maps; and theworks [GM16, GM17a] for scaling limit results for self-avoiding walk and percolation, respectively, on randomplanar maps toward SLE-decorated (cid:112) / D, (cid:101) D are planar domains, ϕ : D → (cid:101) D is a conformal map, and (cid:101) h = h ◦ ϕ − + Q log | ( ϕ − ) (cid:48) | where Q = 2 γ + γ , (1.2)then µ h ( A ) = µ (cid:101) h ( ϕ ( A )) for all Borel sets A ⊆ D . The boundary length measure is similarly preserved undersuch a change of coordinates. A quantum surface is an equivalence class of pairs ( D, h ) where two such pairsare said to be equivalent if they are related as in (1.2). We refer to a representative (
D, h ) of a quantumsurface as an embedding of the quantum surface.One particular type of quantum surface which will be important in this article is the so-called γ -quantumcone. This is an infinite volume surface which is naturally parameterized by C and is marked by two points,called 0 and ∞ , neighborhoods of which respectively have finite and infinite µ h -mass. We will keep trackof the extra marked points by indicating a γ -quantum cone with the notation ( C , h, , ∞ ). In Section 1.4.2below, we will describe a precise method for sampling from the law of h for a particular embedding of a γ -quantum cone into C . This surface naturally arises, however, in the context of any γ -LQG surface ( D, h )with finite volume as follows. Suppose that z ∈ D is sampled from µ h . Then the surface one obtains byadding C to h , translating z to 0, and then rescaling so that µ h assigns unit mass to D converges as C → ∞ to a γ -quantum cone. That is, a γ -quantum cone describes the local behavior of a γ -LQG surface near atypical point chosen from µ h .As explained in [She16a, DMS14], it is very natural to decorate a γ -LQG surface with either an SLE κ , κ = γ ,or an SLE κ (cid:48) , κ (cid:48) = 16 /γ . In the case of a γ -quantum cone ( C , h, , ∞ ), it is particularly natural to decorateit with the space-filling SLE κ (cid:48) process η (cid:48) [MS13] where η (cid:48) is first sampled independently of h (as a curvemodulo time parameterization), then reparameterized by quantum area so that µ h ( η (cid:48) ([ s, t ])) = t − s for all s < t , and then normalized so that η (cid:48) (0) = 0. In this setting, it is shown in [DMS14] that the pair Z = ( L, R )which, for a given time t , is equal to the quantum length of the left and right boundaries of η (cid:48) , evolves asa correlated two-dimensional Brownian motion. Since these quantum boundary lengths are in fact always3nfinite, it is natural to normalize Z so that L = R = 0. By [DMS14, Theorem 9.1] (in the case κ (cid:48) ∈ (4 , κ (cid:48) > L and R are given byVar( L t ) = a | t | , Var( R t ) = a | t | , Cov( L t , R t ) = − a cos θ | t | , θ = 4 πκ (cid:48) (1.3)with a a constant depending only on κ (cid:48) .One of the main results of [DMS14] is that the pair ( L, R ) almost surely determines the pair consisting ofthe γ -quantum cone ( C , h, , ∞ ) and the space-filling SLE κ (cid:48) process η (cid:48) . That is, the latter is a measurablefunction of the former (and it is immediate from the construction that the former is a measurable function ofthe latter). This is natural in the context of discrete models [She16b] which can also be encoded in terms of ananalogous such pair and, in fact, the main result of [She16b] combined with [DMS14] gives the convergence ofFK-decorated random planar maps to CLE decorated LQG with respect to the topology in which two surfacesare close if the aforementioned encoding functions are close. This is the so-called peanosphere topology.As explained in Figure 1, L and R have the interpretation of being the contour functions associated witha pair of infinite trees, and ( C , h, , ∞ ) and η (cid:48) have the interpretation of being the embedding of a certainpath-decorated surface into C which is generated by gluing together the pair of trees encoded by L , R [DMS14].We remark that the construction in [DMS14] deals with the setting of infinite volume surfaces. The setting offinite volume surfaces is the focus of [MS15c] and the corresponding convergence result in the finite volumesetting is established in [GMS15, GS17, GS15]. Our main result is a KPZ formula which allows one to use the representation (
L, R ) of an SLE decoratedquantum cone to compute Hausdorff dimensions for SLE and related processes.
Theorem 1.1.
Let κ (cid:48) > and γ = 4 / √ κ (cid:48) . Let ( C , h, , ∞ ) be a γ -quantum cone and let η (cid:48) be an independentspace-filling SLE κ (cid:48) , parameterized by γ -quantum mass with respect to h and satisfying η (cid:48) (0) = 0 . Assume that h has the circle average embedding (see Definition 1.6 below). Let X be a random Borel subset of C such that X is independent from h (e.g. X could be a set which is determined by the curve η (cid:48) viewed modulo monotonereparameterization). Almost surely, for each Borel set (cid:98) X ⊂ R such that η (cid:48) ( (cid:98) X ) = X , we have dim H ( X ) = (cid:18) γ (cid:19) dim H ( (cid:98) X ) − γ H ( (cid:98) X ) . (1.4)A space-filling SLE κ (cid:48) encodes an entire imaginary geometry of flow lines, which in turn encodes both SLE κ and SLE κ (cid:48) -type paths and a CLE κ (cid:48) (see [MS13] and Section 1.4.3 below). The forthcoming work [MSW17]will show that the imaginary geometry framework also encodes a CLE κ . Therefore, Theorem 1.1 reduces thecomputation of the dimension of any set in the interior of a domain associated with SLE κ or CLE κ for κ (cid:54) = 4to computing the dimension of the corresponding set associated with the correlated Brownian motion Z . Aswe will discuss in Section 1.4 it is often possible to characterize special sets associated with SLE κ and CLE κ as time sets of Z with particular properties. Examples of such sets are1. SLE κ curves for κ (cid:54) = 4 [Bef08, RS05],2. double points of SLE κ (cid:48) for κ (cid:48) > κ (cid:48) for κ (cid:48) > m -tuple points of space-filling SLE κ (cid:48) for m ≥ κ (cid:48) ∈ (4 , κ (cid:48) gasket for κ (cid:48) ∈ (4 ,
8) [MSW14, SSW09]. 4ee Table 1 for a summary of these sets and their dimensions. We will use Theorem 1.1 to calculate theHausdorff dimension of the first five of these sets in Sections 2 and 6. The original proofs relied on rathertechnical two-point estimates for correlations, while our formula provides alternative proofs. In cases wherethe Hausdorff dimension of the SLE set is known, Theorem 1.1 also gives the dimension of the correspondingBrownian motion set. We will use this direction of Theorem 1.1 to calculate the dimension of the Brownianmotion time set corresponding to the CLE κ (cid:48) gasket in Section 2. SLE set SLE dim H BM set BM dim H SLE κ trace, κ ∈ (0 ,
4) 1 + κ L or R κ (cid:48) trace, κ (cid:48) ∈ (4 ,
8) 1 + κ (cid:48) κ (cid:48) κ (cid:48) , κ (cid:48) ∈ (4 ,
8) 3 − κ (cid:48) Simultaneous runninginfima of L and R − κ (cid:48) κ (cid:48) , κ (cid:48) ∈ (4 ,
8) 2 − (12 − κ (cid:48) )(4 + κ (cid:48) )8 κ (cid:48) Composition ofsubordinators κ (cid:48) − κ (cid:48) , κ (cid:48) > κ (cid:48) Running infima of L or R θ − κ (cid:16) ρ + κ (cid:17) (cid:16) ρ − κ (cid:17) , ρ = θπ (cid:16) − κ (cid:17) − − ρ + 2 κ CLE κ (cid:48) gasket, κ (cid:48) ∈ (4 ,
8) 2 − (8 − κ (cid:48) )(3 κ (cid:48) − κ (cid:48) Times not contained in anyleft π/ κ (cid:48) m -tuple points ofspace-filling SLE κ (cid:48) (4 m − − κ (cid:48) ( m − κ (cid:48) − m )8 κ (cid:48) ( m − π -cone times d ( κ (cid:48) , m )Table 1: The sets whose dimension we compute in this paper. Each row shows a set X associated with SLEand a corresponding Brownian motion set (cid:98) X which is contained in ( η (cid:48) ) − ( X ). The dimensions of these setsare related as in Theorem 1.1. See Sections 2 and 6 for proofs that the dimensions in the table are as claimed.The quantity in the bottom right cell is d ( κ (cid:48) , m ) = 1 / − ( m − κ (cid:48) / − / X does not have to be measurable with respect to the space-filling SLE κ (cid:48) . However,Theorem 1.1 does not hold without the hypothesis that X is independent from h . For example, supposewe take X to be the γ -thick points of h (see [HMP10] as well as Section 4). The γ -quantum measure µ h is supported on X [DS11, Proposition 3.4], so since η (cid:48) is parameterized by quantum mass, we havedim H ( η (cid:48) ) − ( X ) = 1. On the other hand, it is shown in [HMP10] that dim H X = 2 − γ /
2, so (1.4) does nothold for this choice of X .Theorem 1.1 is in agreement with the KPZ formula [KPZ88] if the “quantum dimension” of X is defined tobe twice the Hausdorff dimension of the Brownian motion set (cid:98) X . Remark 1.2.
Theorem 1.1 only applies to sets associated with SLE and CLE in the interior of a domain. Inorder to calculate the Hausdorff dimension of sets associated with chordal or radial SLE which intersect theboundary one may apply [RV11, Theorem 4.1], which implies the following one-dimensional KPZ formula.Let h be a free boundary GFF in the upper half-plane and let ν h be the associated boundary measure. Definethe quantum dimension of a set X ⊂ [0 , ∞ ) to bedim H ( (cid:98) X ) , (cid:98) X := { ν h ([0 , x ]) : x ∈ X } . Then it holds a.s. that dim H ( X ) = (cid:18) γ (cid:19) dim H ( (cid:98) X ) − γ H ( (cid:98) X ) . (1.5)5 t C − L t tt η (cid:48) ( t ) Figure 1: The peanosphere construction of [DMS14] shows how to obtain a topological sphere by gluingtogether two correlated Brownian excursions
L, R : [0 , → [0 , ∞ ) (A similar construction works when L, R are two-sided Brownian motions, see [DMS14, Footnote 4]). We choose
C > C − L and R do not intersect. We then define an equivalence relation on the square [0 , × [0 , C ] by identifyingpoints which lie on the same horizontal line segment above the graph of C − L or below the graph of R ; orthe same vertical line segment between the two graphs. As explained in [DMS14], it is possible to check usingMoore’s theorem [Moo28] that the resulting object is a topological sphere decorated with a space-filling path η (cid:48) where η (cid:48) ( t ) for t ∈ [0 ,
1] is the equivalence class of ( t, R t ). The pushforward of Lebesgue measure on [0 , µ on the sphere (i.e., a non-atomic measure which assigns positive massto each open set) and η (cid:48) is parameterized according µ -area, i.e., µ ( η (cid:48) ([ s, t ])) = t − s for all 0 ≤ s < t ≤ peanosphere because the space-filling path η (cid:48) is thepeano curve between the continuum trees encoded by L and R . It is shown in [DMS14] (infinite volumecase) and in [MS15c] (finite volume case) that there is a measurable map which associates a pair ( L, R ) (orequivalently the aforementioned good-measure endowed sphere together with a space-filling path) with anLQG surface decorated with an independent space-filling SLE. That is, the peanosphere comes equippedwith a canonical embedding into the Euclidean sphere where the pushforward of µ encodes an LQG surfaceand η (cid:48) is a space-filling SLE. The embedding of the trees coded by L and R correspond to trees of flow lineswith a common angle in an imaginary geometry [MS13]. The right side of the illustration shows a subset ofthe SLE-decorated LQG surface, where the green region corresponds to points that are visited by η (cid:48) beforesome time t , and where the two trees are embeddings of the trees with contour functions L and R . Thebranches of these continuum random trees correspond to the frontier of the space-filling curve at differenttimes, and the Brownian motions L and R encode the lengths of the left and right, respectively, frontier ofthe SLE curve. The right figure illustrates the embedding of the peanosphere for the regime when κ (cid:48) ∈ [8 , ∞ );for κ (cid:48) ∈ (4 ,
8) the green region on the right figure is not homeomorphic to H since for this range of κ (cid:48) values,space-filling SLE κ (cid:48) is loop-forming.(The function ζ in the statement of [RV11, Theorem 4.1] can be obtained using the scaling properties of thefree boundary GFF and [RV11, Proposition 2.5].) At the end of Section 2 we will include a short exampleshowing how one may use this theorem, combined with results of [DMS14], to obtain the Hausdorff dimensionof the points where a chordal SLE κ ( ρ ), κ ∈ (0 , H intersects the real line.We will now discuss how our version of the KPZ formula relates to other KPZ-type formulas in the literature.The results of [DS11] relate the expected Euclidean mass of the Euclidean δ -neighborhood of a set X tothe expected quantum mass of the so-called quantum δ -neighborhood of X , which is defined in terms ofEuclidean balls of quantum mass δ >
0. The scaling exponents for these dimensions are proven to satisfy theKPZ equation. In [RV11, BJRV13, DRSV14], the authors consider a notion of quantum Hausdorff dimensionin terms of the quantum mass of Euclidean balls covering a set X and obtain KPZ formulas relating the6.s. quantum Hausdorff dimension to the a.s. Euclidean Hausdorff dimension. The KPZ relations in theworks [DS11, RV11, BJRV13, DRSV14] all rely on the Euclidean geometry, since Euclidean balls or squaresare used to cover X . In our formulation, by contrast, we obtain a cover of X by pushing forward a cover ofthe time set (cid:98) X via the curve η (cid:48) , hence the quantum dimension does not rely on the Euclidean geometry. Theversion given in [BGRV14] also does not rely on Euclidean geometry because it is defined in terms of the heatkernel for the Liouville Brownian motion (see [GRV16, Ber15, GRV14, RV15, Jac14]) which is intrinsic to theLQG surface. The nature of [BGRV14], however, is quite different from the present work and, as we shall seein Section 2, the formulation considered here appears to be more amenable to explicit calculation. We will now describe the basic notation and objects that we will use throughout the paper, including quantumcones, quantum wedges and space-filling SLE κ (cid:48) . We refer the reader to [DMS14, MS13, She16a] for moredetails. We adopt the convention of [MS16d, MS16e, MS16a, MS13] that κ denotes a parameter in (0 ,
4] and κ (cid:48) :=16 /κ ∈ (4 , ∞ ). If the κ -value is not restricted to either of the two intervals (0 ,
4] or (4 , ∞ ), we will simplywrite κ . We also use the following notation. Notation 1.3. If a and b are two quantities, we write a (cid:22) b (resp. a (cid:23) b ) if there is a constant C (independentof the parameters of interest) such that a ≤ Cb (resp. a ≥ Cb ). We write a (cid:16) b if a (cid:22) b and a (cid:23) b . Notation 1.4. If a and b are two quantities which depend on a parameter x , we write a = o x ( b ) (resp. a = O x ( b ) ) if a/b → (resp. a/b remains bounded) as x → (or as x → ∞ , depending on context). Wewrite a = o ∞ x ( b ) if a = o x ( b s ) for each s ∈ R . Unless otherwise stated, all implicit constants in (cid:16) , (cid:22) , and (cid:23) and O x ( · ) and o x ( · ) errors involved in the proofof a result are required to satisfy the same dependencies as described in the statement of the result. Fix γ ∈ (0 , γ -LQG surface is an equivalence class of pairs ( D, h ), where D is aplanar domain and h is an instance of some form of the GFF on D , see e.g [DS11, She16a, DMS14], wheretwo such pairs are said to be equivalent if they are related as in (1.2). The surface ( D, h ) is equipped with aquantum area measure that can be formally represented as µ h = e γh ( z ) dz (where dz is Lebesgue measure) aswell as a quantum length measure ν h = e γh ( z ) / dz (where dz is Lebesgue length measure in the case when theboundary is a straight line), which is defined on ∂D as well as on certain curves in the interior of D , includingSLE κ -type curves for κ = γ [She16a]. We refer to a particular choice of equivalence class representative( D, h ) as an embedding of the quantum surface. If (
D, h ) and ( (cid:101) D, (cid:101) h ) are equivalent and ϕ : D → (cid:101) D is theconformal map of (1.2), µ (cid:101) h is almost surely the push-forward of µ h under ϕ , i.e., µ h ( A ) = µ (cid:101) h ( ϕ ( A )) for any A ⊆ D [DS11, Proposition 2.1]. A similar statement is true for ν h .Several types of quantum surfaces of the form ( D, h, x , ..., x k ), where x , ..., x k ∈ D are additional markedpoints, are introduced in [She16a, DMS14]. Two such surfaces ( D, h, x , . . . , x k ) and ( (cid:101) D, (cid:101) h, (cid:101) x , . . . , (cid:101) x k ) aredefined to be equivalent if there exists a conformal map ϕ : D → (cid:101) D where (cid:101) h, h are related as in (1.2) and ϕ ( x j ) = (cid:101) x j for each 1 ≤ j ≤ k . In this paper we will mainly consider α -quantum cones, α < Q , which are aone-parameter family of doubly marked quantum surfaces homeomorphic to C . In Section 2 we will also needsome theory of quantum wedges.Let H ( C ) be the Hilbert space closure modulo a global additive constant of the subspace of functions f ∈ C ∞ ( C ) satisfying (cid:107) f (cid:107) ∇ := ( f, f ) ∇ < ∞ , where ( f, g ) ∇ := (2 π ) − (cid:82) C ∇ f · ∇ g dz for g ∈ C ∞ ( C ) for whichthe integral is well-defined and finite. Let H ( C ) ⊂ H ( C ) be the subspace of functions that are radiallysymmetric about the origin, and let H ( C ) ⊂ H ( C ) be the subspace of functions (modulo of global additive7onstant) which have mean zero about all circles centered at the origin. By [DMS14, Lemma 4.8] we have H ( C ) = H ( C ) ⊕ H ( C ). Recall that a whole-plane GFF h is a modulo additive constant distribution on thecomplex plane (i.e., a continuous linear functional defined on the subspace of functions f ∈ C ∞ ( C ) with (cid:82) C f ( x ) dx = 0) which can be represented as h = (cid:80) n ∈ N α n f n , where ( α n ) n ∈ N is a series of i.i.d. standardnormal random variables, and ( f n ) n ∈ N is an orthonormal basis for H ( C ). Definition 1.5.
Let α ∈ (0 , Q ]. An α -quantum cone is the doubly marked quantum surface ( C , h, , ∞ ),where h = h † + h is a random distribution sampled as follows. The radially symmetric function h † takesthe value A s on ∂B e − s (0), where A s = (cid:101) B s + αs for (cid:101) B a standard two-sided Brownian motion, conditionedsuch that A s ≥ Qs for s <
0. The distribution h is independent of h † , and is given by the projection of awhole-plane GFF onto H ( C ).There is a two-parameter family of embeddings of a quantum cone into ( C , , ∞ ) (i.e., choices of thedistribution h ), corresponding to multiplication of C by a complex number. The distribution h in Definition 1.5is one such choice. Definition 1.6.
For α ∈ (0 , Q ], the circle average embedding of an α -quantum cone is the distribution h ofDefinition 1.5. Remark 1.7.
One of the main reasons why we are interested in the embedding of Definition 1.6 is thatunder this embedding, h | D agrees in law with the restriction to D of a whole-plane GFF plus − α log | · | ,with additive constant chosen so that its circle average over ∂ D vanishes. Indeed, this is a straightforwardconsequence of Definition 1.5.We refer to [DMS14, Section 4.3] for further details regarding quantum cones.For α ≤ Q (with Q as in (1.2)), an α -quantum wedge is a doubly marked quantum surface, which ishomeomorphic H . For α ∈ ( Q, Q + γ ) an α -quantum wedge is a Poissonian chain of finite volume doubly-marked quantum surfaces, each of which is homeomorphic to D and has two marked points. In the first casewe say that the quantum wedge is thick, and in the second case it is thin. A thick wedge can be representedas a quantum surface ( H , h, , ∞ ), where h = h + h † is a decomposition of h into a distribution h of meanzero on all half-circles around the origin, and a radially symmetric function h † . For an α -quantum wedgewith α ≤ Q which is parameterized by H one possible embedding is such that the law of h † on H ∩ ∂B (0 , e − s )is identical to the function A described above in Definition 1.6, except that (cid:101) B s is replaced by (cid:101) B s .If we conformally weld the two boundaries of a quantum wedge according to quantum boundary length weobtain a quantum cone. Conversely, we obtain a quantum wedge if we cut out a surface by consideringan independent whole-plane SLE κ ( ρ ), κ = γ , on top of a quantum cone [DMS14, Theorem 1.12], with ρ depending on the parameter α of the cone. We also obtain a wedge by conformally welding together multiplewedges according to quantum boundary length, and we obtain two independent wedges if we cut a wedge intotwo or several components by a collection of SLE κ ( ρ ) curves for certain values of κ and ρ . Quantum wedgesand cones can be parameterized by their weight W = W ( α ) (which is defined to be γ ( γ/ Q − α ) for awedge and 2 γ ( Q − α ) for a cone) rather than α . The weight of the surfaces is additive under the operationsof gluing/welding and cutting as described above. Here we give a moderately detailed overview of the construction and basic properties of whole-plane space-filling SLE κ (cid:48) from ∞ to ∞ for κ (cid:48) >
4, which was originally defined in [DMS14, Footnote 9], building on thechordal definition in [MS13, Sections 1.2.3 and 4.3]. For κ (cid:48) ≥
8, whole-plane space-filling SLE κ (cid:48) from ∞ to ∞ is just a certain curve from ∞ to ∞ which locally looks like an SLE κ (cid:48) . For κ (cid:48) ∈ (4 , κ (cid:48) from ∞ to ∞ traces points in the same order as a curve which locally looks like SLE κ (cid:48) , but fills in the“bubbles” which it disconnects from its target point with a continuous space-filling loop.To construct whole-plane space-filling SLE κ (cid:48) from ∞ to ∞ , fix a deterministic countable dense subset C ⊂ C and let (cid:98) h be a whole-plane GFF, viewed modulo a global additive multiple of 2 πχ where χ = √ κ (cid:48) / − / √ κ (cid:48) .It is shown in [MS13] that for each z ∈ C , one can make sense of the flow lines η Lz and η Rz of angles π/ π/
2, respectively, started from z . These flow lines are SLE κ (2 − κ ) curves for κ = 16 /κ (cid:48) [MS13, Theorem 1.1].The flow lines η Lz and η Lw (resp. η Rz and η Rw ) started at distinct points in C eventually merge together, suchthat the collection of flow lines η Lz (resp. η Rz ) for z ∈ C form the branches of a tree rooted at ∞ .We define a total order on C by declaring that w comes after z if and only if w lies in a connected componentof C \ ( η Lz ∪ η Rz ) whose boundary is traced by the right side of η Lz and the left side of η Rz . It can be shownusing the same argument as in the chordal case [MS13, Section 4.3] (or alternatively deduced from the chordalcase; c.f. [DMS14, Footnote 9]) that there is a unique space-filling curve η (cid:48) : R → C which traces the points in C in order, is continuous when parameterized by Lebesgue measure, and is such that ( η (cid:48) ) − ( C ) is a denseset of times. The curve η (cid:48) does not depend on the choice of C and is defined to be whole-plane space-fillingSLE κ (cid:48) from ∞ to ∞ .For each fixed z ∈ C , it is a.s. the case that the left and right outer boundaries of η (cid:48) stopped at the first (anda.s. only) time τ z that it hits z are given by the flow lines η Lz and η Rz . For κ (cid:48) ≥
8, these two flow lines donot intersect so C \ η (cid:48) (( −∞ , t ]) for each time t has the topology of the half-plane. For κ (cid:48) ∈ (4 , η Lz and η Rz intersect each other so C \ η (cid:48) (( −∞ , t ]) instead consists of a string of domains with the topologyof the disk, separated by the intersection points. By [DMS14, Footnote 9], if we condition on η Lz and η Rz (equivalently, on η (cid:48) (( −∞ , τ z ]) or η (cid:48) ([ τ z , ∞ )), then the conditional law of η (cid:48) | [ τ z , ∞ ) is that of a chordal SLE κ (cid:48) from 0 to ∞ in η (cid:48) ([ τ z , ∞ )) if κ (cid:48) ≥
8; or a concatenation of independent chordal space-filling SLE κ (cid:48) curves inthe connected components of the interior of η (cid:48) ([ τ z , ∞ )) if κ (cid:48) ∈ (4 , η (cid:48) | ( −∞ ,τ z ] admits a similar description.The curve η (cid:48) is also closely related to the SLE κ (cid:48) ( κ (cid:48) − counterflow lines of (cid:98) h from ∞ to z for any given z ∈ C . In particular, if we parameterize η (cid:48) by capacity as seen from z , so we skip all of the bubbles filledin by η (cid:48) before it hits z , then we a.s. recover the counterflow line from ∞ targeted at z . The collection ofall of these counterflow lines, targeted at a countable dense set of points, forms a whole-plane branchingSLE κ (cid:48) ( κ (cid:48) −
6) process, which can be used to construct a whole-plane CLE κ (cid:48) via a whole-plane analog of theconstruction in [She09]. Hence a whole-plane space-filling SLE κ (cid:48) from ∞ to ∞ encodes a whole-plane CLE κ (cid:48) . Section 2 gives a number of examples of SLE sets of known Hausdorff dimension, for which Theorem 1.1provides an alternative derivation.Section 3 contains various SLE and GFF estimates which we will need for the proof of Theorem 1.1, bothfor the upper bound and for the lower bound. In Section 3.2 we will prove that, with high probability, anysegment of a space-filling SLE κ (cid:48) curve of diameter (cid:15) ∈ (0 , (cid:15) o (cid:15) (1) .An interesting corollary of this result is that space-filling SLE κ (cid:48) is a.s. locally α -H¨older continuous for any α < / A with the α -thick pointsof a GFF h is a.s. equal to dim H A − α /
2. The proof is a generalization of the argument in [HMP10], and isused in the proof of the upper bound of Theorem 1.1.Section 5 contains the proof of Theorem 1.1. Unlike for most Hausdorff dimension calculations, the upperbound for dim H ( X ) is more challenging to prove than the lower bound. Using the results of Section 3, weobtain an estimate for the diameter of η (cid:48) ( I ) for I ⊂ R , in terms of diam( I ) and the thickness of the field at apoint near η (cid:48) ( I ). This leads to an upper bound for the dimension of the intersection of X with the α -thickpoints of h in terms of dim H ( (cid:98) X ). Using the result of Section 4 and optimizing over α , we obtain the upperbound in (1.4). The lower bound in (1.4) is proven via a more direct argument based on moment estimatesfor the quantum measure along with the estimates of Sections 3.2 and 3.3.In Section 6 we use Theorem 1.1 to give a proof for the Hausdorff dimension of the set of m -tuple points ofspace-filling SLE κ (cid:48) , which in the setting of Theorem 1.1 correspond to the image under η (cid:48) of the so-called( m − m -tuple cone time can bedescribed by a “cone vector” in R m +1 consisting of m cone times [Shi85, Eva85] for Z and the time-reversal9f Z , where the end of one cone excursion marks the beginning of another cone excursion in the oppositedirection. We obtain the Hausdorff dimension of the set of cone vectors by standard techniques, including atwo-point estimate for correlations; then obtain the dimension of the set of ( m − R ) by projection. Our result generalizes the result in [Eva85], where the dimension of the set ofsingle cone times is calculated.Finally, in Section 7, we list some open problems related to the results of this paper. Acknowledgements
E.G. was supported by the U.S. Department of Defense via an NDSEG fellowship. N.H. was supported by afellowship from the Norwegian Research Council. J.M. was partially supported by DMS-1204894. Part ofthis work was carried out during the Random Geometry semester at the Isaac Newton Institute, CambridgeUniversity, and the authors would like to thank the institute and the organizers of the program for theirhospitality. The authors would also like to thank Peter M¨orters, Scott Sheffield, and Xin Sun for helpfuldiscussions and an anonymous referee for helpful comments on an earlier version of this paper.
In this section we will use Theorem 1.1 to give alternative proofs of several Hausdorff dimensions alreadyknown in the literature, in addition to a calculation of a new Brownian motion dimension for which thecorresponding SLE dimension is already known. Throughout, we let η (cid:48) be a space-filling SLE κ (cid:48) on top ofan independent γ -quantum cone ( C , h, , ∞ ), κ (cid:48) >
4. We parameterize η (cid:48) by quantum area, and let L and R denote the left and right boundary length process, respectively, relative to time 0. Define Z = ( L, R ),and recall that Z has the law of a two-dimensional Brownian motion with covariances as in (1.3). Beforepresenting the examples we recall the definitions of cone times, cone intervals and cone excursions. Definition 2.1.
Let α ∈ (0 , π ), t ∈ R , and let v ( α (cid:48) ) denote the unit vector in direction α (cid:48) for any α (cid:48) ∈ (0 , π ).Then t is an α -cone time of Z if there is a time s > t such that, for each s (cid:48) ∈ [ t, s ], there exists r s (cid:48) ∈ [0 , ∞ )and an angle α s (cid:48) ∈ [0 , α ], satisfying Z s (cid:48) = Z t + r s (cid:48) v ( α s (cid:48) ). If we do not specify an angle α , we will assume α = π , i.e., a cone time is a π -cone time.For a cone time t of Z we define the function v by v ( t ) = inf { s > t : R s < R t or L s < L t } . (2.1)The interval ( t, v ( t )) is called a cone interval and Z | ( t,v ( t )) is called a cone excursion . We say that t is aright cone time for Z if R v ( t ) > R t (equivalently, L v ( t ) = L t ), and we say that t is a left cone time for Z if L v ( t ) > L t (equivalently, R v ( t ) = R t ).The Hausdorff dimension of SLE κ was first calculated in [RS05] and [Bef08]. For κ ∈ (0 ,
4) we obtain analternative proof by using that the boundary of η (cid:48) ([0 , ∞ )) has the law of an SLE κ curve. Example 2.2.
The Hausdorff dimension of an
SLE κ curve for κ ∈ (0 , is a.s. equal to κ .Proof. Let κ (cid:48) := 16 /κ . If we stop the space-filling SLE κ (cid:48) process η (cid:48) upon reaching 0, the boundary of thealready traced region is given by two flow lines of a whole-plane GFF with angle gap π , see [DMS14, Footnote 9].The marginal law of each of these flow lines is that of a whole-plane SLE κ (2 − κ ), see [MS13, Theorem 1.1]. If η (cid:48) is parameterized by quantum mass, the times at which η (cid:48) traces the left and right boundaries of η (cid:48) ([0 , ∞ ))correspond exactly to the running infima of the left and right boundary length processes L and R , respectively,relative to time 0. This time set has Hausdorff dimension 1 / κ (2 − κ ) curve a.s. has Hausdorff dimension 1 + κ . In fact, thesame argument shows that a.s. every non-trivial segment of a whole-plane SLE κ (2 − κ ) curve has Hausdorffdimension 1 + κ . By local absolute continuity of SLE κ (2 − κ ) and SLE κ away from the self-intersection timesof the former [SW05, equation (9)], an ordinary radial or whole-plane SLE κ also has a.s. Hausdorff dimension1 + κ , and using local absolute continuity again [SW05, Theorems 3 and 6] we deduce the same result forordinary chordal SLE κ . 10he Hausdorff dimension of SLE κ (cid:48) , κ (cid:48) ∈ (4 , κ (cid:48) corresponds to the so-called ancestor free times ( t ( s )) s ≥ of Z . A time s ≥ π/ Z which is contained in [0 , ∞ ). In other words, s is ancestor free if there is no t ∈ [0 , s ) such that L u ≥ L t and R u ≥ R t for all u ∈ ( t, s ]. Example 2.3.
The Hausdorff dimension of an
SLE κ (cid:48) curve for κ (cid:48) ∈ (4 , in C or in a domain whoseboundary has dimension at most κ (cid:48) is a.s. equal to κ (cid:48) .Proof. Consider the space-filling SLE κ (cid:48) η (cid:48) described above. Let ( t ( s )) s ≥ denote the inverse of the local timeat the ancestor free times of ( L, R ) relative to t = 0, as described in [DMS14, Section 1.7.2]. Conditioned on η (cid:48) ([0 , ∞ )) the law of ( η (cid:48) ( t ( s ))) s ≥ is that of a concatenation of independent SLE κ (cid:48) ( κ (cid:48) / − κ (cid:48) / −
4) processesin each of the bubbles (connected components of the interior) of η (cid:48) ([0 , ∞ )), see [DMS14, Lemma 12.4].For anysuch bubble D let D n ⊂ D consist of the points in D at distance at least 1 /n from ∂D . By local absolutecontinuity [SW05], a.s. the intersection of the SLE κ (cid:48) ( κ (cid:48) / − κ (cid:48) / −
4) with D n for sufficiently large n has the same Hausdorff dimension as the intersection of an ordinary SLE κ (cid:48) curve with a sub-domain atpositive distance from the boundary of its domain. Since ∂D has Hausdorff dimension 1 + κ (cid:48) < κ (cid:48) byExample 2.2, it follows that it is sufficient to prove that the image of ( η (cid:48) ( t ( s ))) s ≥ has dimension 1 + κ (cid:48) .By [DMS14, Proposition 1.15], the processes ( L t ( s ) ) s ≥ and ( R t ( s ) ) s ≥ are independent κ (cid:48) / L t ( s ) , R t ( s ) ) s ≥ has dimension κ (cid:48) /
4; see the discussion right after [PT69, Theorem 1.2].Kaufman’s theorem [MP10, Theorem 9.28] implies that the corresponding time set for (
L, R ) has dimension κ (cid:48) /
8. An application of Theorem 1.1 completes the proof.The dimension of the cut points and double points of SLE κ (cid:48) for κ (cid:48) ∈ (4 ,
8) were first calculated in [MW17].Recall that the set of cut points of a curve η is the set { η ( t ) : t ∈ (0 , ∞ ) , η ((0 , t )) ∩ η (( t, ∞ )) = ∅} . The set oflocal cut points of a curve η parameterized by R + is the set { η ( t ) : t > , ∃ s > , η (( t − s, t )) ∩ η (( t, t + s )) = ∅} .The set of cut points of η (cid:48) | [ t, ∞ ) for t ∈ R is contained in the set of points where the left and the right boundariesof η (cid:48) ([ t, ∞ )) meet in the manner described in Figure 2, which in turn corresponds to the set of times whenboth L and R are at a simultaneous running infimum relative to time t . Example 2.4.
The Hausdorff dimension of the sets of cut points and local cut points of a chordal, radial, orwhole-plane
SLE κ (cid:48) for κ (cid:48) ∈ (4 , are each a.s. equal to − κ (cid:48) .Proof. Let ( t ( s )) s ≥ be defined as in the proof of Example 2.3. Let (cid:98) η (cid:48) be the curve from ∞ to 0 which isequal to the time-reversal of ( η (cid:48) ( t ( s ))) s ≥ , which has the law of a whole-plane SLE κ (cid:48) ( κ (cid:48) − (cid:98) η (cid:48) a.s. has dimension 3 − κ (cid:48) /
8. The set of local cutpoints of (cid:98) η (cid:48) is contained in the union over all t ∈ Q ∩ [0 , ∞ ) of the set of points where the left boundary of η (cid:48) ([ t, ∞ )) hits the right boundary of η (cid:48) ([ t, ∞ )) on the left-hand side (except for the point η (cid:48) ( t ) itself) andcontains this set for t = 0. By countable stability of Hausdorff dimension and translation invariance, it sufficesto show that the dimension of the set of points where the left and right boundaries of η (cid:48) ([0 , ∞ )) intersect inthis manner a.s. has dimension 3 − κ (cid:48) / η (cid:48) (parameterized by quantum area) is equal to the set of times whenthe correlated Brownian motions L and R attain a simultaneous running infimum relative to time 0. Asimultaneous running infimum of L and R is the same as a π/ t for the time-reversal of ( L, R )with the property that 0 is contained in the corresponding cone interval. By [Eva85, Theorem 1] (c.f. theproof of [DMS14, Lemmas 9.4 and 9.5]), it follows that the Hausdorff dimension of this set of times is a.s.1 − κ (cid:48) /
8; in fact, the same is a.s. true of its intersection with [0 , s ] for any s >
0. By applying Theorem 1.1,we obtain that the dimension of this set of intersection points of the boundaries of η (cid:48) | [0 , ∞ ) , and hence alsothe set of local cut points of (cid:98) η (cid:48) , is given by 3 − κ (cid:48) /
8; in fact the same is a.s. true for the set of local cutpoints of every non-trivial segment of (cid:98) η (cid:48) .We will now argue by local absolute continuity that the a.s. dimension of the set of local cut points ofwhole-plane, chordal, or radial SLE κ (cid:48) is the same as the a.s. dimension of the set of local cut points of (cid:98) η (cid:48) . Wehave local absolute continuity of the curves when we do not consider points at which the curves hit theirdomain boundary or their past [SW05], and to conclude it is sufficient to argue that the cut point dimension11f the curves does not decrease if we remove points of this kind. Choral SLE κ (cid:48) a.s. does not have globalcut points which intersect the domain boundary, since the left boundary of the SLE κ (cid:48) has the law of anSLE κ ( κ − κ/ −
2) [MS13, Theorem 1.4], which implies by [Dub09, Lemma 15] that the left boundary ofthe SLE κ (cid:48) a.s. does not hit the right domain boundary. It follows from reversibility [MS16e, MS16a] and thedomain Markov property that chordal SLE κ (cid:48) cannot have any local cut points which are also multiple-points orwhich intersect the domain boundary. By local absolute continuity, the same result follows for the radial andwhole-plane cases, i.e., a radial or whole-plane SLE does not have cut points which are also multiple-points orwhich intersect the domain boundary. The curve (cid:98) η (cid:48) can have a local cut point which is also a multiple point,but by local absolute continuity with respect to ordinary SLE κ (cid:48) away from the times it interacts with itsforce point, it has to wrap around the origin between the first time it hits the point and the time when ithas a local cut point there, so since any non-trivial segment of (cid:98) η (cid:48) has the same local cut point dimension of3 − κ (cid:48) /
8, the set of local cut points of this kind does not have a larger dimension than the set of local cutpoints which are not multiple points.We now argue that the dimension of the set of global cut points is also a.s. equal to 3 − κ (cid:48) /
8. By theconformal Markov property and transience of SLE κ (cid:48) , for any t ∈ R and (cid:15) >
0, if we condition on the initialsegment (cid:101) η (cid:48) | t ≤ t of a chordal, radial or whole-plane SLE κ (cid:48) curve (cid:101) η (cid:48) , the global cut points for (cid:101) η (cid:48) | t ≤ t whichlie at distance at least (cid:15) from (cid:101) η (cid:48) ( t ) will be global cut points for (cid:101) η (cid:48) with positive probability. This impliesthat for any ζ > (cid:101) η (cid:48) have dimension at least 3 − κ (cid:48) − ζ with positive probability.Here we subtract a small parameter ζ since the dimension of the cut points of (cid:101) η (cid:48) ([0 , t ]) might be slightlylarger than the dimension of the cut points of (cid:101) η (cid:48) ([0 , t ]) \ B (cid:15) ( (cid:101) η (cid:48) ( t )). By scale invariance, for (say) a chordalSLE κ (cid:48) in H from 0 to ∞ and for any r >
0, the probability that the intersection of the set of global cut pointsof the curve with B r (0) has dimension ≥ − κ (cid:48) − ζ , is independent of r . If (cid:98) h is a GFF whose imaginarygeometry counterflow line [MS16d] is our given SLE κ (cid:48) curve, then the sigma algebra ∩ r> σ ( (cid:98) h | B r (0) ) is trivial(see, e.g., [MS16d, Proposition 3.2]).Since the chordal SLE κ (cid:48) is locally determined by (cid:98) h this implies that the global cut point dimension is ≥ − κ (cid:48) − ζ almost surely. Sending ζ → κ (cid:48) is3 − κ (cid:48) . A similar argument using GFF tail triviality works in the case of whole-plane SLE, and the radialcase follows from the whole-plane case. η (0)( η ( t ( s ))) s ≥ Figure 2: Illustration of Examples 2.3 and 2.4. The figure shows the west-going (resp. east-going) flowline from the origin in red (resp. blue), and the region η (cid:48) (( −∞ , η (cid:48) ( t ( s ))) s ≥ , whose marginal law is a whole-plane SLE κ (cid:48) ( κ (cid:48) − L and R relative to time 0. Example 2.5.
The Hausdorff dimension of the double points of
SLE κ (cid:48) is a.s. equal to − (12 − κ (cid:48) )(4 + κ (cid:48) )8 κ (cid:48) for κ (cid:48) ∈ (4 , and κ (cid:48) for κ (cid:48) ≥ . (2.2)12 roof. For κ (cid:48) ≥ κ (cid:48) has the same dimension as the boundary of η (cid:48) ([0 , ∞ )),since, conditioned on η (cid:48) (( −∞ , η (cid:48) ( t )) t ≥ has the law of a chordal SLE κ (cid:48) . It follows that the set of doublepoints has dimension 1 + 2 /κ (cid:48) , since the left and right boundaries of η (cid:48) ([0 , ∞ )) marginally each have the lawof a whole-plane SLE κ (2 − κ ) from 0 to ∞ , κ = 16 /κ (cid:48) .Now assume κ (cid:48) ∈ (4 , U be a connected component of C \ η (cid:48) (( −∞ , h , and let z (resp. z ) be the first (resp. last) point on ∂U visited by η (cid:48) after time 0. Recallfrom Section 1.4.3 that the conditional law given η (cid:48) (( −∞ , η (cid:48) contained in U is that of aspace-filling chordal SLE κ (cid:48) from z to z . Let (cid:98) η (cid:48) be the curve obtained by skipping the bubbles filled in bythis segment of η (cid:48) , which is an ordinary SLE κ (cid:48) from z to z in U (the curve (cid:98) η (cid:48) can be obtained by skippingthe times contained in reverse π/ (cid:98) η (cid:48) is obtained by skipping the bubbles filled in by η (cid:48) during a certain interval of times, we find thatif s < s are such that z := (cid:98) η (cid:48) ( s ) = (cid:98) η (cid:48) ( s ), then there exists t ∈ Q ∩ [0 , ∞ ) such that the following istrue. There is a connected component U t of C \ η (cid:48) (( −∞ , t ]), such that if τ t is the first time that (cid:98) η (cid:48) enters U t , then s < τ t < s and z is a point of intersection between the chordal SLE κ (cid:48) in the bead U t andthe boundary of the domain (in particular, U t is the connected component of U \ (cid:98) η (cid:48) ([0 , τ t ]) with z on itsboundary). Furthermore, there is a closed arc of ∂U t containing the initial point (cid:98) η (cid:48) ( τ t ) such that everyintersection point of the chordal SLE κ (cid:48) with this arc is a double point of (cid:98) η (cid:48) ; the reason this property does nothold for all intersection points between the chordal SLE κ (cid:48) and ∂U t is that some of these intersection pointswill be contained in ∂U .Since η (cid:48) ( · − t ) d = η (cid:48) modulo rotation and scaling [DMS14, Lemma 9.3], to show that the double pointdimension of (cid:98) η (cid:48) is a.s. given by (2.2), it suffices to show that the dimension of the intersection of (cid:98) η (cid:48) with anynon-trivial arc of ∂U containing z is a.s. given by (2.2).Let ψ : [0 , ∞ ) → C be the parameterization of the left boundary of η (cid:48) (( −∞ , ψ (0) = 0 and such thatfor each 0 ≤ u < v we have that the quantum length of the segment from ψ ( u ) to ψ ( v ) is equal to v − u .Let a > ψ ( a ) is where the left boundary of η (cid:48) (( −∞ , ∂U . Let ϕ : R × [0 , π ] → U be the unique conformal transformation which takes −∞ (resp. + ∞ ) to the initial (resp. terminal) point of (cid:98) η (cid:48) such that the field (cid:98) h = h ◦ ϕ + Q log | ϕ (cid:48) | on R × [0 , π ] has the horizontal translation chosen so that thesupremum of its projection onto the space of functions which are constant on vertical lines is hit at u = 0.For each M ∈ R , let A M = ψ − ( (cid:98) η (cid:48) ∩ ϕ (( −∞ , M ] × { π } )). Let ( (cid:101) L t , (cid:101) R t ) t ∈ R be the time-reversal of ( R t , L t ) t ∈ R ,so ( (cid:101) L t , (cid:101) R t ) t ∈ R is the pair of Brownian motions encoding the time-reversal of η (cid:48) on top of the independentquantum cone. If we define (cid:98) X = { t ≥ (cid:101) R t = inf s ∈ [0 ,t ] (cid:101) R s , ϕ − ( (cid:98) η (cid:48) ( t )) ∈ ( −∞ , M ] × { π }} then η (cid:48) ( (cid:98) X ) = ψ ( A M ).We claim that for any M > A M − a is absolutely continuous with respect to the lawof the range of a stable subordinator of index κ (cid:48) / − U, h | U , (cid:98) η (cid:48) ) viewed as a curve-decorated quantum surface (i.e., moduloconformal maps). Let U Q be the first bead of η (cid:48) ([0 , ∞ )) such that the sum of the quantum masses of theprevious beads (including U Q ) is at least 1 and let (cid:98) η (cid:48)Q be the associated chordal SLE κ (cid:48) curve between itsmarked points. Since U Q is chosen in a manner which does not depend on the particular embedding of thequantum surface parametrized by η (cid:48) ([0 , ∞ )) into C , it follows that the conditional law of the curve-decoratedquantum surface ( U Q , h | U Q , (cid:98) η (cid:48) U Q ) given its quantum area and boundary length is that of a single bead ofa weight-2 − γ / κ (cid:48) curve between its markedpoints. Since U is independent from h , it a.s. holds with positive conditional probability given η (cid:48) (viewedmodulo monotonte parametrization) and ( U, h | U , (cid:98) η (cid:48) ) that ( U Q , (cid:98) η (cid:48)Q ) = ( U, (cid:98) η (cid:48) ). Hence the law of ( U, h | U , (cid:98) η (cid:48) ) isabsolutely continuous with respect to the law of ( U Q , h | U Q , (cid:98) η (cid:48) U Q ).The law of the left and the right boundary length of (cid:98) η (cid:48) parameterized by quantum natural time and rununtil it exits ϕ (( −∞ , M ] × [0 , π ]) is absolutely continuous with respect to a κ (cid:48) / U to the strip as above, the law of the restriction of the field to ( −∞ , M ] × [0 , π ] is13bsolutely continuous with respect to the analogous restricted field for a thick quantum wedge of weight γ − t ≥ (cid:101) R t = inf s ∈ [0 ,t ] (cid:101) R s has the law of the range of a stable subordinator of index1 /
2. Hence the law of (cid:98) X is absolutely continuous with respect to the law of the range of the compositionof two (not necessarily independent) subordinators of index 1 / κ (cid:48) / −
1, respectively. By the uniformdimension transformation result for subordinators [HP74, Theorem 4.1], a.s. dim H ( (cid:98) X ) = κ (cid:48) / − /
2. ByTheorem 1.1, a.s. the dimension of the intersection of (cid:98) η (cid:48) with any non-trivial segment of ∂U is given by (2.2).Recalling the argument at the beginning of the proof, a.s. the set of double points of (cid:98) η (cid:48) is a.s. given by (2.2).Since chordal SLE κ (cid:48) a.s. does not have any boundary double points [MW17, Remark 5.3], the double pointdimension for other types of SLE κ (cid:48) is obtained via local absolute continuity, as in the preceding examples. η θ η θ U (cid:98) η (cid:48) Figure 3: The left and right figure illustrate Example 2.5 and Example 2.7, respectively. Both figures illustrate η (cid:48) stopped at time zero, with the region η (cid:48) (( −∞ , κ (cid:48) ∈ (4 , κ (cid:48) double pointshave the same Hausdorff dimension as the points of intersection between the chordal SLE κ (cid:48) (cid:98) η (cid:48) in U and theleft frontier of η (cid:48) (( −∞ , η (cid:48) , and we assume without loss of generality that the first flow line is given by the right boundary of η (cid:48) ([0 , ∞ )). As we will explain just below, we can sample from the law of the flow line intersection points byconsidering a Bessel process Y which encodes the quantum wedge which lies between the two flow lines. TheBessel process Y that we consider has a different dimension than the Bessel process in [DMS14, Section 4.4],since the excursions of Y encode quantum boundary lengths rather than quantum areas. We derive thedimension of Y in the following lemma. Lemma 2.6.
Let W be a (thin) quantum wedge of weight W ∈ (0 , γ / . There is a Bessel process Y ofdimension d = 4 W/γ with the following property. The ordered sequence of quantum disks of W correspond tothe ordered sequence of the excursions of Y from , and the right quantum boundary length of each quantumdisk is identical to the length of the corresponding excursion of Y from .Proof. By [DMS14, Definition 4.13], W is a Poissonian chain of beads, corresponding to the ordered sequenceof excursions from 0 of a Bessel process (cid:101) Y of dimension (cid:101) d = 1 + 2 W/γ , such that each surface can beparameterized as follows. Let S = R × [0 , π ], and for each excursion e of (cid:101) Y , let ( S , h e ) be a parameterizationof the corresponding bead of W . The distribution h e is given by h e = h † + h , where h † (( t, u )) = (cid:101) X et for all( t, u ) ∈ S , (cid:101) X et is equal to the reparameterization of 2 γ − log( e ) to have quadratic variation 2 dt , and h isindependent of h † and equal in law to the projection of a free boundary GFF onto the space of distributionswith mean 0 on each vertical line. Let a = W/γ − γ/
2. If we take the horizontal translation so that (cid:101) X e reachesits supremum at t = 0, then by [DMS14, Propositions 3.4 and 3.5, Remark 3.7], we have (cid:101) X et = (cid:101) B e, t + at for t ≥
0, and (cid:101) X et = (cid:101) B e, − t − at for t <
0, where (cid:101) B e, and (cid:101) B e, are two independent Brownian motions started from14 b e := 2 γ − log(sup( e )), conditioned on (cid:101) X et ≤ (cid:101) b e for all t ∈ R . (We note that since a <
0, this conditioningcan be made sense of as in [DMS14, Remark 4.3].)The conditional expectation of the right quantum boundary length of ( S , h e ) given ( (cid:101) X et ) t ∈ R is proportional to (cid:90) R exp( γ (cid:101) X et / dt = 4 (cid:90) R exp( γX et ) dt, (2.3)where X et = B e, t + 2 at for t ≥ X et = B e, − t − at for t <
0, and B e, and B e, are two independentBrownian motions started from b e := (cid:101) b e /
2, conditioned on X et ≤ b e for all t ∈ R .We will argue that we obtain a Bessel process of dimension d := 2 (cid:101) d − Wγ if we reparameterize exp( γX et / dt for each e , and concatenate the resulting excursions in the order given by (cid:101) Y .By (2.3) and [DMS14, Remark 4.14, equations (4.3) and (4.4)] this will imply that the collection of conditionalexpected right boundary lengths of W , given the projection of each h e onto the space of functions that areconstant on { t } × [0 , π ] for all t ∈ R , has the law of the excursion lengths of a Bessel process of dimension d .By the same argument as in the proof of [DMS14, Proposition 4.16], this implies that there is a d -dimensionalBessel process Y such that the length of an excursion e is equal to the actual quantum boundary length ofthe corresponding surface, hence we can conclude the proof of the lemma.By [DMS14, Lemma 3.4, Lemma 3.6, Remark 3.7] we would obtain a Bessel process of dimension d by theabove procedure, given that the collection of maxima b e of the processes X et has the right law, since we knowthat the drift ± a of X et corresponds to a Bessel process of dimension d ; see [DMS14, Table 1.1]. Givenan excursion e define (cid:101) e ∗ := sup( e ), and note that e ∗ := ( (cid:101) e ∗ ) / is the maximum of the excursion obtainedby reparameterizing exp( γX et / (cid:101) e ∗ can be described by consideringa Poisson point process of intensity ds ⊗ u (cid:101) d − du , where ds and du denote Lebesgue measure on R + . Arealization of the Poisson point process is a collection of points ( s, (cid:101) e ∗ ), where the second coordinate givesthe maximum value of a Bessel excursion, and the Bessel excursions are ordered chronologically by the firstcoordinate. The collection of points ( s, e ∗ ) = ( s, ( (cid:101) e ∗ ) / ) has the law of a Poisson point process of intensityproportional to ds ⊗ u (cid:101) d − du = ds ⊗ u d − du , hence our wanted result follows. Example 2.7.
Let θ , θ ∈ R and suppose that θ := θ − θ ∈ (0 , π ] . Consider two flow lines η θ i , i ∈ { , } ,of a whole-plane GFF (cid:98) h , started from z ∈ C (in the sense of [MS13]). If θ ∈ (0 , πκ − κ ∧ π ] the Hausdorffdimension of η θ ∩ η θ is a.s. given by − κ (cid:16) ρ + κ (cid:17) (cid:16) ρ − κ (cid:17) , where ρ = θ (2 − κ/ /π − . If κ ≤ and θ ∈ [ πκ − κ , π ] , the flow lines a.s. do not intersect.Proof. By [MS13, Theorem 1.1], η θ has the law of a whole-plane SLE κ (2 − κ ). By [MS13, Theorem 1.11], theconditional law of η θ given η θ is that of a chordal SLE κ ( ρ ; ρ ) from 0 to ∞ in C \ η θ , where ρ i = W i − W = θ (2 − γ / /π ≥ W = W − W ≥ W = 4 − γ . Note that ρ = ρ , with ρ as defined in thestatement of the example. Let ( C , h, , ∞ ) be a weight- W quantum cone (equivalently, a γ -quantum cone)independent of η θ i , i ∈ { , } , and assume w.l.o.g. that z = 0. By [DMS14, Theorem 1.9], the quantumsurface W (resp. W ) having η θ as left (resp. right) boundary and η θ as right (resp. left) boundary, is aquantum wedge of weight W (resp. W ). If κ ≤ θ ∈ [ πκ − κ , π ], W and W are thick wedges, implyingthat η θ ∩ η θ = { } . Assume κ > θ (cid:54)∈ [ πκ − κ , π ]. By Lemma 2.6, there is a Bessel process (cid:98) B of dimension d = 4 W /γ , such that the ordered lengths of its excursions from 0, are identical to the ordered sequence ofthe right boundary lengths of the bubbles. By the comment right after [Ber99, Proposition 2.2], there is asubordinator S of index α = 1 − d/ (cid:98) B is equal to its range.Let η (cid:48) be a whole-plane space-filling SLE κ (cid:48) from ∞ to ∞ as above. The right boundary of η (cid:48) ([0 , ∞ )) has thelaw of an SLE κ (2 − κ ), so we can assume w.l.o.g. that η θ is equal to the right boundary of η (cid:48) ([0 , ∞ )).Let (cid:98) X ⊂ [0 , ∞ ) be the set of times that η (cid:48) | [0 , ∞ ) visits a point in η θ ∩ η θ for the first time. Note that η (cid:48) | [0 , ∞ ) visits a point in η θ exactly when R is equal to its running infimum since time zero, and η (cid:48) visits a point in15he intersection η θ ∩ η θ when the additional condition − R t ∈ S ( R + ) holds. Hence,dim H ( Z t ) t ∈ (cid:98) X = dim H { ( L t , R t ) : − R t ∈ S ( R + ) , R t = inf ≤ s ≤ t R s } . The set of times t where R t = inf ≤ s ≤ t R s , is equal to the range of a stable subordinator S of index α = 1 /
2, such that S ( x ) is the first time that R t hits − x , for any x ≥
0. It follows that (cid:98) X = S ( S ( R + )).By [HP74, Theorem 4.1] it holds a.s. that dim H ( (cid:98) X ) = α α = 1 / − W /γ = 1 / − ( ρ + 2) /κ . ApplyingTheorem 1.1 completes the proof.In our final application of Theorem 1.1 we will use the theorem in the reverse direction as compared to theexamples above. We use the dimension of the CLE κ (cid:48) gasket determined in [SSW09, MSW14] for κ (cid:48) ∈ (4 , π/ π/ Z | [0 , ∞ ) is a time interval ( s, t ) ⊂ [0 , ∞ ) suchthat R u ≥ R s and L u ≥ L s for all u ∈ ( s, t ), and such that R s = R t (resp. L s = L t ). Furthermore a reverse π/ Z | [0 , ∞ ) is a time interval ( s, t ) ⊂ [0 , ∞ ) such that ( − t, − s ) is a cone interval for thetime-reversal ( Z − t ) t ≤ of Z . Example 2.8.
Let κ (cid:48) ∈ (4 , . Consider ( Z t ) t ≥ and let (cid:98) X be the set of times that are not contained in anyleft cone intervals, i.e., (cid:98) X = [0 , ∞ ) \{ u ≥ ∃ left cone interval ( s, t ) , ≤ s < t, s.t. u ∈ ( s, t ) } . (2.4) Then dim H ( (cid:98) X ) = + κ (cid:48) .Proof. It is sufficient to consider a reverse right cone interval ( t , t ), 0 < t < t < ∞ , and prove that the set (cid:98) X (cid:48) = ( t , t ) \{ u ∈ ( t , t ) : ∃ a reverse left cone interval ( s , s ) ⊂ ( t , t ) , s < s , s.t. u ∈ ( s , s ) } (2.5)satisfies dim H ( (cid:98) X (cid:48) ) = + κ (cid:48) . This is sufficient by invariance in law under time-reversal of Brownian motion,since any compact subset of [0 , ∞ ) is a.s. contained in some reverse right π/ , ∞ ) a.s. contains some reverse right π/ η (cid:48) encodes a whole-plane CLE κ (cid:48) . The interior U of the image of the reverseright cone interval ( t , t ) under η (cid:48) is a “bubble” disconnected from ∞ by η (cid:48) , with the boundary traced inthe clockwise direction by η (cid:48) . The restriction of the CLE κ (cid:48) to U has the law of a CLE κ (cid:48) in U . It followsfrom e.g. [She09, MSW14, GM17b] that the interiors of the outermost CLE κ (cid:48) loops in U associated with thespace-filling SLE κ (cid:48) η (cid:48) correspond to outermost reverse left cone excursions of Z | [ t ,t ] . The gasket of theCLE κ (cid:48) in U is the set of points in U not contained in the interiors of any of these loops. The result nowfollows from Theorem 1.1, since we know by [MSW14, SSW09] that the CLE κ (cid:48) gasket a.s. has dimension2 − (8 − κ (cid:48) )(3 κ (cid:48) − / (32 κ (cid:48) ).The final example in this section will be an application of [RV11, Theorem 4.1] to calculate the Hausdorffdimension of the points of intersection between the real line and a chordal SLE κ ( ρ ), κ >
0, in the upperhalf-plane where ρ is in the range of values in which the process does not fill the boundary. This Hausdorffdimension was first obtained in [AS08] for the special case ρ = 0 and κ >
4, and the formula was proved forgeneral values of ρ and κ in [MW17, Theorem 1.6]. Our main result Theorem 1.1 cannot be used in thissetting, since it only applies to SLE and CLE sets in the interior of a domain. Example 2.9.
Let κ > , κ (cid:54) = 4 , and ρ ∈ ( − ∨ ( κ − , κ − , and consider an SLE κ ( ρ ) η on H from to ∞ with force-point at + . Almost surely, dim H ( η ∩ R + ) = 1 − κ ( ρ + 2) (cid:16) ρ + 4 − κ (cid:17) . roof. First we consider the case κ >
4, hence we will write κ (cid:48) instead of κ and η (cid:48) instead of η . Let η (cid:48) bea chordal SLE κ (cid:48) curve from 0 to ∞ in H on top of an independent quantum wedge ( H , h, , ∞ ) of weight γ − γ ρ . Let D be the open subset of H which is between the right boundary of η (cid:48) and [0 , ∞ ).By [DMS14, Theorem 1.17], ( D, h, , ∞ ) has the law of a thin quantum wedge of weight W = γ − γ ρ .Defining (cid:98) X ⊂ [0 , ∞ ) by (cid:98) X := { ν ([0 , x ]) : x ∈ η (cid:48) ∩ R + } it follows by Lemma 2.6 and [Ber99, Proposition 2.2]that (cid:98) X has the law of the range of a stable subordinator of index κ (cid:48) / − − ρ/
2, hence dim H ( (cid:98) X ) = κ (cid:48) / − − ρ/ h since, if h is given the circle-average embedding, say, then the restriction of h to any sub-domain of H bounded awayfrom 0, ∞ , and ∂ D ∩ H is mutually absolutely continuous with respect to the corresponding restriction of afree-boundary GFF on H normalized to have average zero on ∂ D ∩ H .We proceed by the exact same argument when κ ∈ (0 , κ (cid:48) > In this section we will prove various estimates for space-filling SLE and for GFFs which we will need in thesequel.
Throughout this subsection, we fix κ (cid:48) > η (cid:48) be a whole-plane space-filling SLE κ (cid:48) from ∞ to ∞ ,parameterized by Lebesgue measure and satisfying η (cid:48) (0) = 0. Our goal is to prove the following lemma which(together with a multi-scale argument) will be used in the next subsection to argue that η (cid:48) is very unlikely totravel a long distance without absorbing a large Euclidean ball (see Lemma 3.6). Define T ρ := inf { t ≥ η (cid:48) ([0 , t ]) (cid:54)⊂ B ρ (0) } , ∀ ρ > . (3.1)The main result of this subsection is the following lemma. Lemma 3.1.
There are constants δ, p ∈ (0 , depending only on κ (cid:48) such that the following is true. For any (cid:15) ∈ (0 , , it a.s. holds with conditional probability at least p given η (cid:48) | [0 ,T ] that η (cid:48) ([ T , T (cid:15) ]) contains a ballof radius at least δ(cid:15) . The proof of Lemma 3.1 proceeds via a combination of elementary complex analysis and facts from imaginarygeometry [MS16d, MS16e, MS16a, MS13]. See Figure 4 for an illustration and outline of the proof.We remark that in the case when κ (cid:48) ∈ (4 , η (cid:48) | [0 , ∞ ) traces points inthe same order as the associated SLE κ (cid:48) ( κ (cid:48) −
6) curve started from ∞ , and the time reversal of this curveis also an SLE κ (cid:48) ( κ (cid:48) −
6) [MS13, Theorem 1.20]. So, we can reduce our problem to proving that if η (cid:48) is anSLE κ (cid:48) ( κ (cid:48) −
6) from 0 to ∞ , then with uniformly positive conditional probability given η (cid:48) up to the firsttime it exits D , it holds that η (cid:48) forms a bubble which contains a ball of radius δ(cid:15) before exiting B (cid:15) (0).This statement can be proven using basic complex analysis plus the Markov property of SLE κ (cid:48) ( κ (cid:48) − κ (cid:48) > κ (cid:48) ( κ (cid:48) −
6) is not reversible. In fact, the marginal law of η (cid:48) | [0 , ∞ ) is not that ofan SLE κ (cid:48) ( ρ ) for any choice of ρ . So, we instead need to control this curve using interior flow lines of a GFF(which form its left and right boundaries).We now proceed with the proof of Lemma 3.1. First we introduce some notation. For t ≥
0, let K t be thehull generated by η (cid:48) ([0 , t ]), i.e. the union of η (cid:48) ([0 , t ]) and the set of points which it disconnects from ∞ (thishull is just η (cid:48) ([0 , t ]) if κ (cid:48) ≥ (cid:15) η (cid:48) ([0 , T ]) D B (cid:15) (0) f (cid:15) U ∗ x R x L U ∗ f (cid:15) ( x L ) f (cid:15) ( x R ) f (cid:15) ( x L ) f (cid:15) ( x R )1 1 A L A L A R A R Figure 4: Illustration of the proof of Lemma 3.1. We seek to show that η (cid:48) absorbs the ball B δ(cid:15) ( z (cid:15) ) for some δ = δ ( κ (cid:48) ) > B (cid:15) (0). To do this, we study the conformal map f (cid:15) : C \ K T → D taking z (cid:15) to 0 and η (cid:48) ( T ) to 1. The set U ∗ is a neighborhood of a path from 0 to 1 in D whose pre-imageunder f (cid:15) is contained in V (cid:15) ⊂ B (cid:15) (0) (in the figure, f − (cid:15) ( U ∗ ) is contained in B (cid:15)/ ( z (cid:15) ), but in general it mightjust be disconnected from ∞ by K T ∪ B (cid:15)/ ( z (cid:15) )). Consider the flow lines started from a point near z (cid:15) whichform the left and right boundaries of η (cid:48) at the time when it hits this point. These flow lines and their imagesunder f (cid:15) are shown in orange and purple. In the case when f (cid:15) ( x L ) and f (cid:15) ( x R ) are at macroscopic distancefrom 1 (left and middle panels) we can use a local absolute continuity argument to show that with uniformlypositive conditional probability given η (cid:48) | [0 ,T ] , the orange and purple flow lines in the middle figure togetherwith ∂ D form a pocket contained in U ∗ which itself contains f (cid:15) ( B δ(cid:15) ( z (cid:15) )). The pre-image of this pocket under f (cid:15) is contained in η (cid:48) ([ T , T (cid:15) ]). If instead f (cid:15) ( x L ) and f (cid:15) ( x R ) are very close to 1 (as shown in the rightpanel) we instead need to grow flow lines started from f (cid:15) ( x L ) and f (cid:15) ( x R ) (dark green and brown), which canequivalently be described as small segments of f (cid:15) ( η L \ η (cid:48) ([0 , T ])) and f (cid:15) ( η R \ η (cid:48) ([0 , T ])). If only one of f (cid:15) ( x L )or f (cid:15) ( x R ) is very close to 1, we only need to grow a single extra flow line. On a uniformly positive probabilityevent, if we map the complementary connected component containing 0 of the green and brown flow lines to D , the images of the tips will be at uniformly positive distance from 1. This gives us a configuration whichlooks like the one in the middle figure, which allows us to argue that with uniformly positive probability, theunion of all 4 flow lines in the right panel and ∂ D forms a pocket surrounding f (cid:15) ( B δ(cid:15) ( z (cid:15) )). The pre-image ofthis pocket under f (cid:15) will again be contained in η (cid:48) ([ T , T (cid:15) ]).Following [MS16d], we define the constants κ := 16 κ (cid:48) , χ := 2 √ κ − √ κ , λ := π √ κ , λ (cid:48) := π √ κ (cid:48) . (3.2)We also let (cid:98) h be the whole-plane GFF viewed modulo a global additive multiple of 2 πχ which is used toconstruct η (cid:48) as in Section 1.4.3. For z ∈ C we let η Lz (resp. η Rz ) be the flow line of (cid:98) h started from z with angle π/ − π/ η Lz and η Rz are the left and right boundaries of η (cid:48) at the first time it hits z .The set ∂η (cid:48) ([0 , T ]) can be divided into four distinguished arcs, which we denote as follows. • A L (resp. A R ) is the arc of ∂K T traced by η L (resp. η R ). • A L (resp. A R ) is the arc of ∂K T not traced by η L or η R which lies to the left (resp. right) of η (cid:48) ( T ).Using the notation (3.1), we define the σ -algebra F := σ (cid:16) η (cid:48) | [0 ,T ] , (cid:98) h | η (cid:48) ([0 ,T ]) (cid:17) . emma 3.2. The set η (cid:48) ([0 , T ]) is a local set for (cid:98) h in the sense of [SS13, Lemma 3.9]. In particular, theboundary data for the conditional law of h | C \ K T given F on each of the arcs A L , A R , A L , and A R coincideswith the boundary data of the corresponding flow line of (cid:98) h (i.e., it is given by flow line boundary conditions asdescribed in [MS13, Figure 9]).Proof. We first check that η (cid:48) ([0 , T ]) is a local set for (cid:98) h . By [SS13, Lemma 3.9, condition 1], it suffices toshow that for each deterministic open set U ⊂ C , the event { η (cid:48) ([0 , T ]) ∩ U (cid:54) = ∅} is a.s. determined by h | C \ U .For z ∈ C , let S Lz (resp. S Rz ) be the first time that the flow line η Lz (resp. η Rz ) enters U . Since flow lines arelocal sets, each η qz ([0 , S qz ]) for q ∈ { L, R } and z ∈ C is a.s. determined by h | C \ U . Since the outer boundary of η (cid:48) at the first time it hits any given rational z ∈ C is equal to η Lz ∪ η Rz , we see that a.s. η (cid:48) ([0 , T ]) intersects U if and only if there is a z ∈ Q \ D such that η Lz merges into η L ([0 , S L ]) before time S Lz ; and the same is truewith “ R ” in place of “ L ”. This latter event is a.s. determined by h | C \ U .By [SS13, Lemma 3.11] (applied to the local sets η (cid:48) ([0 , T ]) and η Lz , η Rz for z ∈ Q ) and the known boundarydata for interior flow lines of a whole-plane GFF [MS13, Theorem 1.1], we obtain the claimed description ofthe boundary data for the conditional law of (cid:98) h given F .Let z (cid:15) be the point of ∂B (cid:15)/ (0) closest to η (cid:48) ( T ) and let f (cid:15) : ( C ∪ {∞} ) \ K T → D be the conformal mapwhich takes z (cid:15) to 0 and η (cid:48) ( T ) to 1. Let V (cid:15) be the union of B (cid:15)/ ( z (cid:15) ) \ ∂K T and the set of points which itdisconnects from ∞ in C \ K T . Then ∂K T ∩ ∂V (cid:15) is a connected arc of ∂K T . Let I L (resp. I R ) be thesub-arc of ∂K T ∩ ∂V (cid:15) lying to the left (resp. right) of η (cid:48) ( T ) as viewed from η (cid:48) ( T ), looking out from D . Notethat I L (resp. I R ) need not be part of the left (resp. right) outer boundary of K T if all of this left (resp.right) outer boundary is part of ∂V (cid:15) .There is a universal constant q ∈ (0 , /
2) such that conditional on η (cid:48) | [0 ,T ] , a Brownian motion started from z (cid:15) has probability at least q to exit B (cid:15)/ ( z (cid:15) ) at a point outside of B (cid:15)/ ( z (cid:15) ) ∩ D , then make a counterclockwiseloop around B (cid:15)/ ( z (cid:15) ) before re-entering B (cid:15)/ ( z (cid:15) ) or leaving B (cid:15)/ ( z (cid:15) ). If it does so, then such a Brownianmotion first hits K T at a point of I L before exiting V (cid:15) . Symmetrically, Brownian motion started from z (cid:15) hasconditional probability at least q to first hit K T at a point of I R before exiting V (cid:15) .From the above estimates and the conformal invariance of Brownian motion, we infer that there is a universalconstant c ∈ (0 ,
1) such that each point of f (cid:15) ( C \ ( B (cid:15)/ ( z (cid:15) ) ∪ K T )) lies at distance at least c from 0 andeach of the arcs f (cid:15) ( I L ) and f (cid:15) ( I R ) of ∂ D has Euclidean length at least c . In fact, the probability that aBrownian motion started from 0 hits any given ball of radius c centered at a point of D \ B c (0) before exiting D tends to 0 as c →
0, uniformly over all possible choices of center for the ball. Hence the estimate of thefirst paragraph implies that we can find a universal constant c ∈ (0 , c /
2] and a random path α in D from 0to 1 such that B c ( α ) ⊂ f (cid:15) ( V (cid:15) ).Let U be the collection of all simply connected open subsets of D which take the form U = B c/ ( β ) for β asimple piecewise linear path from 0 to 1 in D whose linear segments all connect nearest neighbor points ( c/ Z (by slightly shrinking c , we can assume without loss of generality that 1 /c is an integer, so that 1 ∈ ( c/ Z ).Then U is a finite set and there a.s. exists U ∗ = B c/ ( β ∗ ) ∈ U with U ∗ ⊂ B c ( α ) ⊂ f (cid:15) ( B (cid:15)/ ( z (cid:15) ) \ K T ).By the Koebe quarter theorem, | f (cid:48) (cid:15) ( z (cid:15) ) | (cid:16) (cid:15) − , with universal implicit constant, so by the Koebe growththeorem f − (cid:15) ( B c/ (0)) contains B δ(cid:15) ( z (cid:15) ) for a universal choice of δ ∈ (0 , B c/ (0) is contained in f (cid:15) ( η (cid:48) ([ T , T (cid:15) ])) with uniformly positive conditional probability given η (cid:48) | [0 ,T ] .Recall the imaginary geometry parameters from (3.2). Let (cid:98) h (cid:15) := (cid:98) h ◦ f − (cid:15) − χ arg( f − (cid:15) ) (cid:48) , so that (cid:98) h (cid:15) is similarto a GFF on D with Dirichlet boundary data determined by the images of the distinguished arcs A L , A R , A L , and A R under f (cid:15) (this boundary data is described in Lemma 3.2) except that it possesses a singularityat f (cid:15) ( ∞ ).Let (cid:98) h U ∗ (cid:15) be a GFF on U ∗ with Dirichlet boundary data which coincides with that of (cid:98) h ∗ (cid:15) on ∂U ∗ ∩ ∂ D and whoseboundary data on ∂U ∗ \ ∂ D is 0. As we will see, the laws of (cid:98) h (cid:15) and (cid:98) h U ∗ (cid:15) are mutually absolutely continuouson subsets of U ∗ at positive distance from ∂U ∗ \ ∂ D . Recalling that U ∗ = B c/ ( β ∗ ) for the piecewise linearcurve β ∗ , we define U r ∗ := B rc/ ( β ∗ ) , ∀ r ∈ (0 , . (3.3)19 emma 3.3. Let (cid:98) h (cid:15) (resp. (cid:98) h U ∗ (cid:15) ) be the harmonic part of (cid:98) h (cid:15) | U ∗ (resp. (cid:98) h U ∗ (cid:15) ). For r ∈ (0 , , the conditionallaws of (cid:98) h (cid:15) | U r ∗ and (cid:98) h U ∗ (cid:15) | U r ∗ given F ∨ σ ( (cid:98) h (cid:15) , (cid:98) h U ∗ (cid:15) ) are a.s. mutually absolutely continuous. Furthermore, if M isthe Radon-Nikodym derivative then E ( | log M | | F ) is bounded above by a deterministic constant dependingonly on κ (cid:48) and r .Proof. Fix r (cid:48) ∈ ( r, φ be a smooth bump function which equals 1 on U r ∗ and 0 on U r (cid:48) ∗ , chosen in amanner which depends only on U ∗ . By [MS16d, Lemma 6.4] (applied to the conditional field given F ), foreach w ∈ U r (cid:48) ∗ the conditional law given F of each of (cid:98) h (cid:15) ( w ) and (cid:98) h U ∗ (cid:15) ( w ) is Gaussian with variance boundedabove by a universal constant and mean bounded above in absolute value by a constant depending only on κ (cid:48) and r (for this latter statement, we use that the boundary data for each of (cid:98) h (cid:15) and (cid:98) h U ∗ (cid:15) is bounded).The proof of [MS16d, Proposition 3.4] shows that if we condition on F ∨ σ ( (cid:98) h (cid:15) , (cid:98) h U ∗ (cid:15) ) then the conditionallaw of (cid:98) h (cid:15) | U r ∗ is absolutely continuous with respect to the conditional law of (cid:98) h U ∗ (cid:15) | U r ∗ , with Radon-Nikodymderivative given by M := exp (cid:18) ( (cid:98) h U ∗ (cid:15) , g ) ∇ −
12 ( g, g ) ∇ (cid:19) where g := φ ( (cid:98) h U ∗ (cid:15) − (cid:98) h (cid:15) ) and ( · , · ) ∇ denotes the Dirichlet inner product. Since φ and all its derivatives vanishon ∂U ∗ \ ∂ D and (cid:98) h U ∗ (cid:15) − (cid:98) h (cid:15) is harmonic and vanishes on ∂U ∗ ∩ ∂ D , a short computation using integration byparts shows that ( g, g ) ∇ = (cid:90) U ∗ (cid:18)
12 ∆( φ ( w ) ) − φ ( w )∆ φ ( w ) (cid:19) ( (cid:98) h U ∗ (cid:15) ( w ) − (cid:98) h (cid:15) ( w )) dw. By the above bounds for (cid:98) h (cid:15) and (cid:98) h U ∗ (cid:15) , we infer that E (( g, g ) ∇ | F ) is a.s. bounded above by a constantdepending only on U ∗ , κ (cid:48) , and r . Since E (cid:16) ( (cid:98) h U ∗ (cid:15) , g ) ∇ | F (cid:17) = E (( g, g ) ∇ | F ), also E ( | log M | | F ) a.s. boundedabove by constants depending only on U ∗ , κ (cid:48) , and r . Since there are only finitely many possible realizations U ∈ U of U ∗ , we obtain the statement of the lemma by taking a minimum over all such U . Proof of Lemma 3.1.
Let x L (resp. x R ) be the a.s. unique point of A L ∩ A L (resp. A R ∩ A R ) and with U / ∗ as in (3.3) let H (cid:15) := (cid:110) f (cid:15) ( x L ) , f (cid:15) ( x R ) / ∈ ∂U / ∗ (cid:111) ⊂ F . On H (cid:15) , the conditional law given F of the auxiliary GFF (cid:98) h U ∗ (cid:15) depends only on the choice of U ∗ (i.e., theboundary data of (cid:98) h (cid:15) on does not depend on F ).Let (cid:98) η L,U ∗ (cid:15) (resp. (cid:98) η R,U ∗ (cid:15) ) be the flow line of (cid:98) h U ∗ (cid:15) started from − c/
500 with angle π/ − π/ (cid:98) η L,U ∗ (cid:15) (resp. (cid:98) η R,U ∗ (cid:15) ) at the first time it hits f (cid:15) ( A L(cid:15) ) ∩ ∂U ∗ (resp. f (cid:15) ( A R(cid:15) ) ∩ ∂U ∗ ).Let G U ∗ (cid:15) be the event that the region whose boundary is formed by the left side of (cid:98) η L,U ∗ (cid:15) , the right sideof (cid:98) η R,U ∗ (cid:15) , and ∂ D ∩ ∂U ∗ contains B c/ (0) and is contained in U / ∗ . By [MS13, Lemma 3.9] applied to (cid:98) η L,U ∗ (cid:15) and then [MW17, Lemma 2.5] applied to the conditional law of (cid:98) η R,U ∗ (cid:15) given (cid:98) η L,U ∗ (cid:15) , we infer thata.s. P ( G U ∗ (cid:15) | F ) > H (cid:15) . Since this conditional probability depends only on U ∗ on H (cid:15) and there areonly finitely many possible choices of U ∗ , we can find a p ∈ (0 ,
1) depending only on κ (cid:48) such that a.s. P ( G U ∗ (cid:15) | F ) H (cid:15) ≥ p H (cid:15) .To transfer this to an estimate for (cid:98) h (cid:15) (rather than for (cid:98) h U ∗ (cid:15) ), let (cid:98) η L(cid:15) (resp. (cid:98) η R(cid:15) ) be the flow line of (cid:98) h (cid:15) startedfrom − c/
500 with angle π/ − π/ f (cid:15) ( A L(cid:15) ∪ A L(cid:15) ) (resp. f (cid:15) ( A R(cid:15) ∪ A R(cid:15) ),if this time is finite. Equivalently, (cid:98) η L(cid:15) = f (cid:15) ◦ η Lf − (cid:15) ( − c/ , stopped at the first time it hits merges into the leftboundary of η (cid:48) ([0 , T ]); and similarly for (cid:98) η R(cid:15) . Let G (cid:15) be the event that (cid:98) η L(cid:15) and (cid:98) η R(cid:15) are contained in U / ∗ ; andthe region whose boundary is formed by the left side of η L,U ∗ (cid:15) , the right side of η R,U ∗ (cid:15) , and ∂ D ∩ ∂U ∗ contains B c/ (0).We will now deduce from the previous two paragraphs and Lemma 3.3 that there is a p ∈ (0 , κ (cid:48) , such that a.s. P ( G (cid:15) | F ) H (cid:15) ≥ p H (cid:15) . To see this, define the harmonic parts (cid:98) h (cid:15) , (cid:98) h U ∗ (cid:15) and the20adon-Nikodym derivative M as in Lemma 3.3 (the latter with r = 1 / P ( G U ∗ (cid:15) | F ) H (cid:15) ≥ p H (cid:15) ,there is a p = p ( κ (cid:48) ) ∈ (0 ,
1) such that it a.s. holds with conditional probability at least p given F that P (cid:16) G U ∗ (cid:15) | F ∨ σ ( (cid:98) h (cid:15) , (cid:98) h U ∗ (cid:15) ) (cid:17) H (cid:15) ≥ p H (cid:15) . Since E ( | log M | | F ) is bounded above by a constant depending only on κ (cid:48) , we can find b > − p / F , it holds that P (cid:16) M ≥ b | F ∨ σ ( (cid:98) h (cid:15) , (cid:98) h U ∗ (cid:15) ) (cid:17) H (cid:15) ≥ (1 − p / H (cid:15) . Since flow lines are determined locally by the field, on H (cid:15) it holds with conditional probability at least p / F that P (cid:16) G (cid:15) | F ∨ σ ( (cid:98) h (cid:15) , (cid:98) h U ∗ (cid:15) ) (cid:17) = E (cid:16) M G U ∗ (cid:15) | F ∨ σ ( (cid:98) h (cid:15) , (cid:98) h U ∗ (cid:15) ) (cid:17) ≥ bp . Taking expectations conditional on F proves our claim with p = bp / U ∗ ⊂ f (cid:15) ( B (cid:15)/ ( z (cid:15) ) \ D ), it follows from the definition of space-filling SLE that on G (cid:15) the region inthe definition of G (cid:15) is contained in the interior of f (cid:15) ( η (cid:48) ([ T , T (cid:15) ]). Since we have chosen δ > B δ(cid:15) ( z (cid:15) ) ⊂ f − (cid:15) ( B c/ (0)), we infer from the preceding paragraph that P ( F (cid:15) | F ) H (cid:15) ≥ p H (cid:15) .It remains to treat the case when H (cid:15) does not occur. The idea is to extend K T by growing small segmentsof η L and η R , beyond the ones contained in η (cid:48) ([0 , T ]), to get a larger hull for which an analog of H (cid:15) occurs.See the right panel of Figure 4 for an illustration. For simplicity, suppose that H (cid:15) does not occur and thatboth f (cid:15) ( x L ) and f (cid:15) ( x R ) are contained in ∂U / ∗ (the case when only one of these points is contained in ∂U ∗ istreated similarly by extending only one flow line instead of two). Let ˚ η L(cid:15) (resp. ˚ η R(cid:15) ) be the flow line of (cid:98) h (cid:15) started from x L (resp. x R ) with angle π/ − π/
2) targeted at −
1, say, and let ˚ S L(cid:15) (resp. ˚ S R(cid:15) ) be its exittime from B c/ (1) ⊂ U ∗ . Note that by uniqueness of flow lines (c.f. [MS16d, Theorem 2.4]) ˚ η L(cid:15) | [0 ,S L(cid:15) ] is aninitial segment of f (cid:15) ( η L \ K T ) and similarly with “ R ” in place of “ L ”. Let ˚ D be the connected componentcontaining 0 of D \ (˚ η L(cid:15) ([0 , ˚ S L(cid:15) ]) ∪ ˚ η R(cid:15) ([0 , ˚ S R(cid:15) ])).Fix a small constant a >
0, to be chosen later, and let ˚ E (cid:15) be the event that the harmonic measure from 0 in˚ D of each of the left side of ˚ η L(cid:15) ([0 , S
L(cid:15) ]) and the right side of ˚ η R(cid:15) ([0 , ˚ S R(cid:15) ]) is at least a . Using Lemma 3.3 andtwo applications of [MW17, Lemma 2.4] (which we emphasize does not depend on the particular location ofthe force points) and a similar argument to the one above, we infer that there is a universal choice of a > q ∈ (0 ,
1) such that P ( ˚ E (cid:15) | F ) ≥ ˚ q a.s. on the event (cid:110) f (cid:15) ( x L ) , f (cid:15) ( x L ) ∈ ∂U / ∗ (cid:111) .Let ˚ f (cid:15) : ˚ D → D be the conformal map which fixes 0 and such that ˚ f − (cid:15) (1) is equal to 1 if κ (cid:48) ≥ η L(cid:15) | [0 , ˚ S L(cid:15) ] and the left side of˚ η R(cid:15) | [0 , ˚ S R(cid:15) ] if κ (cid:48) ∈ (4 , h (cid:15) := (cid:98) h (cid:15) ◦ ˚ f − (cid:15) − χ arg( f − (cid:15) ) (cid:48) . The field ˚ h (cid:15) has the same boundary data alongthe image of the left side of ˚ η L(cid:15) ([0 , S
L(cid:15) ]) as it does along A R(cid:15) , and similarly with “ R ” in place of “ L ”.If we apply exactly the same argument as in the case when H (cid:15) occurs but with the field ˚ h (cid:15) in place of the field (cid:98) h (cid:15) , then pull back to ˚ D , we find that that after possibly shrinking p it a.s. holds with conditional probabilityat least p given F that the interior flow lines (cid:98) η L(cid:15) and (cid:98) η R(cid:15) defined above merge into ˚ η L(cid:15) ([0 , ˚ S L(cid:15) ]) and ˚ η R(cid:15) ([0 , ˚ S R(cid:15) ]),respectively, before leaving U ∗ and the region enclosed by these 4 flow lines run up to the merging timecontains B c/ (0). By definition of space-filling SLE, on this event this region is contained in f (cid:15) ( η (cid:48) ([ T , T (cid:15) ]).Hence a.s. P ( F (cid:15) | F ) ≥ p , as required. The goal of this subsection is to prove that it is unlikely that a space-filling SLE κ (cid:48) travels a long distancewithout filling in a big ball. More precisely, Proposition 3.4.
Let η (cid:48) be a space-filling SLE κ (cid:48) from ∞ to ∞ in C . For r ∈ (0 , , R > , and (cid:15) > ,let E (cid:15) = E (cid:15) ( R, r ) be the event that the following is true. For each δ ∈ (0 , (cid:15) ] and each a < b ∈ R such that η (cid:48) ([ a, b ]) ⊂ B R (0) and diam η (cid:48) ([ a, b ]) ≥ δ − r , the set η (cid:48) ([ a, b ]) contains a ball of radius at least δ . Then lim (cid:15) → P ( E (cid:15) ) = 1 . η (cid:48) when it is parameterized by Lebesguemeasure. Corollary 3.5.
Let η (cid:48) be a space-filling SLE κ (cid:48) from ∞ to ∞ in C , parameterized by Lebesgue measure.Almost surely, η (cid:48) is locally H¨older continuous with any exponent < / , and is not locally H¨older continuouswith any exponent > / .Proof. Since η (cid:48) is parameterized by Lebesgue measure, we always have diam η (cid:48) ([ a, b ]) ≥ π − / ( b − a ) / , so η (cid:48) cannot be H¨older continuous for any exponent > / η (cid:48) to ( η (cid:48) ) − ( B R (0)) for some fixed R >
0. Also fix r >
0. By Proposition 3.4, it is a.s. the case that for sufficiently small δ > a < b ∈ R such that η (cid:48) ([ a, b ]) ⊂ B R (0) and diam η (cid:48) ([ a, b ]) ≥ δ − r , the set η (cid:48) ([ a, b ]) contains a ball of radius atleast δ , whence b − a ≥ πδ . Hence for sufficiently small δ , it holds that whenever a, b ∈ ( η (cid:48) ) − ( B R (0)) with a < b and b − a ≤ π / δ , we have diam η (cid:48) ([ a, b ]) ≤ δ (1 − r ) / . This proves the desired H¨older continuity.The key input in the proof of Proposition 3.4 is the following estimate, which is an easy consequence of theresults of Section 3.1. Lemma 3.6.
Let η (cid:48) be a whole-plane space-filling SLE κ (cid:48) from ∞ to ∞ with any choice of parameterization.For z ∈ C , let τ z be the first time η (cid:48) hits z and for ρ ≥ let τ z ( ρ ) be the first time after τ z at which η exits B ρ ( z ) . For (cid:15) ∈ (0 , , let E (cid:15)z ( ρ ) be the event that η (cid:48) ([ τ z , τ z ( ρ )]) contains a Euclidean ball of radius at least (cid:15)ρ .There are constants a , a > depending only on κ (cid:48) such that for each ρ > and each (cid:15) ∈ (0 , , P ( E (cid:15)z ( ρ ) c ) ≤ a e − a /(cid:15) . Proof.
Fix
C > κ (cid:48) . For (cid:15) ∈ (0 , N (cid:15) := (cid:98) ( C(cid:15)ρ ) − (cid:99) and for k ∈ { , . . . , N (cid:15) } , let ρ (cid:15) ( k ) := kC(cid:15)ρ . For k ∈ { , . . . , N (cid:15) } , let F (cid:15)z ( k ) be the event that η (cid:48) ([ τ z ( ρ (cid:15) ( k − , τ z ( ρ (cid:15) ( k ))])contains a ball of radius at least (cid:15)ρ . Then (cid:83) N (cid:15) k =2 F (cid:15)z ( k ) ⊂ E (cid:15)z ( ρ ).By Lemma 3.1 and scale and translation invariance of the law of whole-plane SLE κ (cid:48) , if we choose C > p ∈ (0 ,
1) sufficiently small, depending only on κ (cid:48) , then for k ∈ { , . . . , N (cid:15) − } , P (cid:0) F (cid:15)z ( k + 1) | η (cid:48) | [0 ,τ (cid:15)z ( ρ (cid:15) ( k ))] (cid:1) ≥ p. (3.4)Multiplying this estimate over all k ∈ { , . . . , N (cid:15) − } gives P ( E (cid:15)z ( ρ ) c ) ≤ (1 − p ) N (cid:15) − ≤ a e − a /(cid:15) for an appropriate choice of a , a > (cid:15) > ρ >
0, let S (cid:15) ( ρ ) := B ρ (0) ∩ ( (cid:15) Z ) . (3.5)Lemma 3.6 tells us that with high probability, each segment of the space-filling SLE κ (cid:48) curve η (cid:48) which hasdiameter at least (cid:15) − r and which starts from the first time η (cid:48) hits a point of S (cid:15) ( ρ ) contains a ball of radius atleast (cid:15) . However, it is still possible that there exists a segment of η (cid:48) contained in B R (0) for R < ρ which hasdiameter > (cid:15) − r , fails to contain a ball of radius (cid:15) , and fails to contain any point of S (cid:15) ( ρ ). In the remainderof this subsection, we will rule out this possibility.To this end, we will view the space-filling SLE κ (cid:48) curve η (cid:48) as being coupled with a whole-plane GFF h , definedmodulo a global additive multiple of 2 πχ , as in [MS13]. For z ∈ C , let η ± z be the flow lines of h started from z with angles ∓ π/
2. By (the whole-plane analog of) the construction of space-filling SLE κ (cid:48) in [MS13], theflow lines η ± z form the left and right boundaries of η (cid:48) stopped at the first time it hits z . Lemma 3.7.
Suppose we are in the setting described just above. Fix
R > . For ρ > R , let (cid:101) F ( ρ ) = (cid:101) F ( ρ, R ) be the event that the following is true. For each (cid:15) ∈ (0 , , there exists z , w ∈ S (cid:15) ( ρ ) \ B R (0) such that theflow lines η − w and η − z and the flow lines η + z and η + w merge and form a pocket containing B R (0) before leaving B ρ (0) . Then for each fixed R > , we have P (cid:16) (cid:101) F ( ρ ) (cid:17) → as ρ → ∞ . roof. Let t R (resp. t (cid:48) R ) be the first time η (cid:48) hits (resp. finishes filling in) B R (0). For ρ > ρ > R , let τ ρ bethe first time η (cid:48) hits B ρ (0) and let σ ρ ρ be the first time after τ ρ at which η (cid:48) leaves B ρ (0). Since η (cid:48) is a.s.continuous and a.s. hits every point of C , it follows that there a.s. exists a random ρ > ρ > R such that thefollowing is true.1. τ ρ < t R < t (cid:48) R < σ ρ ρ .2. η (cid:48) ([ τ ρ , t R ]) and η (cid:48) ([ t (cid:48) R , σ ρ ρ ]) each contain a ball of radius 1.For this choice of ρ , ρ and any (cid:15) ∈ (0 , z ∈ S (cid:15) ( ρ ) ∩ η (cid:48) ([ τ ρ , t R ]) and w ∈ S (cid:15) ( ρ ) ∩ η (cid:48) ([ t (cid:48) R , σ ρ ρ ]).By the construction of space-filling SLE, the pocket formed by the flow lines η ± z and η ± w stopped at the firsttime they merge is precisely the set of points traced by η (cid:48) between the first time it hits z and the first timeit hits w . Since η (cid:48) ([ τ τ , σ ρ ρ ]) ⊂ B ρ (0) and η (cid:48) ([ t R , t (cid:48) R ]) ⊃ B R (0), it follows that this pocket contains B R (0)and is contained in B ρ (0). Since ρ is a.s. finite, it follows that P ( ρ < ρ ) → ρ → ∞ . The statement ofthe lemma follows. Lemma 3.8.
Suppose we are in the setting described just above Lemma 3.7. Fix
R > . For (cid:15) ∈ (0 , and ρ > R , let P (cid:15) ( ρ ) be the set of complementary connected components of (cid:91) z ∈S (cid:15) ( ρ ) ( η + z ∪ η − z ) which intersect B R (0) . For r ∈ (0 , , let (cid:101) E (cid:15) ( ρ ) = (cid:101) E (cid:15) ( ρ ; R, r ) be the event that sup P ∈P (cid:15) ( ρ ) diam P ≤ (cid:15) − r . Alsolet (cid:101) F ( ρ ) be defined as in Lemma 3.7. Then for each fixed ρ > R , we have P (cid:16) (cid:101) E (cid:15) ( ρ ) c ∩ (cid:101) F ( ρ ) (cid:17) ≤ ρ o ∞ (cid:15) ( (cid:15) ) at a rate depending only on R .Proof. The proof given here is more or less implicit in the proof of continuity of space-filling SLE in [MS13,Section 4.3], but we give a full proof for completeness. See Figure 5 for an illustration.For ρ > R and z ∈ B ρ (0), let (cid:101) E z(cid:15) ( ρ ) be the event that the following is true. There exists w ∈ S (cid:15) ( ρ ) such that w (cid:54) = z and the curve η − z hits (and subsequently merges with) η − w on its left side before leaving ∂B (cid:15) − r / ( z );and the same is true with η + z in place of η − z and/or “right” in place of “left”. Let (cid:101) E (cid:48) (cid:15) ( ρ ) := (cid:92) z ∈S (cid:15) ( ρ ) (cid:101) E z(cid:15) ( ρ ) . By [MS13, Proposition 4.14] we have P (cid:16) (cid:101) E z(cid:15) ( ρ ) c (cid:17) = o ∞ (cid:15) ( (cid:15) ), so by the union bound, P (cid:16) (cid:101) E (cid:48) (cid:15) ( ρ ) (cid:17) = 1 − ρ o ∞ (cid:15) ( (cid:15) ).Hence to complete the proof it suffices to show that (cid:101) E (cid:48) (cid:15) ( ρ ) ∩ (cid:101) F ( ρ ) ⊂ (cid:101) E (cid:15) ( ρ ).We first argue that on (cid:101) F ( ρ ), the boundary of each P ∈ P (cid:15) ( ρ ) is entirely traced by curves η ± z for z ∈ S (cid:15) ( ρ ).To see this, let z , w ∈ S (cid:15) ( ρ ) be as in the definition of (cid:101) F ( ρ ) and let P be the pocket formed by η ± z and η ± w surrounding B R (0). Then for z / ∈ B ρ (0), the flow lines η ± z cannot cross ∂P without merging into η ± z , hencecannot enter B R (0) without merging into flow lines η ± z for z ∈ S (cid:15) ( ρ ).Now suppose (cid:101) E (cid:48) (cid:15) ( ρ ) ∩ (cid:101) F ( ρ ) occurs and P ∈ P (cid:15) ( ρ ). We must show diam P ≤ (cid:15) − r . Since (cid:101) F ( ρ ) occurs, ∂P consists of either two arcs traced by a pair of flow lines η − z and η + z for some z ∈ S (cid:15) ( ρ ); or four arcs traced by η ± z and η ± w for some z, w ∈ S (cid:15) ( ρ ).Suppose first that we are in the latter case, i.e., that there exists z, w ∈ S (cid:15) ( ρ ) with the property that ∂P contains non-trivial arcs traced by the left side of η − z , the right side of η + z , the right side of η − w , and theright side of η + w . Let I − z be the arc of ∂P traced by the left side of η − z . The curve η − z cannot hit η − v for The statement of [MS13, Proposition 4.14] does not specify the side at which the merging occurs, but the proof shows thatwe can require the merging to occur on a particular side of the curve. v ∈ S (cid:15) ( ρ ) on its left side before η − z finishes tracing I − z (otherwise part of this arc I − z would lie on theboundary of a pocket other than P ). The same is true if we replace η − z with one of the other three arcs of ∂P in our description of ∂P . Since (cid:101) E z(cid:15) ( ρ ) ∩ (cid:101) E w(cid:15) ( ρ ) occurs, each of these four arcs has diameter at most (cid:15) − r .Therefore, diam P ≤ (cid:15) − r . By a similar argument, but with only two distinguished boundary arcs instead offour, we obtain diam P ≤ (cid:15) − r in the case when ∂P is traced by a pair of flow lines η − z and η + z for z ∈ S (cid:15) ( ρ ).Thus (cid:101) E (cid:48) (cid:15) ( ρ ) ∩ (cid:101) F ( ρ ) ⊂ (cid:101) E (cid:15) ( ρ ), as required. z w z z z z P P P P B ρ (0) B R (0) η − z η + z η − w η + w Figure 5: An illustration of the proof of Lemma 3.8. On the event (cid:101) F ( ρ ), there exists points z and w inthe lattice S (cid:15) ( ρ ) such that the flow lines η ± z and η ± w intersect at points shown in orange to form a pocket P which separates B R (0) from ∂B ρ (0). Also shown are four points z , z , z , z ∈ S (cid:15) ( ρ ) and three pockets P , P , P ∈ P (cid:15) ( ρ ) formed by the flow lines started at these points. Flow lines η − z i for i ∈ { , . . . , } are shownin red, and flow lines η + z i are shown in blue. The points where two of these flow lines merge are shown inorange. The pockets P and P are of the type with boundary arcs traced by four flow lines started from twopoints and the pocket P is of the type with boundary arcs traced by two flow lines started from a singlepoint. Proof of Proposition 3.4.
Fix r (cid:48) ∈ (0 , r ). For (cid:15) ∈ (0 , k (cid:15) be the smallest k ∈ N such that 2 − k ≥ (cid:15) . Let E (cid:15) be the event that the following is true. For each k ≥ k (cid:15) and each z ∈ S − k (2 R ) (defined as in (3.5)), theevent E − kr (cid:48) z (2 − k (1 − r (cid:48) ) ) of Lemma 3.6 occurs (i.e., with 2 − k (1 − r (cid:48) ) in place of ρ and 2 − kr in place of (cid:15) ). ByLemma 3.6 and the union bound, we have P (cid:0) E (cid:15) (cid:1) = 1 − o ∞ (cid:15) ( (cid:15) ) . Fix ρ > R to be chosen later and define the event (cid:101) F ( ρ ) as in Lemma 3.7 and the events (cid:101) E (cid:15) ( ρ ) as in Lemma 3.8,the latter with r (cid:48) in place of r . Also let P (cid:15) ( ρ ) be the set of pockets as defined in Lemma 3.8. Let E (cid:15) ( ρ ) := ∞ (cid:91) k = k (cid:15) (cid:101) E − k ( ρ ) .
24y Lemma 3.8, the union bound, and the argument given just above, we have P (cid:16) E (cid:15) ∩ E (cid:15) ( ρ ) ∩ (cid:101) F ( ρ ) (cid:17) = P (cid:16) (cid:101) F ( ρ ) (cid:17) − ρ o ∞ (cid:15) ( (cid:15) ) . Since P (cid:16) (cid:101) F ( ρ ) (cid:17) → ρ → ∞ (by Lemma 3.7), to complete the proof of the proposition it suffices to showthat E (cid:15) ∩ E (cid:15) ( ρ ) ⊂ E (cid:15) for each choice of ρ > R and each sufficiently small (cid:15) ∈ (0 , E (cid:15) ∩ E (cid:15) ( ρ ) occurs and we are given δ ∈ (0 , (cid:15) ] and a < b ∈ R such that η (cid:48) ([ a, b ]) ⊂ B R (0)and diam η (cid:48) ([ a, b ]) ≥ δ − r . Let t ∗ be the smallest t ∈ [ a, b ] such that diam η (cid:48) ([ a, t ∗ ]) ≥ δ − r . Let k δ be thelargest k ∈ N such that 2 − k ≥ δ . Since (cid:101) E − kδ ( ρ ) occurs with r (cid:48) in place of r , it follows that for sufficientlysmall (cid:15) , the set η (cid:48) ([ a, t ∗ ]) is not contained in any single pocket in P − kδ ( ρ ). Whenever η (cid:48) exits a pocket in P − kδ ( ρ ), it hits a point z ∈ S − kδ (2 R ) for the first time. Hence there exists t ∗∗ ∈ [ a, t ∗ ] and z ∈ S − kδ (2 R )such that η (cid:48) hits z for the first time at time t ∗∗ . The set η (cid:48) ([ t ∗∗ , b ]) has diameter at least δ − r . Since E − kδ ( z ) occurs, it follows that this set contains a ball of radius δ . Thus η (cid:48) ([ a, b ]) contains a ball of radius δ ,which concludes the proof. Remark 3.9.
If we assume additional facts about space-filling SLE which are proven in [HS16], our proofyields a more quantitative version of Proposition 3.4, namely that P ( E c(cid:15) ) = o ∞ (cid:15) ( (cid:15) ) . (3.6)Indeed, by [HS16, Proposition 6.2] and scale invariance there exists some ξ = ξ ( κ ) > η (cid:48) isparameterized by Lebesgue measure with η (cid:48) (0) = 0 then for R > P ( B R (0) (cid:54)⊂ η (cid:48) ([ − C, C ])) (cid:22) C − ξ ∀ C > , (3.7)with the implicit constant independent of C . We claim that this estimate implies that the conclusion ofLemma 3.7 (for fixed choice of R ) can be improved to P (cid:16) (cid:101) F ( ρ ) (cid:17) ≤ ρ − ξ + o ρ (1) . (3.8)It is immediate from Lemma 3.6 that except on an event of probability o ∞ C ( C ) as C → ∞ , the set η (cid:48) ([ − C, C ])is contained in B C / oC (1) (0). By (3.7), the set η (cid:48) ([ − C, C ]) contains B R (0) except on an event of probability O C ( C − ξ ). By Lemma 3.6 and translation invariance of η (cid:48) parameterized by Lebesgue measure [HS16,Lemma 2.3], each of η (cid:48) ([ C, C ]) and η (cid:48) ([ − C, − C ]) contains a ball of radius 1 with high probability. Taking C = ρ o ρ (1) and the points z , w ∈ S (cid:15) ( ρ ) in Lemma 3.7 to be contained in these balls of radius 1 yields (3.8).We now obtain (3.6) by taking ρ = (cid:15) − N for large N ∈ N and using (3.8) instead of Lemma 3.7 in the aboveproof of Proposition 3.4. The following lemma will reduce the problem of estimating the γ -quantum measure of a quantum cone (i.e.,the quantum surface appearing in Theorem 1.1) to the problem of estimating the quantum measure inducedby a whole-plane Gaussian free field. Lemma 3.10.
Let α < Q and let h be the circle-average embedding of an α -quantum cone (Definition 1.6).Also let (cid:101) h be a whole-plane GFF normalized so that its circle average over ∂ D is 0. There exists c > depending only on α and γ and a coupling of h and (cid:101) h such that h − (cid:101) h is a.s. a continuous function on C \ { } and for each R > and each M > , P (cid:32) sup z ∈ B R (0) \ B /R (0) | ( h − (cid:101) h )( z ) | > M (cid:33) (cid:22) e − cM (3.9) with the implicit constant depending only on R , α , and γ . h and (cid:101) h are coupled as in the statement of the lemma, the γ -quantum area measures µ h and µ (cid:101) h are a.s. mutually absolutely continuous on B R (0) \ B /R (0) and the estimate (3.9) implies that theRadon-Nikodym derivative satisfies P (cid:32) sup z ∈ B R (0) \ B /R (0) (cid:12)(cid:12)(cid:12)(cid:12) γ log dµ h dµ (cid:101) h ( z ) (cid:12)(cid:12)(cid:12)(cid:12) > M (cid:33) (cid:22) e − cM . (3.10) Proof of Lemma 3.10.
For r > h r (0) and (cid:101) h r (0) be the circle averages of h and (cid:101) h , respectively, over ∂B r (0). By Definition 1.5, the laws of h − h |·| (0) and (cid:101) h − (cid:101) h |·| (0) agree. Furthermore, these distributions areindependent from h |·| (0) and (cid:101) h |·| (0), respectively. The law of (cid:101) h e − s (0) for s ∈ R is that of a standard lineartwo-sided Brownian motion [DS11, Proposition 3.2]. The law of h e − s (0) − αs for s ≥ s < h e − s (0) − αs ≥ Qs . It follows that we can couple h and (cid:101) h in such a way that h − h |·| (0) = (cid:101) h − (cid:101) h |·| (0)and a.s. h r (0) − (cid:101) h r (0) = α log r − for r <
1. Henceforth assume h and (cid:101) h are coupled in this way. We need tobound sup r ∈ [1 ,R ] | h r (0) − (cid:101) h r (0) | .By standard estimates for Brownian motion, we have P (cid:16) sup r ∈ [1 ,R ] | (cid:101) h r (0) | > M (cid:17) (cid:22) e − cM for every choiceof c >
0. By [DMS14, Remark 4.4], we can express the law of h e − s (0) − αs for s ≤ B bea standard linear Brownian motion and let τ be the largest t > B t + ( Q − α ) t = 0 (which isa.s. finite). If we set (cid:98) B t := B t + τ + ( Q − α )( t + τ ), then { (cid:98) B − s } s ≤ d = { h e − s (0) − αs } s ≤ . By the reflectionprinciple, the Gaussian tail bound, and the union bound, for T > P ( τ > T ) ≤ P ( ∃ t > T such that | B t | > ( Q − α ) t ) (cid:22) e − ( Q − α ) T/ . Consequently, we have P (cid:32) sup t ∈ [0 , log R ] | B t + τ + ( Q − α )( t + τ ) | > M (cid:33) ≤ P (cid:32) sup t ∈ [0 , ( Q − α ) − M ] | B t | > M/ (cid:33) + P (cid:18) τ >
12 ( Q − α ) − M − log R (cid:19) (cid:22) e − cM for c > { h e − s (0) − αs } s ≤ and the triangle inequality, we obtain (3.9).In the remainder of this section we will prove several estimates for the quantum measure induced by a whole-plane GFF. The estimates of this section will be used in conjunction with Proposition 3.4 and Lemma 3.10 toestimate the size of the pre-image of the set X under η (cid:48) in the proof of Theorem 1.1. Lemma 3.11.
Let h be a whole plane GFF. Let h be the harmonic part of h | D . For each r ∈ (0 , and each ρ ∈ (0 , , we have P (cid:32) sup z ∈ B ρ (0) | h ( z ) − h (0) | ≥ r log (cid:15) − (cid:33) = o ∞ (cid:15) ( (cid:15) ) . (3.11) Proof.
Let ρ (cid:48) := ρ . By the mean value property of harmonic functions,sup z ∈ B ρ (0) | h ( z ) − h (0) | (cid:22) (cid:90) B ρ (cid:48) (0) | h ( w ) − h (0) | dw, with the implicit constants depending only on ρ . By Jensen’s inequality, for each p > z ∈ B ρ (0) e p | h ( z ) − h (0) | ≤ c (cid:90) B ρ (cid:48) (0) e pc | h ( w ) − h (0) | dw c and c depending only on ρ . By [MS16d, Proposition 6.4], for each w ∈ B ρ (cid:48) (0), h ( w ) − h (0) isa centered Gaussian with variance bounded above by a constant depending only on ρ . Hence for each p > E (cid:32) c (cid:90) B ρ (cid:48) (0) e pc | h ( w ) − h (0) | dw (cid:33) < ∞ . By applying the Chebyshev inequality and letting p → ∞ , we infer (3.11). Lemma 3.12.
Fix r > and R > . Let h be a whole-plane GFF, let µ h be its γ -quantum are measure, andlet ( h (cid:15) ) be its circle average process. For each z ∈ C , each (cid:15) ∈ (0 , , and each δ ∈ (0 , , P (cid:18) µ h ( B (cid:15) ( z )) < δ(cid:15) γ e γh (cid:15) ( z ) (cid:19) = o ∞ δ ( δ ) , (3.12) at a rate depending only on r and R .Proof. The estimate (3.12) is independent of the choice of additive constant for h , can assume without lossof generality that h is normalized so that the circle average h (0) is equal to 0. Fix z ∈ C . For (cid:15) ∈ (0 , φ (cid:15)z ( w ) := (cid:15)w + z be an affine map which takes D to B (cid:15) ( z ) and let (cid:101) h (cid:15)z := h ◦ φ (cid:15)z . By [DS11, Proposition 2.1],we have µ h ( B (cid:15) ( z )) = µ (cid:101) h (cid:15)z + Q log (cid:15) ( B (0)) = (cid:15) γ µ (cid:101) h (cid:15)z ( B (0)) . (3.13)By translation and scale invariance (cid:101) h (cid:15)z has the law of a whole plane GFF, modulo additive constant. Thecircle average of (cid:101) h (cid:15)z at ∂ D is equal to h (cid:15) ( z ). To see this, we observe that the former circle average is equal tothe conditional mean of (cid:101) h (cid:15)z evaluated at 0, given its values outside of D . By conformal invariance of harmonicfunctions and of the zero-boundary GFF, this equals the conditional mean of h at z given its values outside B (cid:15) ( z ), which equals h (cid:15) ( z ).Hence the field (cid:98) h (cid:15)z := (cid:101) h (cid:15)z − h (cid:15) ( z ) agrees in law with h (not just modulo additive constant), and by (3.13), wehave µ h ( B (cid:15) ( z )) = e γh (cid:15) ( z ) (cid:15) γ µ (cid:98) h (cid:15)z ( B (0)) . (3.14)It remains to argue that µ (cid:98) h (cid:15)z ( B (0)) is unlikely to be small; equivalently, µ h ( B (0)) is unlikely to be small.To see this, let h be the conditional mean of h given its values outside D and let ˙ h := h | D − h . Then ˙ h is azero-boundary GFF on D which is conditionally independent from h given h | C \ D , and we have µ h ( D ) ≥ µ ˙ h ( B / (0)) exp (cid:32) − γ sup w ∈ B / (0) | h ( w ) | (cid:33) . (3.15)By [DS11, Lemma 4.5], P (cid:16) µ ˙ h ( B / (0)) < δ / (cid:17) = o ∞ δ ( δ ) . (3.16)Since h (0) = h (0) = 0, Lemma 3.11 implies P (cid:32) exp (cid:32) − γ sup w ∈ B / (0) | h ( w ) | (cid:33) < δ / (cid:33) = o ∞ δ ( δ ) . (3.17)By (3.15), (3.16), and (3.17) we obtain P ( µ h ( D ) < δ ) = o ∞ δ ( δ ). By (3.14), it follows that (3.12) holds. In this subsection we will prove various results which says that the circle average of a whole-plane GFFaround a small circle or the quantum mass of a small ball is unlikely to differ too much from the value wewould expect given the circle average around a larger circle centered at a nearby point. We start with a basiccontinuity estimate for the circle average. 27 emma 3.13.
Fix ρ > R > . Suppose that either h is a zero-boundary GFF on B ρ (0) or h is a whole-planeGFF. Let ( h (cid:15) ) be the circle average process of h . There a.s. exists a modification of h (cid:15) (still denoted by h (cid:15) )such that the following is true. For r > and C > , let (cid:101) C ( C ) = (cid:101) C ( C, R, r ) be the event that | h (cid:15) ( z ) − h (cid:15) (cid:48) ( z (cid:48) ) | ≤ C | ( z, (cid:15) ) − ( z (cid:48) , (cid:15) (cid:48) ) | (1 − r ) / (cid:15) / (3.18) for each (cid:15), (cid:15) (cid:48) ∈ (0 , with ≤ (cid:15)/(cid:15) (cid:48) ≤ and each z, z (cid:48) ∈ B R (0) . Then P (cid:16) (cid:101) C ( C ) (cid:17) → as C → ∞ .Proof. The statement for the case of a zero-boundary GFF on B ρ (0) follows from [HMP10, Proposition 2.1] (c.f.the proof of [MS16f, Proposition 8.4]). In particular, for such a zero-boundary GFF we have P (cid:16)(cid:83) C> (cid:101) C ( C ) (cid:17) =1. If h is a whole-plane GFF, then the law of h | B R (0) is mutually absolutely continuous with respect tothe law of a zero-boundary GFF on B ρ (0) restricted to B R (0) (see, e.g. [MS16d, Proposition 3.2]). Hence P (cid:16)(cid:83) C> (cid:101) C ( C ) (cid:17) = 1 in this case, so P (cid:16) (cid:101) C ( C ) (cid:17) → C → ∞ .We henceforth assume that we have replaced ( h (cid:15) ) with a modification as in Lemma 3.13. Lemma 3.14.
Fix ρ > R > and r ∈ (0 , . Let h be either a zero-boundary GFF on B ρ (0) or a whole-planeGFF and let ( h (cid:15) ) be the circle average process for h . Define the events (cid:101) C ( C ) = (cid:101) C ( C, R, r ) as in Lemma 3.13.For each a > , each c > , and each z, w ∈ B R (0) with | z − w | ≤ c(cid:15) , we have P (cid:16) | h (cid:15) ( w ) − h (cid:15) − r ( z ) | ≥ a log (cid:15) − , (cid:101) C ( C ) (cid:17) (cid:22) (cid:15) a r with the implicit constant depending only on c , C , R , and r .Proof. If z, w ∈ B R (0) with | z − w | ≤ c(cid:15) and (cid:101) C ( C ) occurs, then | h (cid:15) − r ( z ) − h (cid:15) − r ( w ) | (cid:22) . For t >
0, let B t := h e − t (cid:15) − r ( w ) − h (cid:15) − r ( w ). By the calculations in [DS11, Section 3.1], B is a standard linearBrownian motion. Therefore, P (cid:0) | h (cid:15) ( w ) − h (cid:15) − r ( w ) | ≥ a log (cid:15) − − C (cid:1) = P (cid:0) | B r log (cid:15) − | ≥ a log (cid:15) − − C (cid:1) (cid:22) (cid:15) a r . We conclude by means of the triangle inequality.
Lemma 3.15.
Let ρ > R > . Let h be either a zero-boundary GFF on B ρ (0) or a whole-plane GFF and let ( h (cid:15) ) be the circle average process for h . For r ∈ (0 , / and (cid:15) ∈ (0 , , let C (cid:15) = C (cid:15) ( R, r ) be the event that thefollowing is true. For each δ ∈ (0 , (cid:15) ] and each z, w ∈ B R (0) with | z − w | ≤ δ , we have | h δ ( w ) − h δ − r ( z ) | ≤ √ r log δ − . For each r ∈ (0 , / , we have P ( C (cid:15) ) → as (cid:15) → .Proof. Fix
C > (cid:101) C ( C ) = (cid:101) C ( C, R, r ) as in Lemma 3.13. By Lemma 3.14, for each z, w ∈ B R (0) with | z − w | ≤ δ , P (cid:16) | h δ ( z ) − h δ − r ( w ) | ≥ √ r log δ − , (cid:101) C ( C ) (cid:17) (cid:22) δ (3.19)with the implicit constant depending only on C , R , and r . Choose a finite collection S δ of at most O δ ( δ − − r )points in B R (0) such that for each z ∈ B R (0), there exists z (cid:48) ∈ S δ with | z − z (cid:48) | ≤ δ − r . On (cid:101) C ( C ), for such a z and z (cid:48) we have | h δ ( z ) − h δ ( z (cid:48) ) | ≤ C, | h δ − r ( z ) − h δ − r ( z (cid:48) ) | ≤ C.
28y (3.19) and the union bound, on (cid:101) C ( C ) it holds except on an event of probability (cid:22) δ (implicit constantdepending only on C , R , and r ) that | h δ ( z ) − h δ − r ( w ) | ≤ √ r log δ − whenever z, w ∈ S δ with | z − w | ≤ δ . By the triangle inequality, whenever this is the case and δ is sufficientlysmall (depending on r and C ), we have | h δ ( w ) − h δ − r ( z ) | ≤ √ r log δ − (3.20)whenever z, w ∈ B R (0) with | z − w | ≤ δ . For δ >
0, let C (cid:48) δ be the event that this last statement holds, sothat P (cid:16) ( C (cid:48) δ ) c , (cid:101) C ( C ) (cid:17) (cid:22) δ. (3.21)Fix a sequence ( ζ n ) decreasing to 0 such thatlim n →∞ ( ζ n − ζ n +1 ) (1 − r ) / ζ / n +1 = lim n →∞ ( ζ − rn − ζ − rn +1 ) (1 − r ) / ζ (1 − r ) / n +1 = 0 and ∞ (cid:88) n =1 ζ n < ∞ , (3.22)e.g. ζ n = n − q for appropriate q >
1, depending on r . For (cid:15) ∈ (0 , n (cid:15) be the greatest integer n such that ζ n ≥ (cid:15) − r . By (3.21) and the union bound,lim inf ζ → P (cid:32) (cid:101) C ( C ) ∩ ∞ (cid:92) n = n (cid:15) C (cid:48) ζ n (cid:33) = P (cid:16) (cid:101) C ( C ) (cid:17) − o (cid:15) (1) . Since P (cid:16) (cid:101) C ( C ) (cid:17) → C → ∞ (by Lemma 3.13), it suffices to show that for sufficiently small (cid:15) > (cid:101) C ( C ) ∩ ∞ (cid:92) n = n (cid:15) C (cid:48) ζ n ⊂ C (cid:15) . (3.23)To see this, suppose given δ ∈ (0 , (cid:15) ] and each z, w ∈ B R (0) with | z − w | ≤ δ . Let n δ be chosen so that δ ∈ [ ζ n δ +1 , ζ n δ ]. By our choice of ( ζ n ), on (cid:101) C ( C ) we have | h δ ( w ) − h ζ nδ ( w ) | ≤ Co (cid:15) (1) , | h δ − r ( z ) − h ζ − rnδ ( z ) | ≤ Co (cid:15) (1) . By (3.20) with ζ n δ in place of δ , along with the triangle inequality, we obtain (3.23). Proposition 3.16.
Let ρ > R > . Suppose that either h is zero-boundary GFF on B ρ (0) or a whole-planeGFF. Let ( h (cid:15) ) be the circle average process for h and let µ h be its γ -quantum area measure. There is afunction ψ : (0 , ∞ ) → (0 , ∞ ) with lim r → ψ ( r ) = 0 , depending only on γ , such that the following holds. For r ∈ (0 , / , and (cid:15) ∈ (0 , , let G (cid:15) = G (cid:15) ( R, r ) be the event that the following is true. For each δ ∈ (0 , (cid:15) ] andeach z, w ∈ B R (0) with | z − w | ≤ δ , we have µ h ( B δ ( w )) ≥ δ γ + ψ ( r ) e γh δ − r ( z ) . Then P ( G (cid:15) ) → as (cid:15) → .Proof. For
C >
1, define the event (cid:101) C ( C ) = (cid:101) C ( C, R, r ) as in Lemma 3.13. Define the event C (cid:15) = C (cid:15) ( R, r ) as inLemma 3.15. Also fix a sequence ζ n → p >
1. For n ∈ N , choose a finite collection S n of at most O n ( ζ − pn ) points in B R (0) such that each pointof B R (0) lies within distance ζ pn of some point in S n . For (cid:15) ∈ (0 , n (cid:15) be the greatest integer n such that ζ n ≥ (cid:15) − r and let D (cid:15) := ∞ (cid:92) n = n (cid:15) (cid:92) z ∈S n (cid:26) µ h ( B ζ n ( z )) ≥ ζ γ + rn e γh ζn ( z ) (cid:27) .
29y Lemma 3.12 and the union bound, we have P ( D (cid:15) ) → (cid:15) → P (cid:16) (cid:101) C ( C ) ∩ C (cid:15) ∩ D (cid:15) (cid:17) → (cid:15) → C → ∞ , it suffices to show that if p is sufficiently large and (cid:15) is sufficiently small (dependingon p ), then (cid:101) C ( C ) ∩ C (cid:15) ∩ D (cid:15) ⊂ G (cid:15) for an appropriate choice of ψ ( r ) depending only on r and γ . To this end, suppose that (cid:101) C ( C ) ∩ C (cid:15) ∩ D (cid:15) occursand we are given δ ∈ (0 , (cid:15) ] and z, w ∈ A R with | z − w | ≤ δ . By definition of (cid:101) C ( C ) and by (3.22), if p is chosensufficiently large, depending only on r and the choice of sequence ( ζ n ), we can find n δ ∈ N ∩ [ n (cid:15) , ∞ ) and w (cid:48) ∈ S n δ such that δ/ < ζ n δ < δ, | w (cid:48) − w | ≤ ζ pn δ < δ − ζ n δ , and | h δ − r ( z ) − h ζ − rnδ ( z ) | ≤ Co (cid:15) (1) . By the definitions of C (cid:15) and D (cid:15) , we therefore have µ h ( B δ ( w )) ≥ µ h ( B ζ nδ ( w (cid:48) )) (cid:23) δ γ + r e γh ζnδ ( w (cid:48) ) ≥ δ γ + r +5 γ √ r e γh ζ − rnδ ( z ) (cid:23) δ γ + r +5 γ √ r e γh δ − r ( z ) , with the implicit constants depending on C but tending to C -independent constants as (cid:15) →
0. This provesthe statement of the lemma in the case of a whole-plane GFF with ψ ( r ) slightly larger than r + 5 γ √ r . Let h be a GFF on a domain D ⊂ C and let ( h (cid:15) ) be its circle average process. Recall that for α ≥
0, a point z ∈ D is called an α -thick points of h providedlim (cid:15) → h (cid:15) ( z )log (cid:15) − = α. Let T αh be the set of α -thick points of h . Also let (cid:98) T αh := (cid:26) z ∈ C : lim inf (cid:15) → h (cid:15) ( z )log (cid:15) − ≥ α (cid:27) . Thick points are introduced and studied in [HMP10] . In particular, it is proven in [HMP10, Theorem 1.2]that a.s. dim H ( T αh ) = 2 − α /
2. In this section we will adapt the proof of [HMP10, Theorem 1.2] to obtain ageneralization of this fact which gives the a.s. dimension of the intersection of T αh with a general Borel set.The lower bound from this result will be needed in the proof of the upper bound in Theorem 1.1. Theorem 4.1.
Let D ⊂ C be a simply connected domain and let h be a zero-boundary GFF on D . Also let A ⊂ D be a deterministic Borel set. If ≤ α / ≤ dim H A , then almost surely dim H ( T αh ∩ A ) = dim H (cid:16) (cid:98) T αh ∩ A (cid:17) = dim H A − α . If α / > dim H A , then a.s. T αh ∩ A = (cid:98) T αh ∩ A = ∅ . By [BP15, Theorem B.2.5], for each d < dim H ( A ), there exists a closed set A (cid:48) ⊂ A with dim H ( A (cid:48) ) ≥ d .Hence we can assume without loss of generality that A is closed. We make this assumption throughout theremainder of this section.Before we commence with the proof of Theorem 4.1, we observe that dim H ( T αh ∩ A ) and dim H ( T αh ∩ A ) areeach a.s. equal to a constant. The authors of [HMP10] use a different normalization of the GFF from the one used in this paper, so our α -thick points arethe same as their α / emma 4.2. Suppose we are in the setting of Theorem 4.1. There are deterministic constants a, (cid:98) a ≥ suchthat dim H ( T αh ∩ A ) = a and dim H ( T αh ∩ A ) = (cid:98) a a.s.Proof. Let { f j } be an orthonormal basis for the Hilbert space closure for the Dirichlet inner product on theset of compactly supported smooth functions in D , with each f j smooth and compactly supported. We canwrite h = (cid:80) ∞ j =1 a j f j , where the a j ’s are i.i.d. standard Gaussians. Permuting a finite number of coefficientsin this series expansion does not affect whether a given point is an α -thick point for h , nor does it affectthe dimension of T αh ∩ A or (cid:98) T αh ∩ A . By the Hewitt-Savage zero-one law, we obtain the statement of thelemma. In this subsection we will prove the upper bound in Theorem 4.1. It is clear that T αh ⊂ (cid:98) T αh , so we only needto prove an upper bound on the dimension of (cid:98) T αh ∩ A . To do this we will need a couple of basic lemmas. Lemma 4.3.
Suppose we are in the setting of Theorem 4.1. Fix α ∈ (0 , and r > . Almost surely, thereexists a random (cid:15) = (cid:15) ( α, r ) > such that the following is true. If we set A α,r := (cid:26) z ∈ A : h (cid:15) ( z )log (cid:15) − ≥ α − r ∀ (cid:15) ∈ (0 , (cid:15) ] (cid:27) then dim H A α,r ≥ dim H ( (cid:98) T αh ∩ A ) − r .Proof. We have (cid:98) T αh ∩ A ⊂ ∞ (cid:91) n =1 (cid:26) z ∈ A : h (cid:15) ( z )log (cid:15) − ≥ α − r ∀ (cid:15) ∈ (0 , /n ] (cid:27) so the statement of the lemma follows from countable stability of Hausdorff dimension. Lemma 4.4.
Fix r ∈ (0 , / , ρ > R > , and α > . Let h be a zero-boundary GFF on B ρ (0) . For (cid:15) ∈ (0 , and z ∈ B R (0) , let E α,r(cid:15) ( z ) be the event that there is a w ∈ B (cid:15) ( z ) ∩ B ρ − (cid:15) (0) such that h (cid:15) ( w ) ≥ ( α − r ) log (cid:15) − .Also let C (cid:15) = C (cid:15) ( R, r ) be the event of Lemma 3.15. Then for each (cid:101) (cid:15) ≥ (cid:15) , we have P ( E α,r(cid:15) ( z ) ∩ C (cid:101) (cid:15) ) (cid:22) (cid:15) α + o r (1) , with the implicit constant and the o r (1) independent of (cid:15) and uniform for z ∈ B R (0) .Proof. By definition of C (cid:101) (cid:15) , if E α,r(cid:15) ( z ) ∩ C (cid:101) (cid:15) occurs, then h (cid:15) − r ( z ) ≥ (cid:16) α − r − √ r (cid:17) log (cid:15) − . The statement of the lemma follows from the Gaussian tail bound.
Proof of Theorem 4.1, upper bound.
By conformal invariance (see [HMP10, Section 4]) we can assume withoutloss of generality that D = B ρ (0) for some ρ >
0. By countable stability of Hausdorff dimension we canassume without loss of generality that A ⊂ B R (0) for some R ∈ (0 , ρ ). Fix β > dim H A , r ∈ (0 , α ∧ / p ∈ (0 , (cid:15) = (cid:15) ( α, r ) > A α,r be as in Lemma 4.3. By the defining property of A α,r , we can find adeterministic (cid:15) ∈ (0 ,
1) such that with probability at least 1 − p , we have (cid:15) ≥ (cid:15) , in which case h (cid:15) ( z ) ≥ ( α − r ) log (cid:15) − , ∀ (cid:15) ∈ (0 , (cid:15) ] , ∀ z ∈ A α,r . (4.1)Let F α,r be the event that (4.1) holds, so that P ( F α,r ) ≥ − p .For each k ∈ N , we can find a countable collection of balls { B (cid:15) jk ( z jk ) } j ∈ N of radius (cid:15) jk ≤ (cid:15) ∧ − k such that A ⊂ ∞ (cid:91) j =1 B (cid:15) jk ( z jk ) and ∞ (cid:88) j =1 ( (cid:15) jk ) β ≤ − k . B (cid:15) jk ( z jk ) ∩ A α,r (cid:54) = ∅ and F α,r occurs, then the event E α,r(cid:15) jk ( z jk ) of as in Lemma 4.4 occurs.By Lemma 4.4, for each ξ > E C − k ∩ F α,r ∞ (cid:88) j =1 ( (cid:15) jk ) ξ (cid:18) B (cid:15)jk ( z jk ) ∩ A α,r (cid:54) = ∅ (cid:19) ≤ ∞ (cid:88) j =1 ( (cid:15) jk ) ξ + α / o r (1) (4.2)with C − k = C − k ( R, r ) as in Lemma 3.15. If ξ > β − α /
2, then for sufficiently small r , this sum is ≤ − k .Since P ( C − k ) → k → ∞ , if F α,r occurs then it is a.s. the case that for infinitely many k , we have ∞ (cid:88) j =1 ( (cid:15) jk ) ξ (cid:18) B (cid:15)jk ( z jk ) ∩ A α,r (cid:54) = ∅ (cid:19) ≤ − k/ . Therefore dim H A α,r ≤ β − α / F α,r . Since p can be made arbitrarily close to 0, a.s.dim H ( (cid:98) T αh ∩ A ) ≤ dim H A α,r + r ≤ β − α / r. Upon letting β → dim H A and r → H ( (cid:98) T αh ∩ A ) ≤ dim H ( A ) − α /
2. If dim H ( A ) − α / < β sufficiently close to dim H ( A ) and r sufficiently small, the sum (4.2) is ≤ − k/ when ξ = 0. Henceit is a.s. the case that on F α,r , it holds for arbitrarily large k that none of the balls B (cid:15) jk ( z jk ) intersect A α,r .Therefore A α,r = ∅ a.s., so by Lemma 4.3 we have (cid:98) T αh ∩ A = ∅ a.s. In this subsection we will prove the lower bound in Theorem 4.1. It suffices to prove the lower bound fordim H ( T αh ∩ A ). By considering the intersection of A with a dyadic square and re-scaling, we can assumewithout loss of generality that A ⊂ [0 , ⊂ D . We make this assumption in addition to the assumption that A is closed throughout the remainder of this section.Fix α >
0. We make the following definitions (as in [HMP10, Section 3.2]). For n ∈ N , let (cid:15) n := 1 /n ! and t n := log n !. Define the following events for each (cid:15) > z ∈ D , and n, j ∈ N . E z,j := (cid:40) sup (cid:15) ∈ [ (cid:15) j +1 ,(cid:15) j ] | h (cid:15) ( z ) − h (cid:15) j ( z ) − α (log (cid:15) − − log (cid:15) − j ) | ≤ (cid:113) log (cid:15) − j +1 − log (cid:15) − j (cid:41) (4.3) F z,j := (cid:8) | h (cid:15) ( z ) − h (cid:15) j ( z ) | ≤ log (cid:15) − − log (cid:15) − j + 1 ∀ (cid:15) ≤ (cid:15) j (cid:9) (4.4) E n ( z ) := F z,n +1 ∩ n (cid:92) j =1 E z,j (4.5)For n ∈ N , divide [0 , into (cid:15) − n squares of side length (cid:15) n , which intersect only along their boundaries. Let (cid:101) D n be the set of points in [0 , which are centers of these squares and for z ∈ (cid:101) D n , let S n ( z ) be the square ofside length (cid:15) n centered at z . Let D n be the set of z ∈ (cid:101) D n such that S n ( z ) ∩ A (cid:54) = ∅ and let D ∗ n be the set ofthose z ∈ D n for which E n ( z ) occurs. We define the α -perfect-points by P α = P α ( h, A ) := (cid:92) k ≥ (cid:91) n ≥ k (cid:91) z ∈D ∗ n S n ( z ) . (4.6)It is shown in [HMP10, Lemma 3.2] that P α ⊂ T αh a.s. Since we have assumed that A is closed, we a.s. have P α ⊂ T αh ∩ A. (4.7)We next need estimates for the probabilities of the events E n ( z ).32 emma 4.5. For z ∈ (cid:101) D n and n ∈ N , we have P ( E n ( z )) ≥ (cid:15) α / o n (1) n with the o n (1) uniform for z ∈ (cid:101) D n .Proof. Since t (cid:55)→ h e − t ( z ) evolves as a standard linear Brownian motion, [HMP10, Lemma A.3] applied with T = log (cid:15) − j +1 − log (cid:15) − j implies that for each z ∈ (cid:101) D n and j ∈ N , we have P ( E z,j ) ≥ ( (cid:15) j +1 /(cid:15) j ) α / o j (1) . Furthermore, we have P ( F j ( z )) (cid:23)
1. By the Markov property, P ( E n ( z )) = P ( F n +1 ( z )) n (cid:89) j =1 P ( E z,j ) ≥ (cid:15) α / o n (1) n +1 = (cid:15) α / o n (1) n . The following is a restatement of [HMP10, Lemma 3.3].
Lemma 4.6.
There is a constant
C > depending only on D and α such that the following is true. Foreach l ∈ N , each z, w ∈ [0 , with w ∈ S l ( z ) \ S l +1 ( z ) , and each n ≥ l , P ( E n ( z ) ∩ E n ( w )) ≤ C l β − α / l (cid:15) − α / l P ( E n ( z )) P ( E n ( w )) where β l = l (cid:89) k =1 e √ log k . Remark 4.7.
In the setting of Lemma 4.6, we have (cid:15) l = | z − w | o | z − w | (1) and β l = | z − w | o | z − w | (1) , so theestimate of Lemma 4.6 can be re-stated as P ( E n ( z ) ∩ E n ( w )) P ( E n ( z )) P ( E n ( w )) ≤ | z − w | − α / o | z − w | (1) with the o | z − w | (1) depending only on z and w . Proof of Theorem 4.1, lower bound.
Fix d ∈ (0 , dim H ( A )). By Frostman’s lemma [MP10, Theorem 4.30]there exists a Borel probability measure µ on A such that (cid:90) A (cid:90) A | x − y | d dµ ( x ) dµ ( y ) < ∞ . (4.8)We extend µ to C by setting µ ( B ) = µ ( B ∩ A ) for each Borel set B ⊂ C . From (4.8), we obtain that for any a, b ≥ a + b ≤ d , (cid:90) A (cid:90) A | x − y | d dµ ( x ) dµ ( y )= (cid:88) z (cid:54) = w ∈D n (cid:90) S n ( z ) (cid:90) S n ( w ) | x − y | d dµ ( x ) dµ ( y ) + (cid:88) z ∈D n (cid:90) S n ( z ) (cid:90) S n ( z ) | x − y | d dµ ( x ) dµ ( y ) (cid:23) (cid:88) z (cid:54) = w ∈D n | z − w | − a (cid:90) S n ( z ) (cid:90) S n ( w ) | x − y | b dµ ( x ) dµ ( y ) + (cid:88) z ∈D n (cid:15) − an (cid:90) S n ( z ) (cid:90) S n ( z ) | x − y | b dµ ( x ) dµ ( y ) . Hence for any such a and b , (cid:88) z (cid:54) = w ∈D n | z − w | − a (cid:90) S n ( z ) (cid:90) S n ( w ) | x − y | b dµ ( x ) dµ ( y ) + (cid:88) z ∈D n (cid:15) − an (cid:90) S n ( z ) (cid:90) S n ( z ) | x − y | b dµ ( x ) dµ ( y ) (cid:22) . (4.9)33or n ∈ N define a measure ν n on A by dν n ( x ) = (cid:88) z ∈D n E n ( z ) S n ( z ) ( x ) P ( E n ( z )) dµ ( x ) . Observe that E ( ν n ( A )) = (cid:88) z ∈D n µ ( S n ( z )) = µ ( A ) = 1 . By Lemmas 4.5 and 4.6, E (cid:0) ν n ( A ) (cid:1) = (cid:88) z (cid:54) = w ∈D n P ( E n ( z ) ∩ E n ( w )) P ( E n ( z )) P ( E n ( w )) µ ( S n ( z )) µ ( S n ( w )) + (cid:88) z ∈D n µ ( S n ( z )) P ( E n ( z )) (cid:22) (cid:88) z (cid:54) = w ∈D n µ ( S n ( z )) µ ( S n ( w )) | z − w | α / o | z − w | (1) + (cid:88) z ∈D n µ ( S n ( z )) (cid:15) α / o n (1) n . By (4.9) applied with b = 0, this is bounded above by an n -independent constant provided d − α / > b > E (cid:18)(cid:90) A (cid:90) A | x − y | b dν n ( x ) dν n ( y ) (cid:19) = (cid:88) z (cid:54) = w ∈D n P ( E n ( z ) ∩ E n ( w )) P ( E n ( z )) P ( E n ( w )) (cid:90) S n ( z ) (cid:90) S n ( w ) | x − y | b dµ ( x ) dµ ( y )+ (cid:88) z ∈D n P ( E n ( z )) (cid:90) S n ( z ) (cid:90) S n ( z ) | x − y | b dµ ( x ) dµ ( y ) (cid:22) (cid:88) z (cid:54) = w ∈D n | z − w | − α / o | z − w | (1) (cid:90) S n ( z ) (cid:90) S n ( w ) | x − y | b dµ ( x ) dµ ( y )+ (cid:88) z ∈D n (cid:15) − α / o n (1) n (cid:90) S n ( z ) (cid:90) S n ( z ) | x − y | b dν n ( x ) dν n ( y ) . By (4.9), this is bounded above by an n -independent constant provided b < d − α /
2. It follows from theusual argument (see the proof of [HMP10, Lemma 3.4]) that for such a b , it holds with positive probabilitythat we can find a weak subsequential limit ν of the measures ν n such that ν is supported on P α , ν ( P α ) > (cid:90) P α (cid:90) P α | x − y | b dν ( x ) dν ( y ) < ∞ . By (4.7) and [MP10, Theorem 4.27], it holds with positive probability that dim H ( A ∩ T αh ) ≥ b . By Lemma 4.2,this probability is in fact equal to 1 for each b < d − α /
2. Therefore dim H ( A ∩ T αh ) ≥ d − α / In this subsection we will prove the upper bound in Theorem 1.1.
Proof of Theorem 1.1, upper bound.
We start with some reductions. By the countably stability of Hausdorffdimension, we can assume without loss of generality that a.s. X ⊂ B R (0) \ B /R (0) for some deterministic R >
0. By Lemma 3.10, we can couple h with a whole-plane GFF (cid:101) h normalized so that its circle average over ∂ D is 0 in such a way that the γ -quantum area measures µ h and µ (cid:101) h are a.s. mutually absolutely continuous on B R (0) \ B /R (0), with Radon-Nikodym derivative bounded above and below by (random) positive constants.34n such a coupling, it holds that for each interval I ⊂ R with η (cid:48) ( I ) ⊂ B R (0) \ B /R (0) that | I | (cid:16) µ (cid:101) h ( η (cid:48) ( I )) withrandom but I -independent implicit constants. In particular, dim H ( η (cid:48) ) − ( X ) is unchanged if we parameterize η (cid:48) by µ (cid:101) h instead of µ h . Hence we can assume that h is a whole-plane GFF normalized so that its circleaverage over ∂ D is 0, instead of the circle average embedding of a γ -quantum cone. We make this assumptionthroughout the remainder of the proof.Let α ∈ (0 ,
2] and r ∈ (0 , / h to B R (0) isabsolutely continuous with respect to the law of the corresponding restriction of a zero-boundary GFF h on B R (0), minus a random constant C equal to the circle average of h over ∂ D . Since the set X is independentfrom h + C , it follows from Theorem 4.1 and Lemma 4.3 that we can find a random set X α,r ⊂ X and arandom (cid:15) > H X α,r ≥ dim H X − α − r (5.1)and a.s. h (cid:15) ( z ) + C ≥ ( α − r ) log (cid:15) − , ∀ (cid:15) ∈ (0 , (cid:15) ] ∀ z ∈ X α,r . By decreasing (cid:15) we can arrange that in fact h (cid:15) ( z ) ≥ ( α − r ) log (cid:15) − , ∀ (cid:15) ∈ (0 , (cid:15) ] ∀ z ∈ X α,r . (5.2)Now let (cid:98) X ⊂ R be as in the theorem statement for our given choice of X . We will prove an upper bound fordim H ( X α,r ) in terms of dim H ( (cid:98) X ). Let (cid:98) X α,r := ( η (cid:48) ) − ( X α,r ) ∩ (cid:98) X. For (cid:15) >
0, let E (cid:15) = E (cid:15) ( R, r ) and G (cid:15) = G (cid:15) ( R, r ) be defined as in Propositions 3.4 and 3.16, respectively. Also let S (cid:15) := (cid:40) sup z ∈ B R (0) h ( z ) ≤ (cid:112) log (cid:15) − (cid:41) . Note that P ( E (cid:15) ∩ G (cid:15) ∩ S (cid:15) ) → (cid:15) → ζ >
0, we can find a countable collection I ζ of intervals oflength at most ζ , each of which contains a point of (cid:98) X α,r , such that (cid:98) X α,r ⊂ (cid:91) I ∈I ζ I and (cid:88) I ∈I ζ diam( I ) dim H ( (cid:98) X )+ r ≤ ζ. (5.3)We claim that on E (cid:15) ∩ G (cid:15) ∩ S (cid:15) , there a.s. exists a random ζ > ζ ∈ (0 , ζ ] and each choiceof (cid:98) X ⊂ R and cover I ζ as above thatdiam η (cid:48) ( I ) ≤ diam( I ) (2+ γ / − γα ) − + o r (1) ∀ I ∈ I ζ , (5.4)with the o r (1) deterministic and independent of I . Indeed, let (cid:15) be as in (5.2) and set ζ = (cid:15) ∧ (cid:15) . Suppose E (cid:15) ∩ G (cid:15) ∩ S (cid:15) occurs, ζ ∈ (0 , ζ ], (cid:98) X and I ζ are as above, and I ∈ I ζ . Let δ I := diam η (cid:48) ( I ) ∧ (cid:15) . By definition of E (cid:15) , we can find z ∈ X α,r and w ∈ η (cid:48) ( I ) ∩ B δ I ( z ) such that B δ − rI ( w ) ⊂ η (cid:48) ( I ). By definition of G (cid:15) , we have µ h ( η (cid:48) ( I )) ≥ δ γ + o r (1) I e γh δI ( z ) . Since z ∈ X α,r , (5.2) and (5.4) imply that if ζ ≤ ζ , then e γh δI ( z ) ≥ δ − γα + γrI . Hence for such a ζ , we have µ h ( η (cid:48) ( I )) ≥ δ γ − γα + o r (1) I . Since η (cid:48) is parameterized by quantum mass, we infer thatdiam( I ) ≥ δ γ − γα + o r (1) I . I ) ≤ ζ / , this implies (5.4).By (5.3), { η (cid:48) ( I ) : I ∈ I ζ } is a cover of X α,r . Moreover, if we are given r (cid:48) > r > r (cid:48) , then for ζ ∈ (0 , ζ ] we have by (5.3) and (5.4) that (cid:88) I ∈I ζ (diam η (cid:48) ( I ))( dim H ( (cid:98) X )+ r (cid:48) ) (cid:16) γ − γα (cid:17) ≤ ζ. Hence for such a choice of r , whenever E (cid:15) ∩ G (cid:15) ∩ S (cid:15) occurs it holds for every choice of (cid:98) X as in the theoremstatement that dim H ( X α,r ) ≤ (cid:16) dim H ( (cid:98) X ) + r (cid:48) (cid:17) (cid:18) γ − γα (cid:19) . By letting (cid:15) → (cid:98) X simultaneously. By (5.1), it is a.s.the case that for each choice of (cid:98) X as in the theorem statement, we havedim H ( X ) ≤ (cid:16) dim H ( (cid:98) X ) + r (cid:48) (cid:17) (cid:18) γ − γα (cid:19) + α r. The right side is minimized when r = r (cid:48) = 0 by taking α = γ dim H ( (cid:98) X ). Since r is arbitrary and r (cid:48) can bemade as small as we like by shrinking r , this yields the upper bound in (1.4). We will prove the lower bound in Theorem 1.1 by covering X with balls B of radius (cid:15) B , such that (cid:80) B (cid:15) dB issmall for some d > dim H ( X ). We obtain a cover of the time set (cid:98) X by considering the pre-images of these ballsunder η (cid:48) . The length of the intervals covering (cid:98) X is estimated by considering the quantum mass of the ballsin our cover via the circle average process, and by bounding the number of time intervals corresponding toeach ball in the cover of X . The proof also relies on Proposition 3.4, Lemma 3.11, and [RV14, Theorem 2.11]. Lemma 5.1.
For each
R > , r ∈ (0 , , z ∈ B R (0) , ˜ (cid:15) > , and (cid:15) > , the ball B (cid:15) − r ( z ) can be written asa union of sets of the form η (cid:48) ( I ) which intersect only along their boundaries, where I is an interval thatis not contained in any larger interval I (cid:48) satisfying η (cid:48) ( I (cid:48) ) ⊂ B (cid:15) − r ( z ) . On the event E ˜ (cid:15) − r = E ˜ (cid:15) − r ( r, R ) ofProposition 3.4, the number of such sets that intersect B (cid:15) ( z ) , is bounded above by (cid:15) − r for all sufficientlysmall (cid:15) , i.e., for (cid:15) < (cid:15) ( r, ˜ (cid:15) ) .Proof. On E ˜ (cid:15) − r , every time interval I of the form above with η (cid:48) ( I ) ∩ B (cid:15) ( z ) (cid:54) = ∅ and (cid:15) > η (cid:48) ( I )) ≥ ( (cid:15) − r − (cid:15) ) − r = (cid:15) o r (1) . Since η (cid:48) ( I ) ⊂ B (cid:15) − r ( z ) and B (cid:15) − r ( z ) has area (cid:15) − r = (cid:15) o r (1) ,the lemma follows. The exact exponent − r is obtained by dividing the area of B (cid:15) − r ( z ), by the bound forthe area of η (cid:48) ( I ). Lemma 5.2.
Let h be a whole-plane GFF normalized such that h (0) = 0 , and let R > and z ∈ B R (0) .For ≤ β < γ and (cid:15) ∈ (0 , , we have E [ µ h ( B (cid:15) ( z )) β ] ≤ (cid:15) f ( β )+ o (cid:15) (1) , where f ( β ) = (2 + γ ) β − γ β and the o (cid:15) (1) depends on α , β , and R , but not on z .Proof. Defining φ (cid:15)z ( w ) = (cid:15)w + z we have h ◦ φ (cid:15)z = ˙ h + h , for ˙ h a zero-boundary GFF in B (0) and h harmonicin B (0) and independent of ˙ h . As explained after (3.13) the average of h around ∂B (0) is equal to the circleaverage h (cid:15) ( z ). By the coordinate change formula for quantum surfaces we have E (cid:0) µ h ( B (cid:15) ( z )) β (cid:1) = (cid:15) (2+ γ / β E (cid:0) µ ˙ h + h ( B (0)) β (cid:1) ≤ (cid:15) (2+ γ / β E (cid:0) e γβh (cid:15) ( z ) × µ ˙ h ( B (0)) β × sup w ∈ B (0) e γβ ( h ( w ) − h (cid:15) ( z )) (cid:1) . r >
0, and define the event A r,(cid:15) by A r,(cid:15) = { sup w ∈ B (0) e γ ( h ( w ) − h (cid:15) ( z )) > (cid:15) − r } By Lemma 3.11, we have P ( A r,(cid:15) ) = o ∞ (cid:15) ( (cid:15) ). Since h (cid:15) ( z ) is Gaussian with variance at most (1 + o (cid:15) (1)) log (cid:15) − ,for each (cid:101) β > E (cid:0) e ˜ βh (cid:15) ( z ) (cid:1) = (cid:15) − ˜ β + o (cid:15) (1) . (5.5)By [RV14, Theorem 2.11], µ ˙ h ( B (0)) has finite moments of all orders < /γ . Choose p j > j = 1 , , , βp < γ and (cid:80) j =1 p − j = 1. By H¨older’s inequality, E (cid:0) e γβh (cid:15) ( z ) × µ ˙ h ( B (0)) β × sup w ∈ B (0) e γβ ( h ( w ) − h (cid:15) ( z )) A r,(cid:15) (cid:1) ≤ E (cid:0) e γβp h (cid:15) ( z ) (cid:1) p − × E (cid:0) µ ˙ h ( B (0)) βp (cid:1) p − × E (cid:0) sup w ∈ B e γβp ( h ( w ) − h (cid:15) ( z )) (cid:1) p − × P (cid:0) A r,(cid:15) (cid:1) p − ≤ (cid:15) − γ β p + o (cid:15) (1) P (cid:0) A r,(cid:15) (cid:1) p − = o ∞ (cid:15) ( (cid:15) ) , which implies E (cid:0) µ h ( B (cid:15) ( z )) β (cid:1) ≤ (cid:15) (2+ γ / − r ) β E (cid:0) (cid:15) γβh (cid:15) ( z ) µ ˙ h ( B (0)) β (cid:1) + o ∞ (cid:15) ( (cid:15) ) = (cid:15) (2+ γ / − r ) β − γ β + o (cid:15) (1) by independence of ˙ h and h (cid:15) .We are now ready to prove the lower bound of our main theorem: Proof of Theorem 1.1, lower bound.
As in the proof of the upper bound in Section 5.1, we can assume withoutloss of generality that X ⊂ B R (0) \ B /R (0) for some fixed R > h with a whole-planeGFF normalized so that its circle average over ∂ D is 0.If dim H ( X ) = 2 the lower bound clearly holds, so we assume dim H ( X ) <
2. Let β ∈ [0 , γ ) be the (random)solution of dim H ( X ) = f ( β ), and let β ∈ ( β , γ ). Then choose some d ∈ (dim H ( X ) , f ( β )), and some r > d < − r + (1 − r ) f ( β ) . (5.6)Then choose a sequence { δ n } n ∈ N , such that δ n ∈ (0 ,
1) for each n and (cid:80) ∞ n =1 δ rn < ∞ . Since d > dim H ( X ),we can find, for each n ∈ N , a random collection of balls B n , measurable with respect to σ ( X ) and covering X , such that (cid:88) B ∈B n (cid:15) dB < δ n , where (cid:15) B denotes the radius of B . For any B ∈ B n , let I ( B ) denote the set of intervals I as defined inLemma 5.1, such that η (cid:48) ( I ) intersects B , and let B (cid:48) be the ball of radius (cid:15) − rB centered at the same point as B . Let ˜ (cid:15) >
0, and assume without loss of generality that (cid:15) B < (cid:15) ( r, ˜ (cid:15) ) for all B ∈ B n and n ∈ N as defined inLemma 5.1. Define the random variable Z n by Z n = E ˜ (cid:15) − r (cid:88) B ∈B n (cid:88) I ∈I ( B ) | I | β , where E ˜ (cid:15) − r = E ˜ (cid:15) − r ( r, R ) is the event of Proposition 3.4. By using (cid:80) I ∈I ( B ) | I | ≤ µ ( B (cid:48) ) and |I ( B ) | < (cid:15) − rB ,we have Z n ≤ E ˜ (cid:15) − r (cid:88) B ∈B n (cid:15) − rB µ ( B (cid:48) ) β . It follows from Lemma 5.2 and (5.6) that for all sufficiently large n , E ( Z n | X ) ≤ (cid:88) B ∈B n (cid:15) − r +(1 − r ) f ( β )+ o n (1) B ≤ (cid:88) B ∈B n (cid:15) dB < δ n .
37y Chebyshev’s inequality, P ( Z n > δ − rn ) ≤ δ rn , so by the Borel-Cantelli lemma the event { Z n > δ − rn } happens at most finitely often almost surely. It follows that Z n → (cid:15) → n →∞ (cid:88) B ∈B n (cid:88) I ∈I ( B ) | I | β = 0 . This completes the proof, since ∪ B ∈B n ,I ∈I ( B ) I is a cover of any set (cid:98) X as in the theorem statement. SLE κ (cid:48) for κ (cid:48) ∈ (4 ,
8) does not have triple points, and for κ (cid:48) ≥ κ (cid:48) , κ (cid:48) ∈ (4 , κ (cid:48) ↓
4, the maximal multiplicity of the path a.s. tends to ∞ . In thissection we will calculate the a.s. Hausdorff dimension of m -tuple points for space-filling SLE κ (cid:48) for m ≥
3. Weremark that this dimension could also have been derived from [MW17, Theorem 1.4] by arguing that the m -tuple points of space-filling SLE κ (cid:48) have the same dimension as the m -tuple points of an SLE κ (cid:48) ( κ (cid:48) − η (cid:48) be a whole–plane space-filling SLE κ (cid:48) parameterized by quantum area withrespect to an independent γ -quantum cone ( C , h, , ∞ ). Let Z = ( L, R ) denote the associated quantumboundary length processes, and recall that Z is a two-dimensional Brownian motion with covariance givenby (1.3).Let m ≥
3. Except for the countable number of triple points corresponding to local minima of L or R , thereis a bijection between m -tuple points of SLE κ (cid:48) and the ( m − Z , which will be definedjust below. Recall the definition of an ordinary ( π/ v ( t ) defined in (2.1). Let t be a cone time of Z (resp. cone time of the time-reversal of Z ), and definethe function u (resp. v R and u R ) by v R ( t ) = sup { s < t : R s < R t or L s < L t } ,u ( t ) = inf (cid:26) s > t : inf s (cid:48) ∈ [ t,s ] R s (cid:48) < R t and inf s (cid:48) ∈ [ s,t ] L s (cid:48) < L t (cid:27) ,u R ( t ) = sup (cid:26) s < t : inf s (cid:48) ∈ [ s,t ] R s (cid:48) < R t and inf s (cid:48) ∈ [ s,t ] L s (cid:48) < L t (cid:27) . An m -tuple cone time is a generalization of a cone time (see Figure 6). Definition 6.1.
Let m ≥ t ∈ R m − , t = ( t , . . . , t m − ). Then t is an ( m − t m − > t such that the following properties are satisfied for any j ∈ { , . . . , m − } . For j even (resp. odd) t j is a cone time of Z , and we have either t j +1 = v ( t j ) and t j − = u ( t j ), or t j +1 = u ( t j ) and t j − = v ( t j ). For j odd (resp. even) t j is a cone time of the time-reversal of Z , and we have either t j +1 = v R ( t j ) and t j − = u R ( t j ), or t j +1 = u R ( t j ) and t j − = v R ( t j ).Let (cid:101) T ( m ) ⊂ R m − denote the set of ( m − T ( m ) ⊂ (cid:101) T ( m ) denote the set of conevectors satisfying (I), and for which t is running infimum of L .We say that t ∈ R is an ( m − t = t for some ( m − t . Let (cid:101) T ( m ) ⊂ R denote the set of ( m − T ( m ) ⊂ (cid:101) T ( m ) denote the set of cone times correspondingto elements of T ( m ). Remark 6.2.
Note that η (cid:48) ( t j ) = η (cid:48) ( t ) for all j ∈ { , . . . , m − } . Also note that, for t ∈ T ( m ), we have t j − , t j < t j − , L j − = L j − and R j − = R j for all relevant j , in particular t m − is a running infimumof R (resp. L ) for odd (resp. even) m . Furthermore, note that we have chosen to let an ( m − m − t in the vector in addition to the ( m −
2) cone times, while we have not included t m − . Wechose not to include t m − in order to simplify the calculation of some probabilities in Sections 6.1 and 6.2,while we did include t since it will be needed for one of our regularity conditions in Section 6.2. By symmetry (cid:101) T ( m ) (resp. (cid:101) T ( m )) and T ( m ) (resp. T ( m )) have the same Hausdorff dimension almost surely.38 t t t t L t C − R t tt t t t t t t L t C − R t tt t t t t t t L t C − R t L t − L t R t − R t t t t t L t − L t R t − R t t t t t t t t L t − L t R t − R t t t t t t t t Figure 6: Points of multiplicity m can be illustrated as a rectangular path of m vertical line segments and m − m = 4 and m = 7. Each horizontal line segment intersects L (or C − R ) at its end points, and L (or C − R ) is above (or below) the line segment between the intersection times, implying that L (or R ) hasa local running infimum where the line segment ends, and that its time-reversal has a local running infimumwhere the line segment starts. The local running infima correspond to a sequence of π -cone times of theplanar Brownian motion ( L, R ), see the bottom figures. The ( m −
2) times corresponding to π -cone timesare marked with red dots, while the times t and t m − , which are running infima of L or R , and correspondto the same spatial point of η (cid:48) , are marked with blue dots.We now state the main results of this section. Theorem 6.3.
Let κ (cid:48) ∈ (4 , . Let m ∈ [3 , (2 κ (cid:48) − / ( κ (cid:48) − ∩ N . Then a.s. dim H (cid:101) T ( m ) = dim H (cid:101) T ( m ) = 12 − ( m − (cid:18) κ (cid:48) − (cid:19) . (6.1) If m > (2 κ (cid:48) − / ( κ (cid:48) − , (cid:101) T ( m ) and (cid:101) T ( m ) are empty a.s. Throughout this section, we let θ = 4 πκ (cid:48) be as in (1.3). Remark 6.4.
By a linear transformation it follows that (6.1) also gives the dimension of the set of ( m − θ -cone times of standard planar Brownian motion. 39heorem 6.3 and Theorem 1.1 imply: Corollary 6.5.
Let κ (cid:48) ∈ (4 , and m ∈ [3 , (2 κ (cid:48) − / ( κ (cid:48) − ∩ N . The Hausdorff dimension of m -tuplepoints of space-filling SLE κ (cid:48) is a.s. equal to (4 m − − κ (cid:48) ( m − κ (cid:48) − m )8 κ (cid:48) . If m > (2 κ (cid:48) − / ( κ (cid:48) − , the set of m -tuple points is a.s. empty.Proof. The corollary follows from Theorem 1.1 if we can show that, excluding the countable number of triplepoints corresponding to local minima of L or R , there is a bijection between m -tuple points and ( m − m − m -tuple point.Conversely, note that an m -tuple point of η (cid:48) not corresponding to an element of (cid:101) T ( m ), would correspond toeither (a) a vector t ∈ R m − for which at least one of the elements t ∈ R is a local minimum of L or R , or(b) a vector t ∈ R m − for which at least one of the elements t ∈ R is a running infimum of L and a runninginfimum of the time-reversal of R , or vice-versa. The set of vectors satisfying (a) is empty (except in the case m = 3, when it is countable), since the set of local minima is countable, and the set of vectors satisfying (b)is empty by [Shi88, Theorem 1].For δ ∈ (0 , m − ), let T ( δ, m ) = { t ∈ T ( m ) : | t i − t j | > δ, t j ∈ (0 , , ∀ i, j ∈ { , . . . , m − }} . By the self-similarity of Brownian motion and the stability of Hausdorff dimension under countable unions, itis sufficient to calculate the dimension of T ( δ, m ) for fixed δ . For n ∈ N , define (cid:15) n = ( n !) − . Let k ∈ N be thesmallest integer such that (cid:15) k < δ . For m ≥ δ ∈ (0 , m − ), and n ∈ N , define D = D δ,m and D n = D δ,m,n by D = (cid:8) t = ( t , . . . , t m − ) ∈ (0 , m − : t j − , t j < t j − , | t i − t j | > δ, ∀ i, j ∈ { , . . . , m − } (cid:9) ,D n = D ∩ ( (cid:15) n Z ) . We will consider δ , k , m , κ (cid:48) as fixed throughout the rest of the section, and all implicit constants and decayrates might depend on these constants. The upper bound is based on an estimate for the probability that an ( m − L, R ) near the approximate cone times, and one type of eventconcerning the behavior of (
L, R ) in the time interval between the cone times.Let
C > < r (cid:28)
1. For j ∈ { , . . . , m − } , n ∈ N and t ∈ D n , let A n,j,C,r ( t ) = (cid:110) L t j + s ≥ L t j − C(cid:15) − rn , R t j + s ≥ R t j − C(cid:15) − rn , ∀ s ∈ (0 , (cid:15) k + m +1 ) (cid:111) if j > (cid:110) L t j − s ≥ L t j − C(cid:15) − rn , R t j − s ≥ R t j − C(cid:15) − rn , ∀ s ∈ (0 , (cid:15) k + m +1 ) (cid:111) if j odd, (cid:110) L t j + s ≥ L t j − C(cid:15) − rn , ∀ s ∈ (0 , (cid:15) k + m +1 ) (cid:111) if j = 0 . For j ∈ { , , . . . , (cid:98) ( m − / (cid:99)} , n ∈ N and t ∈ D n , let B n,j,C,r ( t ) be the event that • | X t j − − X t ∗ | ≤ C(cid:15) − rn and inf s ∈ [ t ∗ ,t j − ] X s ≥ X t j − − C(cid:15) − rn for each ( X, t ∗ ) ∈ { ( L, t j − ) , ( R, t j ) } if 2 j + 1 < m ; or 40 | L t j − − L t j − | ≤ C(cid:15) − rn and inf s ∈ [ t j − ,t j − ] L s ≥ L t j − − C(cid:15) − rn if 2 j + 1 = m .Recall that t j − , t j < t j − for all j ∈ { , , . . . , (cid:98) ( m − / (cid:99)} , hence the intervals considered above arewell-defined.Define E n ( t ) = m − (cid:92) j =0 A n,j,C,r ( t ) ∩ (cid:98) ( m − / (cid:99) (cid:92) j =1 B n,j,C,r ( t ) . For t ∈ D n we say that t is an n -approximate ( m − -tuple cone vector if the event E n ( t ) occurs.The event A n,j,C,r ( t ) occurs when there is an approximate cone time for Z or the time-reversal of Z attime t j . The events B n,j,C,r ( t ) ensure that the L or R coordinate of two pairs of approximate cone times t j − , t j − and t j − , t j are approximately identical.We will need the following two lemmas both for the proof of the upper bound and for the proof of the lowerbound of Theorem 6.3. The first lemma is [Shi85, equation (4.3)]. Lemma 6.6.
Let
L, R be correlated Brownian motions satisfying (1.3) . For any t ∈ R and (cid:15) > , P (cid:0) X s ≥ X t − (cid:15) , ∀ s ∈ [ t, t + 1] , X = L, R (cid:1) (cid:16) (cid:15) π θ , with the implicit constant depending only on θ . Our second lemma will be used to estimate the conditional probabilities of the events B n,j,C,r ( t ) given theother events we are interested in. We condition on ( L, R ) restricted to intervals near each time t i , and theevent (cid:101) H n ( M ) is introduced to ensure that the path of ( L, R ) restricted to these intervals is not too irregular,and that (
L, R ) does not violate the conditions of the event B n,j,C,r ( t ) in these intervals. Lemma 6.7.
Let s ≤ (cid:15) k +1 . For j ∈ { , , . . . , (cid:98) ( m − / (cid:99)} let F j be the σ -algebra generated by1. ( L t , R t ) for t ∈ (0 , t j − − s ) ,2. ( L t , R t ) − ( L t i , R t i ) for i (cid:54) = 2 j − and t ∈ ( t i − (cid:15) k +1 , t i + (cid:15) k +1 ) , and3. ( L t , R t ) − ( L t j − , R t j − ) for t ∈ ( t j − − s, t j − + (cid:15) k +1 ) .Let { H n } n ∈ N be a sequence of events measurable with respect to F j . For M > and n ∈ N let (cid:101) H n ( M ) be theevent that the following is true. • M s > X t j − − s − X t j − > M − s and M s > X t j − − s − X t ∗ > M − s for each ( X, t ∗ ) ∈{ ( L, t j − ) , ( R, t j ) } ; and • inf t (cid:48) ∈ [ t j − − s,t j − ] X t (cid:48) ≥ X t j − − C(cid:15) − rn and inf t (cid:48) ∈ [ t ∗ ,t j − − s ] L t (cid:48) ≥ L t ∗ − C(cid:15) − rn for each ( X, t ∗ ) ∈{ ( L, t j − ) , ( R, t j ) } .Assume there is a constant M > independent of n and t such that the conditional probability of (cid:101) H n ( M ) given H n is at least M − for all n ∈ N and t ∈ D n . Then P (cid:0) B n,j,C,r ( t ) (cid:12)(cid:12) H n (cid:1) (cid:16) (cid:40) (cid:15) − rn for j + 1 < m,(cid:15) − rn for j + 1 = m, (6.2) with the implicit constants depending only on δ, C, m, θ, s and M .If s = (cid:15) l for l ∈ { k + 1 , . . . , n − } , and M can be chosen independently of l, n and t , then P (cid:0) B n,j,C,r ( t ) (cid:12)(cid:12) H n (cid:1) (cid:16) (cid:40) (cid:15) − rn (cid:15) − o n (1) l for j + 1 < m,(cid:15) − rn (cid:15) − + o n (1) l for j + 1 = m, (6.3) with the implicit constant depending only on δ, C, m, θ and M . roof. First we will prove (6.2). Let 2 j + 1 < m . In order for B n,j,C,r ( t ) to occur, we must have | ( X t j − − X t j − − s ) + ( X t j − − s − X t j − − s ) + ( X t j − − s − X t ∗ ) | ≤ C(cid:15) − rn ∀ ( X, t ∗ ) ∈ { ( L, t j − ) , ( R, t j ) } (6.4)and inf s ∈ [ t ∗ ,t j − ] X s ≥ X t j − − C(cid:15) − rn , ∀ ( X, t ∗ ) ∈ { ( L, t j − ) , ( R, t j ) } . (6.5)The first and third terms in the left-hand side of (6.4) is measurable with respect to F j , while the secondterm is independent of F j . The second term is a normally distributed random variable with variance oforder s . The probability of (6.4) conditioned on F j is therefore equal to the probability that two jointlyGaussian random variables with variance of order s take values in two given intervals of length 2 C(cid:15) − rn . If2 j + 1 = m , the same result holds, only with one Gaussian random variable, instead of two Gaussian randomvariables. By the upper bounds in the first event defining (cid:101) H n ( M ) the estimate (6.2) follows with (6.4) insteadof B n,j,C,r ( t ) on the left-hand side.To complete the proof of (6.2) we need to show that (6.5) happens with uniformly positive probabilityconditioned on H n and (6.4). This follows by using that L t j − − s − L t j − , L t j − − s − L t j − , and thecorresponding quantities for R , have a macroscopic magnitude on the event (cid:101) H n ( M ).The estimate (6.3) follows by small modifications of the argument above using Brownian scaling. We onlyconsider the case 2 j + 1 < m , since the case 2 j + 1 = m is similar. Again we need two jointly Gaussianrandom variables of variance s to take values in bounded intervals of length 2 C(cid:15) / n , and the upper boundsin the first event defining (cid:101) H n ( M ) imply that this probability is of order (cid:15) − rn (cid:15) − l . The event (cid:101) H n ( M ) alsoensures that (6.5) occurs with uniformly positive probability conditioned on H n and (6.4).The following lemma implies the upper bound of Theorem 6.3: Lemma 6.8.
For any n ∈ N and t ∈ D n , P (cid:0) E n ( t ) (cid:1) (cid:16) (cid:15) +( m − π θ + ) − cr where c > is a constant depending only on δ, C, m and θ .Proof. Since | t i − t j | > δ for any two i, j ∈ { , . . . , m − } , the Markov property of Brownian motion implies thatthe events A n,j,C,r ( t ) for j ∈ { , . . . , m − } are independent. By Lemma 6.6 we have P ( A n,j,C,r ( t )) (cid:16) (cid:15) π θ (1 − r ) n for j = 1 , . . . , m −
2, and P ( A n, ,C,r ( t )) (cid:16) (cid:15) − rn . The events B n,j,C,r ( t ) are not independent of each otherand of the events ( A n,i,C,r ( t )) i =0 ,...,m − , but as we will see in the remainder of the proof we have sufficientindependence to obtain a good estimate for conditional probabilities.Let j ∈ { , . . . , (cid:98) ( m − / (cid:99)} . Since t j − , t j < t j − by the definition of D , the event (cid:32) m − (cid:92) i =0 A n,i,C,r ( t ) (cid:33) ∩ (cid:92) i : t i − L, R ) is ( − r )-H¨older continuous with H¨older norm at most C/ 2. Assume s ∈ T ( δ, m ), and let t ∈ D n be the element of D n that minimizes (cid:107) s − t (cid:107) , where (cid:107) · (cid:107) denotese.g. the L ∞ norm. If A C,r occurs, then t is an n -approximate ( m − A C,r T ( δ, m ) ⊂ (cid:91) t ∈ D ∗ n S n ( t ) . (6.7)First assume m ≤ κ (cid:48) − κ (cid:48) − , and let d > − ( m − π θ − ). By Lemma 6.8, for any C > r , E (cid:18) A C,r (cid:88) t ∈ D ∗ n diam( S n ( t )) d (cid:19) (cid:22) (cid:88) t ∈ D n (cid:15) dn P ( E n ( t )) = (cid:15) − ( m − n × (cid:15) dn × (cid:15) +( m − π θ + )+ crn → n → ∞ . Chebyshev’s inequality and the Borel-Cantelli lemma together imply that a.s.lim n →∞ A C,r (cid:88) t ∈ D ∗ n diam( S n ( t )) d = 0 . (6.8)By (6.7) the set { S n ( t ) } t ∈ D ∗ n, gives a cover for T ( δ, m ) on the event A C,r , hence dim H ( T ( δ, m )) ≤ d a.s. if A C,r occurs. Since P ( A C,r ) → C → ∞ , we a.s. have dim H ( T ( δ, m )) ≤ d .By definition T ( m ) = Proj ( T ( m )), where Proj : R m − → R is the projection t (cid:55)→ t . Since coordinate projec-tion is Lipschitz continuous, dim H ( T ( m )) ≤ dim H ( T ( m )). Since dim H ( T ( m )) = sup δ ∈ Q ,δ ∈ (0 ,m − ) dim H ( T ( m, δ )),we obtain the desired upper bound by letting d → − ( m − π θ − ), and using that dim H ( (cid:101) T ( m )) =dim H ( T ( m )) and dim H ( (cid:101) T ( m )) = dim H ( T ( m )).Now we will consider the case m > κ (cid:48) − κ (cid:48) − . Note that (6.8) still holds in this case, and that we can choose d = 0. When d = 0, the left hand side of (6.8) counts the number of elements in D ∗ n , and it follows that D ∗ n is empty for all sufficiently large n . By (6.7) we can conclude that T ( δ, m ), hence (cid:101) T ( m ) and (cid:101) T ( m ), areempty. We will now prove the lower bound of Theorem 6.3. The proof will be by standard methods, and relies on anestimate for the correlation of the two events that t and s are approximate cone vectors, see Proposition 6.12.In order to obtain sufficient independence of these two events, we will work with a slightly modified definitionof approximate cone vectors. Let t ∈ D n . For j ∈ { , , . . . , m − } define the events A n,j ( t ) = A n,j, , ( t ) , B n,j ( t ) = B n,j, , ( t ) ,F j ( t ) = (cid:110) inf s ∈ [0 ,(cid:15) l ] ( X t j − X t j − s ) ∈ (cid:15) l ( − l, − l − / ) , ∀ l ≥ k + 1 , ∀ ( X, t ∗ ) ∈ { ( L, t j − ) , ( R, t j ) } (cid:111) , j > , (cid:110) inf s ∈ [0 ,(cid:15) l ] ( X t j + s − X t j ) ∈ (cid:15) l ( − l, − l − / ) , ∀ l ≥ k + 1 , ∀ ( X, t ∗ ) ∈ { ( L, t j − ) , ( R, t j ) } (cid:111) , j odd , (cid:110) inf s ∈ [0 ,(cid:15) l ] ( L t j − L t j − s ) ∈ (cid:15) l ( − l, − l − / ) , ∀ l ≥ k + 1 (cid:111) , j = 0 , and for j ∈ { , , . . . , (cid:98) ( m − / (cid:99)} , define the event B ∗ n,j ( t ) = (cid:26) inf s ∈ [ t ∗ + (cid:15) n ,t j − − (cid:15) n ] X s > X t j − + 3 (cid:15) n , ∀ ( X, t ∗ ) ∈ { ( L, t j − ) , ( R, t j ) } (cid:27) . G ( t ) by G ( t ) = (cid:26) R t +2 (cid:15) k +2 < inf t ∈ [ t − (cid:15) k +2 ,t + (cid:15) k +2 ] R t − (cid:15) k +2 (cid:27) , and finally define the event (cid:101) E n ( t ) by (cid:101) E n ( t ) = m − (cid:92) j =0 A n,j ( t ) ∩ (cid:98) ( m − / (cid:99) (cid:92) j =1 B n,j ( t ) ∩ (cid:98) ( m − / (cid:99) (cid:92) j =1 B ∗ n,j ( t ) ∩ m − (cid:92) j =0 F j ( t ) ∩ G ( t ) . We say that t ∈ D n is a perfect n -approximate ( m − -tuple cone vector if the event (cid:101) E n ( t ) occurs. Let D ∗ n, P denote the set of all t ∈ D n such that (cid:101) E n ( t ) occurs.The set T P ( δ, m ) of perfect ( m − T P ( δ, m ) := (cid:92) k ≥ (cid:91) n ≥ k (cid:91) t ∈ D ∗ n, P S n ( t ) . As we will see below, the events B ∗ n,j ( t ) imply that if s (cid:54) = t , both events (cid:101) E n ( t ) and (cid:101) E n ( s ) can only happen ifwe have t j < s j for all even j and t j > s j for all odd j , or vice-versa. The events F j ( t ) will imply that bothevents (cid:101) E n ( t ) and (cid:101) E n ( s ) can only happen if all the elements of the vector t − s are of approximately the samemagnitude, and the regularity condition G ( t ) will imply that both events (cid:101) E n ( t ) and (cid:101) E n ( s ) cannot happen if | t i − s j | is very small for some i (cid:54) = j . Lemma 6.9. The set of perfect ( m − -tuple cone vectors is contained in the set of ( m − -tuple conevectors, i.e., T P ( δ, m ) ⊂ T ( δ, m ) . Proof. Assume t (cid:54)∈ T ( δ, m ). Then at least one of the following conditions are satisfied: (i) there is a j ∈ { , ..., m − } such that t j is not a cone time for Z or for the time-reversal of Z , (ii) there is an even (resp.odd) j ∈ { , . . . , m − } such that v ( t j ) (cid:54) = t j ± or u ( t j ) (cid:54) = t j ± (resp. v R ( t j ) (cid:54) = t j ± or u R ( t j ) (cid:54) = t j ± ), or (iii) t (cid:54)∈ { u ( t ) , v ( t ) } . For each n ∈ N let s n ∈ D n be a vector such that (cid:107) t − s n (cid:107) is minimized. In either case(i)-(iii) continuity of L and R imply that (cid:101) E n ( s ) cannot occur for sufficiently large n . Hence t (cid:54)∈ T P ( δ, m ).The following lemma combined with Proposition 6.12 will imply the lower bound of Theorem 6.3. Lemma 6.10. For any n ∈ N and t ∈ D n , P ( (cid:101) E n ( t )) (cid:16) (cid:15) +( m − π θ + ) n with the implicit constants depending only on δ, m and θ .Proof. Let N and N be normal random variables with correlation cos( θ ) and variance a , with a as in (1.3).For any l ≥ k + 1 and j ∈ { , ..., (cid:98) ( m − / (cid:99)} even, P (cid:16) inf s ∈ [0 ,(cid:15) l ] ( R t j − R t j − s ) (cid:54)∈ (cid:15) l ( − l, − l − ) or inf s ∈ [0 ,(cid:15) l ] ( L t j − L t j − s ) (cid:54)∈ (cid:15) l ( − l, − l − ) (cid:17) = P (cid:16) | N | (cid:54)∈ ( l − , l ) or | N | (cid:54)∈ ( l − , l ) (cid:17) (cid:16) l − . By using this estimate, a similar estimate for odd j , and independence of the events F j ( t ), the Borel-Cantellilemma implies that (cid:84) m − j =0 F j ( t ) happens with positive probability.44 j − t j − t j (cid:15) / n (cid:15) / n L t − L t j + (cid:15) / n R t − R t j + (cid:15) / n t j t j − (cid:15) l t j − (cid:15) l − Figure 7: The left figure illustrates regularity condition B ∗ n,j ( t ). The three green dots correspond to times t j − + (cid:15) n , t j + (cid:15) n and t j − − (cid:15) n , respectively. We want the curve to be bounded away from the boundaryof the cone for most of the cone excursion, hence preventing approximate cone vectors s with cone excursionsthat are partially inside and partially outside the corresponding excursion of t . For example, if (cid:101) E n ( t ) ∩ (cid:101) E n ( s )occurs, t (cid:54) = s , and t j < s j < s j +1 , we want s j +1 < t j +1 . The right figure illustrates regularity condition F j ( t ).The red curve is Z | [ t j − (cid:15) l ,t j ] , and the blue curve is Z | [ t j − (cid:15) l − ,t j ] . The vertical red arrow shows the absolute valueof inf s ∈ [ t j − (cid:15) l ,t j ] ( R s − R t j ), and the vertical blue arrow shows the absolute value of inf s ∈ [ t j − (cid:15) l − ,t j ] ( R s − R t j );the horizontal arrows show the same values for L . Regularity condition F j ( t ) implies that the modulus ofthese infima decrease in a certain way as we increase l . Hence, if both events (cid:101) E n ( t ) and (cid:101) E n ( s ) occur we knowthe approximate value of R t j − R s j and L t j − L s j in terms of t j − s j . This will help us establish that allelements of the vector t − s are of approximately the same order whenever both t and s are approximateperfect cone vectors.The events A n,j ( t ) are independent of each other and of the events F j ( t ), and P ( A n,j ( t )) (cid:16) (cid:15) π θ n by Lemma 6.6.The event G ( t ) is independent of F j ( t ) and A n,j ( t ) for j > 0. Conditioned on F ( t ) ∩ A n, ( t ) the event G ( t )has uniformly positive probability, since the value of R t +2 (cid:15) k +2 − R t + (cid:15) k +2 is independent of F ( t ) ∩ A n, ( t ).For j = 1 , , . . . , (cid:98) ( m − / (cid:99) , let B ∗ , loc n,j ( t ) be the event that the following is true. • inf s ∈ [ t j − − (cid:15) k + m +1 ,t j − − (cid:15) n ] X s > X t j − + 3 (cid:15) n and X t j − − (cid:15) k + m +1 − X t j − > (cid:15) k + m +1 for each ( X, t ∗ ) ∈{ ( L, t j − ) , ( R, t j ) } . • inf s ∈ [ t ∗ + (cid:15) n ,t ∗ + (cid:15) k + m +1 ] X s > X t ∗ +3 (cid:15) n and X t ∗ + (cid:15) k + m +1 − X t ∗ > (cid:15) k + m +1 for each ( X, t ∗ ) ∈ { ( L, t j − ) , ( R, t j ) } .Note that the occurrence of B ∗ , loc n,j ( t ) implies that the conditions defining B ∗ n,j ( t ) are satisfied near t j − , t j − and t j . The event B ∗ , loc n,j ( t ) is independent of F i ( t ) for all i , since the event F i ( t ) depends on thebehavior of Z right before (resp. after) t i for i even (resp. odd), while the event B ∗ , loc n,j ( t ) depends on thebehavior of Z right before (resp. after) t j − (resp. t j − and t j ). The event B ∗ , loc n,j ( t ) is also independent ofthe events A n,i ( t ) for i (cid:54) = 2 j − , j − , j . Furthermore B ∗ , loc n,j ( t ) is independent of G ( t ) for j > 1, and theprobability of B ∗ , loc n, ( t ) changes only by a constant order factor when conditioning on G ( t ). By Brownianscaling, P (cid:18) inf s ∈ [ t j + (cid:15) n ,t j + (cid:15) k + m +1 ] R s > R t j + 3 (cid:15) n , R t j + (cid:15) k + m +1 − R t j > (cid:15) k + m +1 | A n, j ( t ) (cid:19) (cid:23) P inf s ∈ [ t j + (cid:15) n ,t j + (cid:15) k + m +1 ] R s > R t j + 3 (cid:15) n , R t j + (cid:15) k + m +1 − R t j > (cid:15) k + m +1 | inf s ∈ [ t j + (cid:15) n ,t j + (cid:15) k + m +1 ] R s ≥ R t j − (cid:15) n (cid:23) , B ∗ , loc n,j ( t ). It follows that P m − (cid:92) j =0 A n,j ( t ) ∩ F j ( t ) ∩ G ( t ) ∩ (cid:98) ( m − / (cid:99) (cid:92) j =1 B ∗ , loc n,j ( t ) (cid:16) (cid:15) +( m − π θ n . By (6.2) of Lemma 6.7 with s = (cid:15) k + m +1 we get further P m − (cid:92) j =0 A n,j ( t ) ∩ F j ( t ) ∩ G ( t ) ∩ (cid:98) ( m − / (cid:99) (cid:92) j =1 B ∗ , loc n,j ( t ) ∩ B n,j ( t ) (cid:16) (cid:15) +( m − π θ + ) n . (6.9)It remains to show that the same estimate holds if we replace B ∗ , loc n,j ( t ) by B ∗ n,j ( t ). It is straightforward toobtain the estimate with (cid:22) and B ∗ n,j ( t ) instead of B ∗ , loc n,j ( t ), since (cid:101) E ( t ) ⊂ E ( t ) for C = 1 , r = 0, and we havethe estimate of Lemma 6.8. To obtain (6.9) with (cid:23) , and with B ∗ n,j ( t ) instead of B ∗ , loc n,j ( t ), we need to show thatthe inequalities of B ∗ n,j ( t ) hold with uniformly positive probability also at a macroscopic distance from thecone times t j − , t j − , t j , conditioned on the event of (6.9). Note that all the events we condition on, exceptfor B n,j , only concern the behavior of the curve near the approximate cone times, while B n,j says that the L and R coordinate of pairs of cone times are close, and that the curve is above this L or R coordinate in the timeinterval between the cone times. The event B ∗ , loc n,j ( t ) ensures that R t j + (cid:15) k + m +1 , R t j − − (cid:15) k + m +1 , L t j − + (cid:15) k + m +1 and L t j − − (cid:15) k + m +1 have a macroscopic distance from R t j , R t j − , L t j − and L t j − , respectively. Theoccurrence of B ∗ n,j , conditioned on the event of (6.9), therefore corresponds to the event that two approximateBrownian bridges of duration of order 1 between two given pairs of points at (possibly different) height oforder 1, are always larger than 3 (cid:15) n . This happens with probability (cid:23) n and t . Hence theconditional probability of B ∗ n,j given the event of (6.9) is positive uniformly in n and t . Lemma 6.11. Let t , s ∈ D n , t (cid:54) = s , and assume the event (cid:101) E n ( t ) ∩ (cid:101) E n ( s ) occurs for some n ≥ m + k . Thenat least one of the two following conditions are satisfied:(I) All i, j ∈ { , . . . , m − } satisfy | s i − t j | ≥ (cid:15) k + m .(II) There is an l ∈ ( k, n − such that | s j − t j | ∈ [ (cid:15) l + m +1 , (cid:15) l +2 ] for all j ∈ { , . . . , m − } . We furthermorehave either s j < t j for all odd j and t j < s j for all even j , or s j < t j for all even j and t j < s j for allodd j .Proof. If (I) is not satisfied, at least one of the following conditions must be satisfied: (i) 0 < min j | s j − t j | <(cid:15) k + m , (ii) s j = t j for some j , or (iii) | s j − t i | < (cid:15) k + m for some i (cid:54) = j . We will show that (i) implies (II), andthat (ii) and (iii) cannot occur, hence complete the proof of the lemma.Case (i): Assume (i) holds, but (I) does not hold. Furthermore, assume the first condition of (II) does nothold, i.e., there is no l > k such that | s j − t j | ∈ [ (cid:15) l + m +1 , (cid:15) l +2 ] for all j . Then we can find i, j ∈ { , . . . , m − } and l (cid:48) ≥ k + 2 such that | s j − t j | < (cid:15) l (cid:48) +1 , | s i − t i | > (cid:15) l (cid:48) , and | i − j | = 1. Assume w.l.o.g. that j is even, i = j + 1, and t j < s j < s j +1 ; all other cases can be treated similarly.We claim that s j +1 < t j +1 . Assume the opposite, i.e., t j (cid:48) +1 < s j (cid:48) +1 for j = 2 j (cid:48) . Note that this implies t j (cid:48) +1 ∈ ( s j (cid:48) , s j (cid:48) +1 ), since t j (cid:48) < t j (cid:48) +1 , | s j (cid:48) − t j (cid:48) | < (cid:15) l (cid:48) +1 and | t j (cid:48) +1 − t j (cid:48) | > δ . Regularity condition B ∗ n,j (cid:48) +1 ( t ) gives a contradiction to B n,j (cid:48) +1 ( s ), since s j (cid:48) ∈ ( t j (cid:48) , t j (cid:48) +1 ), and L s j (cid:48) ≤ L s j (cid:48) +1 + (cid:15) n < L t j (cid:48) +1 − (cid:15) n . This implies the claim.Since 0 < min j (cid:48) | s j (cid:48) − t j (cid:48) | < (cid:15) l (cid:48) +1 , we have n > l (cid:48) + 1, which implies (cid:15) n < (cid:15) l (cid:48) +1 . Observe that the event B n,j ( t ), hence (cid:101) E n ( t ), cannot occur if (cid:101) E n ( s ) occurs, sinceinf s ∈ [ t j ,t j +1 ] L s ≤ inf s ∈ [ s j +1 ,s j +1 + (cid:15) l (cid:48) ] L s < inf s ∈ [ s j − (cid:15) l (cid:48) +1 ,s j ] L s − (cid:15) l (cid:48) +1 +( L s j +1 − L s j ) < inf s ∈ [ s j − (cid:15) l (cid:48) +1 ,s j ] L s − (cid:15) n ≤ L t j − (cid:15) n , F j ( s ) and F j +1 ( s ). We have obtained a contradiction, hence there isan l > k such that | s j − t j | ∈ [ (cid:15) l + m +1 , (cid:15) l +2 ] for all j . Note that l + 2 ≤ n , since 0 < min j (cid:48) | s j (cid:48) − t j (cid:48) | ≤ (cid:15) l +2 .By the same argument as when deriving s j +1 < t j +1 above, regularity conditions B ∗ n,j (cid:48) +1 ( t ) and B ∗ n,j (cid:48) +1 ( s )for j (cid:48) ∈ { , . . . , (cid:98) ( m − / (cid:99)} imply that s j < t j for all odd j , and t j < s j for all even j , or vice versa.Case (ii): Since t (cid:54) = s , we can find j such that t j = s j and either s j − (cid:54) = t j − or s j +1 (cid:54) = t j +1 . Assumew.l.o.g. that j = 2 j (cid:48) is even, and that s j +1 > t j +1 . By definition of D we have t j +1 ∈ ( s j , s j +1 ). We get acontradiction from the regularity condition B ∗ n,j (cid:48) +1 ( s ), since L t j (cid:48) = L s j (cid:48) ≤ L s j (cid:48) +1 + (cid:15) n < L t j (cid:48) +1 − (cid:15) n ≤ L t j (cid:48) − (cid:15) n . (6.10)Case (iii): Assume w.l.o.g. that j > i . By F j ( t ), F j ( s ), F i ( t ) and F i ( s ), i and j have the same parity. By thesame argument as in case (ii), we have s i (cid:54) = t j . By the same argument as for Case (i), and by induction on l ,we have | s i − l − t j − l | < (cid:15) k + m − l for all l ≤ i , hence | s − t j − i | < (cid:15) k +2 . By the regularity condition G ( s ), theevent A n,j − i ( t ), hence (cid:101) E n ( t ), cannot occur, conditioned on (cid:101) E n ( s ). This gives us the desired contradiction,hence case (iii) cannot occur. Proposition 6.12. For any s , t ∈ D n , we have P ( (cid:101) E n ( t ) ∩ (cid:101) E n ( s )) ≤ (cid:107) t − s (cid:107) − − ( m − π θ + )+ o (cid:107) t − s (cid:107) (1) P ( (cid:101) E n ( t )) P ( (cid:101) E n ( s )) . (6.11) Proof. We will prove the assertion separately for the two cases of Lemma 6.11.Case (I): The 2( m − 1) events A n,j ( t ) and A n,j ( s ) are all independent, and by Lemma 6.6, P ( A n,j ( t )) = P ( A n,j ( s )) (cid:16) (cid:15) π θ n for j > , P ( A n, ( t )) = P ( A n, ( s )) (cid:16) (cid:15) n . Let ( B i ) i =1 ,..., (cid:98) ( m − / (cid:99) denote an ordering of the events B n,j ( t ), B n,j ( s ), where B j (cid:48) ( t (cid:48) ) comes before B j (cid:48)(cid:48) ( t (cid:48)(cid:48) ), t (cid:48) , t (cid:48)(cid:48) ∈ { t , s } , if t (cid:48) j (cid:48) − < t (cid:48)(cid:48) j (cid:48)(cid:48) − . By (6.2) of Lemma 6.7 with s = (cid:15) k + m +1 , P (cid:32) B j | (cid:32) m − (cid:92) i =0 A n,i ( t ) (cid:33) ∩ (cid:32) m − (cid:92) i =0 A n,i ( s ) (cid:33) ∩ (cid:32) j − (cid:92) i =1 B i (cid:33)(cid:33) (cid:16) (cid:40) (cid:15) n if B j = B n,i ( t ) or B j = B n,i ( s ) , i + 1 < m,(cid:15) n if B j = B n,i ( t ) or B j = B n,i ( s ) , i + 1 = m. It follows that P ( (cid:101) E n ( s ) ∩ (cid:101) E n ( t )) ≤ m − (cid:89) j =0 P ( A n,j ( t )) × m − (cid:89) j =0 P ( A n,j ( s )) × (cid:98) ( m − / (cid:99) (cid:89) j =1 P (cid:32) B j | (cid:32) m − (cid:92) i =0 A n,i ( t ) (cid:33) ∩ (cid:32) m − (cid:92) i =0 A n,i ( s ) (cid:33) ∩ (cid:32) j − (cid:92) i =1 B i (cid:33)(cid:33) (cid:16) (cid:15) m − π θ + ) n (cid:16) P ( (cid:101) E n ( s )) P ( (cid:101) E n ( t )) . Case (II): Assume w.l.o.g. that s < t , which implies s j < t j for all odd j and t j < s j for all even j . Since (cid:15) l + m +1 = (cid:15) o l (1) l , and by Lemma 6.6, Brownian scaling and the Markov property of Brownian motion, we have P ( A n,j ( t ) | A n,j ( s )) = (cid:40) (cid:15) − π θ + o l (1) l (cid:15) π θ n for j > ,(cid:15) − + o l (1) l (cid:15) n for j = 0 . By using this estimate, P ( A n,j ( s )) (cid:16) (cid:15) π θ n for j > P ( A n, ( s )) (cid:16) (cid:15) n , and that A n,j ( s ) and A n,j ( t ) areindependent of A n,i ( s ) and A n,i ( t ) for i (cid:54) = j , we get further P (cid:32)(cid:32) (cid:92) i even A n,i ( t ) (cid:33) ∩ (cid:32) m − (cid:92) i =0 A n,i ( s ) (cid:33)(cid:33) = (cid:15) − − π θ (cid:98) ( m − / (cid:99) + o l (1) l (cid:15) π θ ( m − (cid:98) ( m − / (cid:99) ) n . (6.12)47y (6.2) of Lemma 6.7 with s = (cid:15) k + m +1 , for j ∈ { , . . . , (cid:98) ( m − / (cid:99)} we have P B n,j ( s ) | (cid:32) m − (cid:92) i =0 A n,i ( s ) (cid:33) ∩ (cid:92) i even or i> j − A n,i ( t ) ∩ (cid:92) i : t i − >t j − B n,i ( t ) ∩ B n,i ( s ) (cid:16) (cid:40) (cid:15) n for 2 j + 1 < m,(cid:15) n for 2 j + 1 = m. (6.13)Next we will show that P B n,j ( t ) ∩ A n, j − ( t ) | (cid:32) m − (cid:92) i =0 A n,i ( s ) (cid:33) ∩ (cid:92) i even or i> j − A n,i ( t ) ∩ (cid:92) i : t i − >t j − B n,i ( t ) ∩ B n,i ( s ) ∩ B n,j ( s ) (cid:22) (cid:40) (cid:15) π θ +1 n (cid:15) − ( π θ +1)+ o l (1) l for 2 j + 1 < m,(cid:15) π θ + n (cid:15) − ( π θ + )+ o l (1) l for 2 j + 1 = m. (6.14)Let (cid:101) A n, j − ( t ) = (cid:110) X t j − − s ≥ X t j − − (cid:15) n , ∀ s ∈ (0 , (cid:15) l + m +1 / , ∀ X = { L, R } (cid:111) , and note that A n, j − ( t ) ⊂ (cid:101) A n, j − ( t ). By Lemma 6.6 P ( (cid:101) A n, j − ) = (cid:15) − π θ + o l (1) l (cid:15) π θ n . (6.15)Let s = (cid:15) l + m +1 / 2, and note that t j − − s j − ≥ s . Since both (cid:101) A n, j − ( t ) and the events we condition onin (6.14) are measurable with respect to the σ -algebra generated by1. ( L t , R t ) for t ≤ t j − − s and2. ( L t , R t ) − ( L t j − , R t j − ) for t ∈ ( t j − − s, t j − ),the estimate (6.3) from Lemma 6.7 implies that P B n,j ( t ) | (cid:32) m − (cid:92) i =0 A n,i ( t ) (cid:33) ∩ (cid:92) i even or i> j − A n,i ( s ) ∩ (cid:92) i>j B n,i ( t ) ∩ B n,i ( s ) ∩ B n,j ( s ) ∩ (cid:101) A n, j − ( t ) = (cid:40) (cid:15) n (cid:15) − o l (1) l for 2 j + 1 < m,(cid:15) n (cid:15) − + o l (1) l for 2 j + 1 = m. By using this estimate and (6.15), we obtain (6.14). By multiplying equations (6.13) and (6.14), taking theproduct over j ∈ { , . . . , (cid:98) ( m − / (cid:99)} , and then multiplying by (6.12), we get P ( (cid:101) E n ( t ) ∩ (cid:101) E n ( s )) ≤ (cid:15) − − ( m − π θ + )+ o l (1) l × (cid:15) m − π θ + ) n = (cid:107) t − s (cid:107) − − ( m − π θ + )+ o (cid:107) t − s (cid:107) (1) P ( (cid:101) E n ( t )) P ( (cid:101) E n ( s )) . Proof of lower bound in Theorem 6.3: By Lemma 6.10, Proposition 6.12 and [MWW16, Proposition 4.8] wehave P (dim H ( T P ( δ, m )) ≥ (cid:101) d ) > , (6.16)for any (cid:101) d < − ( m − π θ − ). [MWW16, Proposition 4.8] is stated for D ⊂ C instead of D ⊂ R m − , but theproof carries through in exactly the same manner if considering the domain D instead, only with 2 replaced48y ( m − 1) in the statement of the proposition. Also, [MWW16, Proposition 4.8] is stated for events thatare defined for all points in the domain, while our events (cid:101) E n ( t ) are only defined for t ∈ D n . We obtain newevents E (cid:48) n ( t ) defined for all t ∈ D as follows. Given t ∈ D we let s ∈ D n be the vector such that (cid:107) t − s (cid:107) is minimized, with some (unspecified) rule to break ties. We define E (cid:48) n ( t ) to be the event such that E (cid:48) n ( t )occurs if and only if (cid:101) E n ( s ) occurs.By the scale invariance and the Markov property of Brownian motion, the event of (6.16) almost surelyoccurs. By letting (cid:101) d → − ( m − π θ − ) it follows that, almost surely,dim H ( T P ( δ, m )) ≥ − ( m − (cid:18) π θ − (cid:19) . (6.17)By Lemma 6.9 and T ( δ, m ) ⊂ (cid:101) T ( m ) we get a lower bound for one of the two considered sets of Theorem 6.3:dim H ( (cid:101) T ( m )) ≥ − ( m − (cid:18) π θ − (cid:19) . Let Proj : R m − → R be projection t (cid:55)→ t , and define T P ( δ, m ) := Proj ( T P ( δ, m )). We will argue that a.s.dim H ( T P ( δ, m )) ≤ dim H ( T P ( δ, m )) . (6.18)Assume d, s > s (cid:28) 1, satisfy d − s > dim H ( T P ( δ, m )). Given any (cid:15) > 0, we can find a sequence of intervals( I i ) i ∈ N , such that T P ( δ, m ) ⊂ ∪ i ∈ N I i and (cid:80) i ∈ N | I i | d − s < (cid:15) . Let ( l i ) i ∈ N be such that | I i | ∈ ( (cid:15) l i +1 , (cid:15) l i ).Let s , t ∈ T P ( δ, m ), t (cid:54) = s . One of the two cases (I) or (II) of Lemma 6.11 must hold. Therefore we can find asequence ( (cid:101) I i ) i ∈ N of rectangles (cid:101) I i ⊂ R m − , such that Proj ( (cid:101) I i ) = I i , diam( (cid:101) I i ) (cid:22) (cid:15) l i − m +2 = | I i | o | Ii | (1) , and T P ( δ, m ) ⊂ ∪ i ∈ N (cid:101) I i . We have (cid:80) i ∈ N diam( (cid:101) I i ) d ≤ (cid:80) i ∈ N | I i | d − s < (cid:15) when max i | I i | is sufficiently small, hencefor all sufficiently small (cid:15) . It follows that dim H ( T P ( δ, m )) ≤ d , and by letting s → d → dim H ( T P ( δ, m )),we see that (6.18) holds almost surely. The lower bound for the second of the considered sets of Theorem 6.3,follows by dim H (cid:0)(cid:101) T ( m ) (cid:1) ≥ dim H (cid:0) T P ( δ, m ) (cid:1) ≥ dim H (cid:0) T P ( δ, m ) (cid:1) ≥ − ( m − (cid:18) π θ − (cid:19) . In this section we will list some open problems relating to the results of this paper.1. Let Z be a correlated Brownian motion as in (1.3), run for positive time, and let (cid:98) X be the set oftimes t ≥ Z . The Hausdorff dimension of (cid:98) X iscomputed in a very indirect manner in Example 2.8. Can one obtain this dimension directly? If so, onewould obtain a new proof of the dimension of the gasket of CLE κ (cid:48) for κ (cid:48) ∈ (4 , κ (cid:48) ∈ (4 , 8) and let Γ be a CLE κ (cid:48) in a simply connected domain D ⊂ C . We define the thin gasket ofΓ is to be the set T of points in D which are not disconnected from ∂D by any loop in Γ. Equivalently, T is the closure of the union of the outer boundaries of the outermost CLE loops, where the outerboundary of a loop (cid:96) is the boundary of the set of points disconnected from ∂D by (cid:96) . The thin gasketdiffers from the ordinary gasket of [MSW14, SSW09] in that the ordinary gasket includes points whichare disconnected from ∂D by some loop in Γ but which are not actually surrounded by this loop. Whatis the a.s. Hausdorff dimension of T ?One can make a reasonable guess as to what this dimension should be, as follows. Suppose we takeas our ansatz that the quantum scaling exponent ∆ of T in the KPZ formula is a linear functionof κ (cid:48) . This assumption seems reasonable, as it is satisfied with T replaced by the ordinary gasket, theSLE κ (cid:48) or SLE /κ (cid:48) curve, and the double and cut points of SLE. By SLE duality, as κ (cid:48) → 8, the outerboundaries of CLE loops start to look like SLE curves. Therefore we should expect dim H T → / (cid:48) → 8. On the other hand, as κ (cid:48) → 4, the thin gasket starts to look like the ordinary gasket, so bythe results of [NW11, SSW09], we should expect dim H T → / κ (cid:48) → 4. These guesses lead to theprediction that ∆ = κ (cid:48) which, in turn, by the KPZ formula this yields the prediction (16 − κ (cid:48) ) forthe dimension.In the peanosphere setting, consider the restriction of the whole-plane CLE κ (cid:48) associated with the curve η (cid:48) to some bubble U disconnected from ∞ by η (cid:48) . This restriction has the law of a CLE κ (cid:48) in U andits thin gasket can be described by an explicit (but rather complicated) functional of the Brownianmotion Z = ( L, R ). Such a description is implicit in [GM17b] since this paper describes the set ofpoints disconnected from ∂U by each CLE κ (cid:48) loop in U . One could obtain dim H T by computing theHausdorff dimension of this Brownian motion set and applying Theorem 1.1. Alternatively, one couldattempt to make rigorous an argument of the sort given in [DMS14, Section 11]. As a third possibility,one could take a direct approach, possibly using imaginary geometry [MS16d, MS16e, MS16a, MS13] toregularize the events in the two-point estimate as in [MW17, GMS17b].3. Can one describe various multifractal quantities associated with the SLE κ curve in terms of theBrownian motion Z = ( L, R ) in the peanosphere construction of [DMS14], such as the multifractalspectrum [GMS17b, Dup00], the winding spectrum [DB02, DB08], various notions of higher multifractalspectra [Dup03], the multifractal spectrum at the tip [JVL12], the optimal H¨older exponent [Lin08,JVL11], or the integral means spectrum [BS09a, GMS17b]? If so, can these quantities be computedusing Theorem 1.1 or some variant thereof? References [ABJ15] T. Alberts, I. Binder, and F. Johansson Viklund. A Dimension Spectrum for SLE BoundaryCollisions. ArXiv e-prints , January 2015, 1501.06212.[Aru15] J. Aru. KPZ relation does not hold for the level lines and SLE κ flow lines of the Gaussian freefield. Probab. Theory Related Fields , 163(3-4):465–526, 2015, 1312.1324. MR3418748[AS08] T. Alberts and S. Sheffield. Hausdorff dimension of the SLE curve intersected with the real line. Electron. J. Probab. , 13:no. 40, 1166–1188, 2008, 0711.4070. MR2430703 (2009e:60025)[Bef08] V. Beffara. The dimension of the SLE curves. Ann. Probab. , 36(4):1421–1452, 2008, math/0211322.MR2435854 (2009e:60026)[Ber96] J. Bertoin. L´evy processes , volume 121 of Cambridge Tracts in Mathematics . CambridgeUniversity Press, Cambridge, 1996. MR1406564 (98e:60117)[Ber99] J. Bertoin. Subordinators: examples and applications. In Lectures on probability theory andstatistics (Saint-Flour, 1997) , volume 1717 of Lecture Notes in Math. , pages 1–91. Springer,Berlin, 1999. MR1746300 (2002a:60001)[Ber15] N. Berestycki. Diffusion in planar Liouville quantum gravity. Ann. Inst. Henri Poincar´e Probab.Stat. , 51(3):947–964, 2015, 1301.3356. MR3365969[BGRV14] N. Berestycki, C. Garban, R. Rhodes, and V. Vargas. KPZ formula derived from Liouville heatkernel. ArXiv e-prints , June 2014, 1406.7280.[BJRV13] J. Barral, X. Jin, R. Rhodes, and V. Vargas. Gaussian multiplicative chaos and KPZ duality. Comm. Math. Phys. , 323(2):451–485, 2013, 1202.5296. MR3096527[BP15] C. J. Bishop and Y. Peres. Fractal sets in Probability and Analysis. In preparation, 2015.[BS09a] D. Beliaev and S. Smirnov. Harmonic measure and SLE. Comm. Math. Phys. , 290(2):577–595,2009, 0801.1792. MR2525631 (2011c:60265)50BS09b] I. Benjamini and O. Schramm. KPZ in one dimensional random geometry of multiplicativecascades. Comm. Math. Phys. , 289(2):653–662, 2009, 0806.1347. MR2506765 (2010c:60151)[DB02] B. Duplantier and I. Binder. Harmonic measure and winding of conformally invariant curves. Phys. Rev. Lett. , 89(264101), 2002, cond-mat/0208045.[DB08] B. Duplantier and I. Binder. Harmonic measure and winding of random conformal paths: ACoulomb gas perspective. Nucl.Phys. B , 802:494–513, 2008, 0802.2280.[DK88] B. Duplantier and K.-H. Kwon. Conformal invariance and intersections of random walks. Phys.Rev. Lett. , 61:2514–2517, Nov 1988.[DMS14] B. Duplantier, J. Miller, and S. Sheffield. Liouville quantum gravity as a mating of trees. ArXive-prints , September 2014, 1409.7055.[DRSV14] B. Duplantier, R. Rhodes, S. Sheffield, and V. Vargas. Renormalization of critical Gaussianmultiplicative chaos and KPZ relation. Comm. Math. Phys. , 330(1):283–330, 2014, 1212.0529.MR3215583[DS11] B. Duplantier and S. Sheffield. Liouville quantum gravity and KPZ. Invent. Math. , 185(2):333–393, 2011, 1206.0212. MR2819163 (2012f:81251)[Dub09] J. Dub´edat. Duality of Schramm-Loewner evolutions. Ann. Sci. ´Ec. Norm. Sup´er. (4) , 42(5):697–724, 2009, 0711.1884. MR2571956 (2011g:60151)[Dup98] B. Duplantier. Random walks and quantum gravity in two dimensions. Phys. Rev. Lett. ,81(25):5489–5492, 1998. MR1666816 (99j:83034)[Dup00] B. Duplantier. Conformally invariant fractals and potential theory. Phys. Rev. Lett. , 84(7):1363–1367, 2000, cond-mat/9908314.[Dup03] B. Duplantier. Higher conformal multifractality. J. Stat. Phys. , 110(3–6):691–738, 2003, cond-mat/0207743.[Eva85] S. N. Evans. On the Hausdorff dimension of Brownian cone points. Math. Proc. CambridgePhilos. Soc. , 98(2):343–353, 1985. MR795899 (86j:60185)[GHM16] E. Gwynne, N. Holden, and J. Miller. Dimension transformation formula for conformal mapsinto the complement of an SLE curve. ArXiv e-prints , March 2016, 1603.05161.[GHMS17] E. Gwynne, N. Holden, J. Miller, and X. Sun. Brownian motion correlation in the peanospherefor κ > Annales de l’Institut Henri Poincar´e , to appear, 2017, 1510.04687.[GHS16] E. Gwynne, N. Holden, and X. Sun. Joint scaling limit of a bipolar-oriented triangulation andits dual in the peanosphere sense. ArXiv e-prints , March 2016, 1603.01194.[GKMW16] E. Gwynne, A. Kassel, J. Miller, and D. B. Wilson. Active spanning trees with bending energyon planar maps and SLE-decorated Liouville quantum gravity for κ ≥ ArXiv e-prints , March2016, 1603.09722.[GM16] E. Gwynne and J. Miller. Convergence of the self-avoiding walk on random quadrangulations toSLE / on (cid:112) / ArXiv e-prints , August 2016, 1608.00956.[GM17a] E. Gwynne and J. Miller. Convergence of percolation on uniform quadrangulations with boundaryto SLE on (cid:112) / ArXiv e-prints , January 2017, 1701.05175.[GM17b] E. Gwynne and J. Miller. Convergence of the topology of critical Fortuin-Kasteleyn planar mapsto that of CLE κ on a Liouville quantum surface. In preparation, 2017.[GM17c] E. Gwynne and J. Miller. Scaling limit of the uniform infinite half-plane quadrangulation in thegromov-hausdorff-prokhorov-uniform topology. Electron. J. Probab. , 22:1–47, 2017, 1608.00954.51GMS15] E. Gwynne, C. Mao, and X. Sun. Scaling limits for the critical Fortuin-Kasteleyn model on arandom planar map I: cone times. ArXiv e-prints , February 2015, 1502.00546.[GMS17a] E. Gwynne, J. Miller, and S. Sheffield. The Tutte embedding of the mated-CRT map convergesto Liouville quantum gravity. ArXiv e-prints , May 2017, 1705.11161.[GMS17b] E. Gwynne, J. Miller, and X. Sun. Almost sure multifractal spectrum of SLE. Duke MathematicalJournal , to appear, 2017, 1412.8764.[GRV14] C. Garban, R. Rhodes, and V. Vargas. On the heat kernel and the Dirichlet form of LiouvilleBrownian motion. Electron. J. Probab. , 19:no. 96, 25, 2014, 1302.6050. MR3272329[GRV16] C. Garban, R. Rhodes, and V. Vargas. Liouville Brownian motion. Ann. Probab. , 44(4):3076–3110,2016, 1301.2876. MR3531686[GS15] E. Gwynne and X. Sun. Scaling limits for the critical Fortuin-Kasteleyn model on a randomplanar map III: finite volume case. ArXiv e-prints , October 2015, 1510.06346.[GS17] E. Gwynne and X. Sun. Scaling limits for the critical Fortuin-Kasteleyn model on a random planarmap II: local estimates and empty reduced word exponent. Electronic Jorunal of Probability ,22:Paper No. 45, 1–56, 2017, 1505.03375.[HMP10] X. Hu, J. Miller, and Y. Peres. Thick points of the Gaussian free field. Ann. Probab. , 38(2):896–926, 2010, 0902.3842. MR2642894 (2011c:60117)[HP74] J. Hawkes and W. E. Pruitt. Uniform dimension results for processes with independent increments. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete , 28:277–288, 1973/74. MR0362508 (50 ArXiv e-prints , October2016, 1610.05272.[Jac14] H. Jackson. Liouville Brownian Motion and Thick Points of the Gaussian Free Field. ArXive-prints , December 2014, 1412.1705.[JVL11] F. Johansson Viklund and G. F. Lawler. Optimal H¨older exponent for the SLE path. DukeMath. J. , 159(3):351–383, 2011, 0904.1180. MR2831873[JVL12] F. Johansson Viklund and G. F. Lawler. Almost sure multifractal spectrum for the tip of anSLE curve. Acta Math. , 209(2):265–322, 2012, 0911.3983. MR3001607[KMSW15] R. Kenyon, J. Miller, S. Sheffield, and D. B. Wilson. Bipolar orientations on planar maps andSLE . ArXiv e-prints , November 2015, 1511.04068.[KPZ88] V. Knizhnik, A. Polyakov, and A. Zamolodchikov. Fractal structure of 2D-quantum gravity. Modern Phys. Lett A , 3(8):819–826, 1988.[Le 13] J.-F. Le Gall. Uniqueness and universality of the Brownian map. Ann. Probab. , 41(4):2880–2960,2013, 1105.4842. MR3112934[Lin08] J. R. Lind. H¨older regularity of the SLE trace. Trans. Amer. Math. Soc. , 360(7):3557–3578,2008. MR2386236 (2009f:60048)[LSW01a] G. F. Lawler, O. Schramm, and W. Werner. Values of Brownian intersection exponents. I.Half-plane exponents. Acta Math. , 187(2):237–273, 2001. MR1879850 (2002m:60159a)[LSW01b] G. F. Lawler, O. Schramm, and W. Werner. Values of Brownian intersection exponents. II. Planeexponents. Acta Math. , 187(2):275–308, 2001. MR1879851 (2002m:60159b)[LSW02] G. F. Lawler, O. Schramm, and W. Werner. Values of Brownian intersection exponents. III.Two-sided exponents. Ann. Inst. H. Poincar´e Probab. Statist. , 38(1):109–123, 2002. MR1899232(2003d:60163) 52LSW03] G. Lawler, O. Schramm, and W. Werner. Conformal restriction: the chordal case. J. Amer.Math. Soc. , 16(4):917–955 (electronic), 2003, math/0209343v2. MR1992830 (2004g:60130)[LSW17] Y. Li, X. Sun, and S. S. Watson. Schnyder woods, SLE(16), and Liouville quantum gravity. ArXiv e-prints , May 2017, 1705.03573.[Mie13] G. Miermont. The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. , 210(2):319–401, 2013, 1104.1606. MR3070569[Moo28] R. L. Moore. Concerning upper semi-continuous collections of continua. Trans. Amer. Math.Soc. , 27(4):416–428, 1928.[MP10] P. M¨orters and Y. Peres. Brownian motion . Cambridge Series in Statistical and ProbabilisticMathematics. Cambridge University Press, Cambridge, 2010. With an appendix by OdedSchramm and Wendelin Werner. MR2604525 (2011i:60152)[MS13] J. Miller and S. Sheffield. Imaginary geometry IV: interior rays, whole-plane reversibility, andspace-filling trees. Probab. Theory Related Fields , to appear, 2013, 1302.4738.[MS15a] J. Miller and S. Sheffield. An axiomatic characterization of the Brownian map. ArXiv e-prints ,June 2015, 1506.03806.[MS15b] J. Miller and S. Sheffield. Liouville quantum gravity and the Brownian map I: The QLE(8/3,0)metric. ArXiv e-prints , July 2015, 1507.00719.[MS15c] J. Miller and S. Sheffield. Liouville quantum gravity spheres as matings of finite-diameter trees. ArXiv e-prints , June 2015, 1506.03804.[MS16a] J. Miller and S. Sheffield. Imaginary geometry III: reversibility of SLE κ for κ ∈ (4 , Ann.Math. , 184(2):455–486, 2016, 1201.1498.[MS16b] J. Miller and S. Sheffield. Liouville quantum gravity and the Brownian map II: geodesics andcontinuity of the embedding. ArXiv e-prints , May 2016, 1605.03563.[MS16c] J. Miller and S. Sheffield. Liouville quantum gravity and the Brownian map III: the conformalstructure is determined. ArXiv e-prints , August 2016, 1608.05391.[MS16d] J. Miller and S. Sheffield. Imaginary geometry I: interacting SLEs. Probab. Theory RelatedFields , 164(3-4):553–705, 2016, 1201.1496. MR3477777[MS16e] J. Miller and S. Sheffield. Imaginary geometry II: Reversibility of SLE κ ( ρ ; ρ ) for κ ∈ (0 , Ann. Probab. , 44(3):1647–1722, 2016, 1201.1497. MR3502592[MS16f] J. Miller and S. Sheffield. Quantum Loewner evolution. Duke Math. J. , 165(17):3241–3378, 2016,1312.5745. MR3572845[MSW14] J. Miller, N. Sun, and D. B. Wilson. The Hausdorff dimension of the CLE gasket. Ann. Probab. ,42(4):1644–1665, 2014, 1403.6076. MR3262488[MSW17] J. Miller, S. Sheffield, and W. Werner. CLE percolations. Forum Math. Pi , 5:e4, 102, 2017,1602.03884. MR3708206[MW17] J. Miller and H. Wu. Intersections of SLE Paths: the double and cut point dimension of SLE. Probab. Theory Related Fields , 167(1-2):45–105, 2017, 1303.4725. MR3602842[MWW15] J. Miller, S. S. Watson, and D. B. Wilson. The conformal loop ensemble nesting field. Probab.Theory Related Fields , 163(3-4):769–801, 2015, 1401.0217. MR3418755[MWW16] J. Miller, S. S. Watson, and D. B. Wilson. Extreme nesting in the conformal loop ensemble. Ann. Probab. , 44(2):1013–1052, 2016, 1401.0218. MR347446653NW11] S. Nacu and W. Werner. Random soups, carpets and fractal dimensions. J. Lond. Math. Soc. ,83(2):789–809, 2011.[PT69] W. E. Pruitt and S. J. Taylor. Sample path properties of processes with stable components. Z.Wahrscheinlichkeitstheorie verw. Geb. , 12:267–289, 1969.[RS05] S. Rohde and O. Schramm. Basic properties of SLE. Ann. of Math. (2) , 161(2):883–924, 2005,math/0106036. MR2153402 (2006f:60093)[RV11] R. Rhodes and V. Vargas. KPZ formula for log-infinitely divisible multifractal random measures. ESAIM Probab. Stat. , 15:358–371, 2011, 0807.1036. MR2870520[RV14] R. Rhodes and V. Vargas. Gaussian multiplicative chaos and applications: A review. Probab.Surv. , 11:315–392, 2014, 1305.6221. MR3274356[RV15] R. Rhodes and V. Vargas. Liouville Brownian motion at criticality. Potential Anal. , 43(2):149–197,2015, 1311.5847. MR3374108[Sch00] O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J.Math. , 118:221–288, 2000, math/9904022. MR1776084 (2001m:60227)[She09] S. Sheffield. Exploration trees and conformal loop ensembles. Duke Math. J. , 147(1):79–129,2009, math/0609167. MR2494457 (2010g:60184)[She16a] S. Sheffield. Conformal weldings of random surfaces: SLE and the quantum gravity zipper. Ann.Probab. , 44(5):3474–3545, 2016, 1012.4797. MR3551203[She16b] S. Sheffield. Quantum gravity and inventory accumulation. Ann. Probab. , 44(6):3804–3848, 2016,1108.2241. MR3572324[Shi85] M. Shimura. Excursions in a cone for two-dimensional Brownian motion. J. Math. Kyoto Univ. ,25(3):433–443, 1985. MR807490 (87a:60095)[Shi88] M. Shimura. Meandering points of two-dimensional Brownian motion. Kodai Math. J. , 11:169–176,1988.[SS13] O. Schramm and S. Sheffield. A contour line of the continuum Gaussian free field. Probab.Theory Related Fields , 157(1-2):47–80, 2013, math/0605337. MR3101840[SSW09] O. Schramm, S. Sheffield, and D. B. Wilson. Conformal radii for conformal loop ensembles. Comm. Math. Phys. , 288:43–53, 2009.[SW05] O. Schramm and D. B. Wilson. SLE coordinate changes. New York J. Math. , 11:659–669(electronic), 2005, math/0505368. MR2188260 (2007e:82019)[SW12] S. Sheffield and W. Werner. Conformal loop ensembles: the Markovian characterization and theloop-soup construction. Ann. of Math. (2) , 176(3):1827–1917, 2012, 1006.2374. MR2979861[WW15] M. Wang and H. Wu. Remarks on the intersection of SLE κ ( ρ ) curve with the real line. ArXive-prints , July 2015, 1507.00218.[Zha08] D. Zhan. Duality of chordal SLE. Invent. Math. , 174(2):309–353, 2008, 0712.0332. MR2439609(2010f:60239)[Zha10] D. Zhan. Duality of chordal SLE, II.