A note on the Brownian loop measure

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1 A note on the Brownian loop measure Gregory F. Lawler Department of Mathematis University of Chiago Marh 23, 2009 Abstrat The Brownian loop measure is a onformally invariant measure on loops that also satisfies the restrition property. In studying the Shramm-Loewner evolution (SLE), a quantity that arises is the measure of loops in a domain D that interset both V and V 2. If V, V 2 are nonpolar, and D = C this measure is infinite. We show the existene of a finite normalized quantity that an be used in its plae. The motivation for studying this question omes from bulk SLE with boundary onditions, but this paper only disusses the loop measure. Introdution The Brownian loop measure [4] in C is a sigma-finite measure on unrooted loops in C that satisfies two important properties: onformal invariane and the restrition property. It arose in the study of the Shramm-Loewner evolution (SLE). An important quantity for SLE is Λ(V,V 2 ;D), whih denotes the measure of the set of loops in a domain D that interset both V and V 2. This quantity omes up in omparison of SLE in two different domains. If µ D (z,z 2 ) denotes the hordal SLE κ (κ 4) measure for z to z 2 in a domain D and D D is a subdomain that agrees with D in neighborhoods of z and z 2, then dµ D(z,z 2 ) { } dµ D (z,z 2 ) (γ) = exp Λ(γ,D \ D;D), 2 Researh supported by National Siene Foundation grant DMS

2 where (3κ 8)(6 κ) = 2κ denotes the entral harge. If V,V 2 are nonpolar disjoint losed subsets of the Riemann sphere and D is a domain whose boundary is nonpolar, then this quantity is positive and finite (see Corollary 4.6). It is also onformally invariant in the sense that if f : D f(d) is a onformal transformation, then Λ(f(V ),f(v 2 );f(d)) = Λ(V,V 2 ;D). In trying to generalize these ideas to SLE in the bulk, see [2], one is tempted to write similar quantities of the form Λ(V,V 2 ; C). However this quantity as defined is infinite. The purpose of this note is to prove the existene of a normalized quantity Λ (V,V 2 ) that has many of the properties one wants. Theorem.. If V,V 2 are disjoint nonpolar losed subsets of the Riemann sphere, then the limit exists where Λ (V,V 2 ) = lim r 0+ [Λ(V,V 2 ; O r ) log log(/r)], () O r = {z C : z > r}. Moreover, if f is a Möbius transformation of the Riemann sphere, Λ (f(v ),f(v 2 )) = Λ (V,V 2 ). We ould write the assumption disjoint nonpolar losed subsets of the Riemann sphere as disjoint losed subsets of C, at least one of whih is ompat, suh that Brownian motions hits both subsets at some positive time. Invariane of Λ under Möbius transformations implies that the definition () does not hange if we shrink down at a point on the Riemann sphere other than the origin. In other words if O r (z) = z + O r and D R denotes the open disk of radius R about the origin, Λ (V,V 2 ) = lim r 0+ [Λ(V,V 2 ; O r (z)) log log(/r)], (2) 2

3 Λ (V,V 2 ) = lim R [Λ(V,V 2 ; D R ) log log R]. (3) The goal of this paper is to prove Theorem.. Theorem 4.7 establishes the existene of the limit in (). Theorem 4. proves the alternate forms (2) and (3). If f is a Möbius transformation, then onformal invariane of the loop measure implies Λ(V,V 2 ; O r ) = Λ(f(V ),f(v 2 );f(o r )). Invariane of Λ under dilations, translations, and inversions an be dedued from this and (), (2), and (3), respetively. The proof really only uses standard arguments about planar Brownian motion but we need to ontrol the error terms. In order to make the paper easier to understand, we have split it into three setions. The first setion onsiders estimates for planar Brownian motion. Readers who are well aquainted with planar Brownian motion may wish to skip this setion and refer bak as neessary. This setion does assume knowledge of planar Brownian motion as in [, Chapter 2]. The next setion disusses the Brownian (boundary) bubble measure and gives estimates for it. The Brownian loop measure is a measure on unrooted loops, but for omputational purposes it is often easier to assoiate to eah unrooted loop a partiular rooted loop yielding an expression in terms of Brownian bubbles. The last setion proves the main theorem by giving estimates for the loop measure. We will use the following notation: D r = {z : z < r}, D = D, O r = {z : z > r}, O = O, O r (w) = w + O r, A r,r = D R O r = {z : r < z < R}, A R = A,R, C r = D r = O r = { z = r}, C r (w) = O r (w) = {z : w z = r}. We say that a subset V of C is nonpolar if it is hit by Brownian motion. More preisely, V is nonpolar if for every z C, the probability that a Brownian motion starting at z hits V is positive. Sine Brownian motion is reurrent we an replae is positive with equals one. For onveniene, we will all a domain (onneted open subset) D of C nonpolar if D is nonpolar. 3

4 2 Lemmas about Brownian motion If B t is a omplex Brownian motion and D is a domain, let τ D = inf{t : B t D}. A domain D is nonpolar if and only if P z {τ D < } = for every z. In this ase we define harmoni measure of D at z D by h D (z,v ) = P z {B τd V }. If V is smooth then we an write h D (z,v ) = V h D (z,w) dw, where h D (z,w) is the Poisson kernel. If z D \ V and D is smooth near z, we define the exursion measure of V in D from z by E D (z,v ) = E(z,V ;D) = n h D (z,v ), where n = n z,d denotes the unit inward normal at z. If V is smooth, we an write E D (z,v ) = h D (z,w) dw, V where h D (z,w) := n h D (z,w) is the exursion or boundary Poisson kernel. (Here the derivative n is applied to the first variable.) One an also obtain the exursion Poisson kernel as the normal derivative in both variables of the Green s funtion; this establishes symmetry, h D (z,w) = h D (w,z). If f : D f(d) is a onformal transformation, then (assuming smoothness of f at boundary points at whih f is taken) h D (z,v ) = h f(d) (f(z),f(v )), h D (z,w) = f (w) h f(d) (f(z),f(w)), E D (z,v ) = f (w) E f(d) (f(z),f(v )), h D (z,w) = f (z) f (w) h f(d) (f(z),f(w)). The exat form of the Poisson kernel in the unit disk shows that there is a suh that for all z /2, w = 2π h D (z,w) z. 4

5 By taking an inversion, we get that if z 2, 2π h O (z,w) z. (4) It is standard that h AR (z,c R ) = log z, < z < R. (5) log R In partiular, E AR (,C R ) = log R, E A R (R,C ) = R log R, (6) If V D is smooth, let h D (z,w;v ) = h D (z,w)/h D (z,v ) for w V. In other words, h D (z,w;v ) is the density of the exit distribution of a Brownian motion onditioned so that it exits at V. We similarly define h D (z,w;v ). Lemma 2.. There exists < suh that the following holds. Suppose R > 0 and D is a domain with A R D O. Then 2π h D (z,w) log R R, w =,z D O R/2. (7) Remark The onlusion of this lemma is very reasonable. If a Brownian motion starting at a point z far from the origin exits D at C, then the hitting distribution is almost uniform. This uses the fat that D D R is the same as A R. The important result is the estimate of the error term. Proof. Assume w =. Let τ = τ D and let = D O. It suffies to prove the estimate for z = R/2. For every ζ R/2, (4) gives Note that Using (8), we get 2πh O (ζ,w) R. (8) h O (z,w) = h D (z,w) + E z [h O (B τ,w);b τ ]. 2πE z [h O (B τ,w);b τ ] = h D (z, )[ + O(R )]. Therefore, 2π h D (z,w) = h D (z,c ) + O(R ). (9) 5

6 Sine h D (z,c ) is bounded below by the probability of reahing C before C R, (5) implies h D (z,c ) log 2 log R, and hene (9) implies 2π h D (z,w) = h D (z,c ) [ + O ( log R R )]. (0) Corollary 2.2. There exists < suh that if R 2, z =, w = R, Then h (z,w) AR 2πR log R R2. () Proof. Reall that h Ar (z,w) = h Ar (w,z). We know from (6) that h AR (w,ζ) dζ = C R log R. Also, by definition, h AR (w,z) = h A R (w,z) R log R. Note that h AR (w,z) is bounded by the minimum and maximum values of h AR (ŵ,z) over ŵ = R/2, and hene (7) gives h AR (ŵ,z) = ( ) log R 2π + O. R Corollary 2.3. Suppose R >,D is a domain ontaining D R, ˆD is a nonpolar domain ontaining O with smooth boundary and D = D ˆD. For w ˆD, let q w = 2π h D(e iθ,w)dθ, 2π 0 and let q = ˆD q w dw = 2π h D(e iθ, 2π ˆD)dθ, 0 Then if z D with z R/2 and w D, h D(z,w) = q w q h D O(z,C ) [ + O ( log R R )]. 6

7 Remark The impliit onstants in the O( ) term are uniform and do not depend on z,w,d, D. The key fat is that (up to the error term) h D(z,w) fators into two terms q w /q and h D O (z,c ). Proof. Let τ = τ D and Let σ = τ D O. Then (0) implies that if z D, z R/2, [ ( )] h D(z, ˆD) log R = q h D O (z,c ) + O. R Similarly, h D(z,w) = q w h D O (z,c ) [ + O ( log R R )]. Lemma 2.4. Suppose D is a nonpolar domain ontaining D. If 0 < s <, let D s = D O s. Then if s < r /2 and z = r, log r log s h D s (z,c s ) log r log s [ ] p log 2, ( p)log(/r) where p = p D = sup h D/2 ( w,c /2 ) <. w = Remark The inequality p D < follows immediately from the fat that D is nonpolar and ontains D. Proof. Let T = T s = inf{t : B t C s C } and σ = σ s,r = inf{t T : B t C r }. Then if z /2, By (5), P z {B τds C s } = P z {B T C s } + P z {B T C,B τds C s }. whih gives the lower bound. Let P z {B T C s } = log r log s, q = q(r,s,d) = sup P z { } B τds C s, z =r u = u(r,d) = sup P w { } B τdr C r. w = 7

8 Then, P z {B T C,B τds C s } P z {σ < τ Ds B T C }P z {B τds C s B T C,σ < τ Ds } uq. Applying this to the maximizing z, gives q log r log s + uq, q log r ( u) log s. By (5), the probability that a Brownian motion starting on C /2 reahes C r before reahing C is log 2/log(/r). Using a similar argument as in the previous paragraph, we see that u p log 2 log(/r) + p u, u p p log 2 log(/r). (2) Proposition 2.5. Suppose D is a nonpolar domain ontaining the origin. Then there exists = D < suh that if 0 < r /2, D r = D O r, and z D, z, h Dr (z,c r ) log(/r). (3) Also, if w = r, h Dr (z,w) r log(/r). (4) Proof. Find 0 < β < /2 suh that D O 2β is nonpolar. It suffies to prove (3) for r < β. Sine D O 2β is nonpolar, there exists q = q D,β > 0 suh that for every z 2β, the probability that a Brownian motion starting at z leaves D before reahing C β is at least q. If r < β, the probability that a Brownian motion starting at C β reahes C r before reahing C 2β is p(r) = log 2/log(2β/r) log(/r). Let Q(r) = sup z 2β h Dr (z,c r ). Then arguing similarly to the previous proof, we have Q(r) ( q) [p(r) + [ p(r)]q(r)] p(r) + ( q)q(r), whih yields Q(r) p(r)/q. This gives (3) and (4) follows from h Dr (z,w) h D2r (z,c 2r ) sup h Or (ζ,w). ζ =2r 8

9 Proposition 2.6. There exists < suh that the following holds. Suppose z = /2 and 0 < s < r < /8. Let D s,r = O s O r (z). Then for w, log(rs) log r h D s,r (w,c s ) log(/r). (5) Proof. Without loss of generality, we assume z = /2. Let L denote the line {x + iy : x = /4}. Let τ = τ Ds,r, T the first time a Brownian motion reahes C r C r (z), and σ the first time after T that the Brownian motion returns to L. By symmetry, for every w L, Using Lemma 2.4, we get P w {B T C r } = 2. P w {τ < σ B T C r } = log r log s Therefore, for every w L, [ ( + O P w {τ < σ;b τ O r (z)} = 2, P w {τ < σ;b τ O s } = log r 2log s [ ( + O log(/r) log(/r) )]. )]. This establishes (5) for w L. If w, then the probability of reahing O r (z) before reahing L is O(/log(/r)) and the probability of reahing O s before reahing L is O(/log(/s)). Using this we get (5) for w. Remark The end of the proof uses a well known fat. Suppose one performs independent trials with three possible outomes with probabilities p, q, p q, respetively. Then the probability that an outome of the first time ours before one of the seond type is p/(p + q). 3 Brownian bubble measure If D is a nonpolar domain and z D is an analyti boundary point (i.e., D is analyti in a neighborhood of D), the Brownian bubble measure m D (z) of D at z is a sigma-finite measure on loops γ : [0,t γ ] C with γ(0) = γ(t γ ) = z and γ(0,t γ ) D. It an be defined as the limit as ǫ 0+ of π ǫ h D (z + ǫn,z) times the probability measure on paths obtained from 9

10 starting a Brownian motion at z + ǫn and onditioning so that the path leaves D at z. Here n = n z,d is the inward unit normal. If D D agrees with D in a neighborhood of z, then the bubble measure at D, m D(z) is obtained from m D (z) by restrition. This is also an infinite measure but the differene m D (z) m D(z) is a finite measure. We will denote its total mass by m(z;d, D) = m D (z) m D(z). The normalization of m is hosen so that m(0; H, H D) =. (6) Remark The fator of π in the bubble measure was put in so that (6) holds. However, the loop measure in the next setion does not have this fator so we will have to divide it out again. For this paper, it would have been easier to have defined the bubble measure without the π but we will keep it in order to math definitions elsewhere. From the definition, we see that if D D is smooth m(z;d, D) = π n h D(z,w)h D (w,z) dw. D D This is also equal to π n f(z) for the funtion f(ζ) = h D (ζ,z) h D(ζ,z). Let h D, (V,z),h D,+ (V,z) denote the infimum and supremum, respetively, of h D (w,z) over w V. Then a simple estimate is h D, ( D D,z) Lemma 3.. If R >, let m(z;d, D) π E D(z, D D) h D,+( D D,z). (7) ρ(r) = m(; O,A R ). There exists < suh that for all R 2, ρ(r) 2log R R log R. Remark Rotational invariane implies that m(z; O,A R ) = ρ(r) for all z =. 0

11 Proof. By (6), and by (4), We now use (7). E(; O,A R ) = log R. 2π h O (w,) = + O(R ). The next lemma generalizes this to domains D with A R D O. The result is similar but the error term is a little larger. Note that the q in the next lemma equals if D = O. Lemma 3.2. Suppose R 2 and D is a domain satisfying A R D O. Let q be the probability that a Brownian motion started uniformly on C R exits D at C, i.e., Then if w =, q = q(r,d) = 2πR m(w;d,a R ) = q 2 log R m(w; O,D) = q 2 log R + O C R h D (z,c ) dz. [ + ( log R R ( q log R + R log R Proof. By definition, m(w;d,a R ) = π h AR (w,z)h D (z,w) dz. C R By (), we know that By (7), we know that h AR (w,z) = 2π R log R h D (z,w) = 2π h D(z,C ) [ + O [ + O ( log R R ( log R R )], (8) ). (9) )]. )]. Combining these gives (8), and (9) follows from Lemma 3. and m(w; O,A R ) = m(w;d,a R ) + m(w; O,D).

12 Corollary 3.3. There exists < suh that the following is true. Suppose R 2 and D is a domain with A R D O. Suppose D O R is nonpolar and hene p = p R,D := sup h D\OR/2 (z,c R/2 ) <. z =R Then, if w =, m(w; O,D) 2 log R ( p) log 2 R. Proof. Let q be as in the previous lemma. By (2) we see that q and hene the result follows from (9). p log 2 ( p) log R. Remark We will used saled versions of this orollary. For example, if D is a nonpolar domain ontaining D, r < /2, D r = D O r, and w = r, r2 m(w; O r,d r ) 2 log(/r) ( p)log 2 (/r), where p = sup h D/2 (z,c /2 ). z = Proposition 3.4. Suppose V is a nonpolar losed set, z 0, and z, 0 V. For 0 < r,s <, let D s,r = O s O r (z). Then as s,r 0+, if w = s, π m(w;d s,r,d s,r \ V ) = 2πs 2 log(/s) where δ r,s = (log(/r)) + (log(/s)). log r log(rs) [ + O (δ r,s)], Remark The impliit onstants in the O( ) term depend on V,z but not on w. 2

13 Proof. We will use (7) and write δ = δ r,s. By saling we may assume z = 2 and let d = min{2,dist(0,v ),dist(2,v )}. We will only onsider r,s d/2. By (6), E(w,C d ;A s,d ) = s E(w/s,C d/s ;A d/s ) = Using this and (5) we an see that For ζ V, (5) gives Therefore. by (7) E(w,V ;D s,r \ V ) = s log(/s) h Ds,r (ζ,c s ) = log r log(rs) h Ds,r (ζ,w) = log r 2πs log(rs) s log(/s) [ + O(δ)]. [ + O(δ)]. [ + O(δ)]. (20) [ + O(δ)]. The next proposition is the analogue of Proposition 3.4 with z =. Proposition 3.5. Suppose V is a nonpolar ompat set, with 0 V. For 0 < s,r <, let D s,r = O s D /r. Then, for as s,r 0+, if w = s, π m(w;d s,r,d s,r \ V ) = 2πs 2 log(/s) where δ r,s = (log(/r)) + (log(/s)). log r log(rs) [ + O (δ r,s)], Proof. The proof is the same as the previous proposition. In fat, it is slighly easier beause (20) is justified by (5). 3

14 4 Brownian loop measure The Brownian loop measure is a measure on unrooted loops. It is this measure that is onformally invariant. For omputational purposes it is useful to write the measure in terms of the bubble measure. The following expression is obtained by assigning to eah unrooted loop the root losest to the origin, see [2]. µ = µ C = π 2π 0 0 m Or (re iθ )r dr dθ. (2) To be preise, we are onsidering the right hand side as a measure on unrooted loops. If D is a subdomain, then µ D is defined by restrition. If D r D, then the Brownian loop measure in D restrited to loops that interset D r an be written as π 2π r 0 0 m Ds (se iθ )sdr dθ, (22) where D s = D O s. If r < r, then the Brownian loop measure of loops in O r that interset D r is given by 2π r m Ds (se iθ )sdr dθ. π 0 r Using this and appropriate properties of the bubble measure we an onlude the following. Lemma 4.. For every 0 < s < r < and d > 0, the loop measure of the set of loops in O s of diameter at least d that interset D r is finite. Remark This result is not true for s = 0. The Brownian loop measure of loops in C of diameter greater than d that interset the unit disk is infinite. See, e.g., Lemma 4.3 below. Conformal invariane implies that the Brownian loop measure of loops in A r,2r that separate the origin from infinity is the the same for all r. It is easy to see that this measure is positive and the last lemma shows that it is finite. It follows that the measure of the set of loops that surround the origin is infinite. If V,V 2,... are losed subsets of the Riemann sphere and D is a nonpolar domain, then Λ(V,V 2,...,V k ;D) 4

15 is defined to be the loop measure of the set of loops in D that interset all of V,...,V k. Note that Λ(V,V 2,...,V k ;D) = Λ(V,V 2,...,V k+ ;D) + Λ(V,V 2,...,V k ;D \ V k+ ). (23) If V,V 2,...,V k are the traes of simple urves that inlude the origin, then the omment in the last paragraph shows that for all r > 0, Λ(V,V 2,...,V k ; D r ) =. Lemma 4.2. Suppose V,V 2 are losed sets and D is a domain. Let Then V j = A e j,e j, Oj = O e j, D j = D O j. Λ(V,V 2 ;D) = j= Λ(V,V 2,V j+ ;D j ). (24) Proof. For eah unrooted loop, onsider the point on the loop losest to the origin. The measure of the set of loops for whih the distane to the origin is exatly e j for some integer j is 0. For eah loop, there is a unique j suh that the loop is in O j but not in O j+. Exept for a set of loops of measure zero, suh a loop intersets V j+ but does not interset V k for k < j+, and hene eah loop is ounted exatly one on the right-hand side of (24). Lemma 4.3. There exists < suh that if 0 < s <,R 2, [ ] log(r/s) Λ(C,C R ; O s ) log log R R log R. In partiular, there exists < suh that if R 2/s > 2, Λ(C,C R ; O s ) log(/s) log R log2 (/s) log 2 R. Proof. By (22), rotational invariane, and the saling rule, we get Λ(C,C R ; O s ) = 2 s r m(r; O r,a r,r )dr = 2 s r ρ(r/r)dr, where ρ is as in Lemma 3.. From that lemma, we know that ( ) ρ(r/r) = 2log(R/r) + O r, R log(r/r) 5

16 and hene Λ(C,C R ; O s ) = O ( ) + R log R s r (log R log r) dr. The first assertion follows by integrating and the seond from the expansion [ ] log(r/s) log = log(/s) ( log 2 ) log R log R + O (/s) log 2. R Lemma 4.4. Suppose V is a losed, nonpolar set with 0 V and α > 0. There exists = V,α < suh that for r < min{,dist(0,v )}/4, Λ(V, O αr \ O r ; O r ) log α log(/r) log 2 (/r). Proof. By saling, we may assume that dist(0,v ) =. It suffies to prove the result for r suffiiently small. By (22), we have Λ(V, O αr \ O r ; O r ) = π 2π αr 0 r m(se iθ ; O s,d s )sds dr, where D s = O s \ V. By Corollary 3.3, for r s αr, [ ( m(se iθ ; O s,d s ) = 2s 2 + O log(/s) Therefore, Λ(V, O αr \ O r ; O r ) = [ ( + O log(/r) )] αr log(/r) r )]. ds s log(/s). Also, αr r ds s log(/s) ( ) = log log r log α = log(/r) + O ( log log αr ( log 2 (/r) ) ). 6

17 Lemma 4.5. Suppose V,V 2 are disjoint, nonpolar losed subsets of the Riemann sphere with 0 V. Then there exists = V,V 2 < suh that for all r dist(0,v )/2, Λ(V,C r ; C \ V 2 ) log(/r). (25) Proof. Constants in this proof depend on V,V 2. Without loss of generality assume 0 V 2 and let D r = O r \ V 2. We will first prove the result for r r 0 = [dist(0,v ) dist(0,v 2 )]/2. By (22), we have Λ(V,C r ; C \ V 2 ) = m(z;d π z,d z \ V )da(z). (26) By (4), h Dr (w,z) z r By omparison with an annulus, we get Using (7), we then have E D z \V (z,v ) r log(/r), w V, z = r. π m(z;d z,d z \ V ) By integrating, we get (25) for r r 0. Let r = dist(0,v )/2 and note that, z = r. z log(/ z ) z 2 log 2 (/ z ). Λ(V,C r ; C \ V 2 ) Λ(V,C r0 ; C \ V 2 ) + Λ(V,C r ; O r0 \ V 2 ). Using Lemma 4. we an see that Therefore, Λ(V,C r ; O r0 \ V 2 ) <. Λ(V,C r ; C \ V 2 ) <, and we an onlude (25) for r 0 r r with a different onstant. Corollary 4.6. Suppose V,V 2 are disjoint losed subsets of the Riemann sphere and D is a nonpolar domain. Then Λ(V,V 2 ;D) <. 7

18 Proof. Assume 0 V. Lemma 4.5 shows that Λ(V, D s ;D) < for some s > 0. Note that Λ(V,V 2 ;D) Λ(V, D s ;D) + Λ(V,V 2 ; O s ). Sine at least one of V,V 2 is ompat, Lemma 4. implies that Λ(V,V 2 ; O s ) <. Theorem 4.7. Suppose V,V 2 are disjoint, nonpolar losed subsets of the Riemann sphere. Then the limit exists. Λ (V,V 2 ) = lim r 0+ [Λ(V,V 2 ; O r ) log log(/r)] (27) Proof. Without loss of generality, assume that dist(0,v ) 2 and let O k = O e k. Let ˆV 2 V 2 be a nonpolar losed subset with 0 ˆV 2. Constants in the proof depend on V,V 2. Sine Λ(V,V 2 ; O r ) inreases as r dereases to 0, it suffies to establish the limit lim k [ Λ(V,V 2 ; O k ) log k Repeated appliation of (23) shows that if k, Λ(V,V 2 ; O k ) = Λ(V,V 2 ; O 0 ) + Similarly, for fixed k, (23) implies ]. k Λ(V,V 2, O j \ O j ; O j ). j= Λ(V, O k \ O k ; O k ) Λ(V,V 2, O k \ O k ; O k ) From Lemma 4.4, we an see that = Λ(V, O k \ O k ; O k \ V 2 ) Λ(V, O k \ O k ; O k \ ˆV 2 ). Λ(V, O k \ O k ; O k ) = k + O ( k 2 ), 8

19 and hene the limit lim k log k + k Λ(V, O k \ O k ; O k ) exists and is finite. By Lemma 4.5, we see that Λ(V, O j \ O j ; O j \ ˆV 2 ) = Λ(V, D k ;D \ ˆV 2 ) k, j=k j= and hene [ Λ(V, O j \ O j ; O j ) Λ(V,V 2, O j \ O j ; O j ) ] k, j=k where the onstant depends on V and ˆV 2 but not otherwise on V 2. Remark It follows from the proof that Λ (V,V 2 ) = Λ(V,V 2 ; O k ) log k + O ( ), k where the O( ) terms depends on V and ˆV 2 but not otherwise on V 2. As a onsequene we an see that if 0 V and V 2,r = V 2 { z r}, then lim r 0+ Λ (V,V 2,r ) = Λ (V,V 2 ). (28) The definition of Λ in (27) seems to make the origin a speial point. Theorem 4. shows that this is not the ase. Lemma 4.8. Suppose V is a nonpolar losed subset, z 0 and 0 V. Let α > 0. There exists,r 0 (depending on z,v,α) suh that if 0 < r < r 0, Λ(V, C \ O r ; O αr (z)) log 2 log(/r). Proof. We will first assume z V. For s r, let D s = O s O αr (z). As in (22), Λ(V, C \ O r ; O αr (z)) = m(w;d π w,d w \ V )da(w). w r 9

20 By Lemma 3.4, if w = s r, π m(w;d s,d s \ V ) = and therefore, Λ(V, C \ O r ; O αr (z)) = log r 2πs 2 log(/s) r A straightforward omputation gives log r r 0 0 log r log(rs) ds s log(/s) log(rs) [ ( + O ds = log 2. s log(/s) log(rs) [ ( + O log(/r) )], log(/r) )]. This finishes the proof for z V. If z V and t > 0, let V V be a losed nonpolar set with z V. Then (23) implies Λ(V, C \ O r ; O αr (z)) = Λ(V, C \ O r ; O αr (z)) + Λ(V \ V, C \ O r ; O αr (z) \ V ). Sine the previous paragraph applies to V it suffies to show that ( ) Λ(V \ V, C \ O r ; O αr (z) \ V ) = O. log(/r) We an write Λ(V \ V, C \ O r ; O αr (z) \ V ) = π By using (8) we an see that and hene w r m(w;d s \ V,D s \ V ) Λ(V \ V, C \ O r ; O αr (z) \ V ) r 0 m(w;d s \ V,D s \ V )da(w). log 2 (/s), ds s log 2 (/s) log(/r). The following is the equivalent lemma for z =. It an be proved similarly or by onformal transforamtion. 20

21 Lemma 4.9. Suppose V is a nonpolar losed subset, and 0 V. Let α > 0. There exists,r 0 (depending on V,α) suh that if 0 < r < r 0, Λ(V, C \ Or ; D α/r ) log 2 We extend this to k losed sets. log(/r). Lemma 4.0. Suppose V,...,V k are losed nonpolar subsets of C \ {0}. Let α > 0. There exists,r 0 (depending on z,α,v,...,v k ) suh that if 0 < r < r 0, Λ(V,...,V k, C \ O r ; O αr (z)) log 2 Λ(V,...,V k, C \ O r ; D α/r ) log 2 Proof. If k = 2, inlusion-exlusion implies log(/r). log(/r). Λ(V V 2, C \ O r ; O αr (z)) + Λ(V,V 2,,C \ O r ; O αr (z)) = Λ(V, C \ O r ; O αr (z)) + Λ(V 2,,C \ O r ; O αr (z)). Sine Lemma 4.8 applies to V V 2,V,V 2, we get the result. The ases k > 2 and z = are done similarly. Theorem 4.. Suppose V,V 2 are disjoint, nonpolar losed subsets of the Riemann sphere and z C. Then Moreover, Λ (V,V 2 ) = lim r 0+ [Λ(V,V 2 ; O r (z)) log log(/r)]. Λ (V,V 2 ) = lim R [Λ(V,V 2 ;D R ) log log R]. Proof. We will assume 0 V. Using (27), we see that it suffies to prove that lim r 0+ [Λ(V,V 2 ; O r (z)) Λ(V,V 2 ; O r )] = 0. Note that Λ(V,V 2 ; O r (z)) Λ(V,V 2 ; O r ) = Λ(V,V 2, C \ O r ; O r (z)) Λ(V,V 2, C \ O r (z); O r ). 2

22 Lemma 4.0 implies ( Λ(V,V 2, C \ O r ; O r (z)) = log 2 + O log(/r) ), (29) where the onstants in the error term depend on z,v,v 2. Similarly, using translation invariane of the loop measure, we an see that ( ) Λ(V,V 2, C \ O r (z); O r ) = log 2 + O. log(/r) If V,V 2,...,V k are pairwise disjoint nonpolar losed sets of the Riemann sphere, we define similarly Λ (V,...,V k ) = lim r 0+ [Λ(V,V 2,...,V k ; O r ) log log(/r)]. One an prove the existene of the limit in the same way or we an use the relation Λ (V,...,V k ) = Λ (V,...,V k+ ) + Λ(V,...,V k ; C \ V k+ ). Referenes [] G. Lawler (2005). Conformally Invariant Proesses in the Plane, Amer. Math. So. [2] G. Lawler, Partition funtions, loop measure, and versions of SLE, to appear in J. Stat. Phys. [3] G. Lawler, O. Shramm, W. Werner (2003), Conformal restrition: the hordal ase, J. Amer. Math. So. 6, [4] G. Lawler, W. Werner (2004), The Brownian loop soup, Probab. Theory Related Fields 28,