Table Of ContentCoulomb gas integrals for commuting SLEs:
Schramm’s formula and Green’s function
Jonatan Lenells and Fredrik Viklund
7
KTH Royal Institute of Technology
1
0
2 January 16, 2017
n
a
J
Abstract
3
1 We use methods developed in conformal field theory to produce martingale ob-
servables for systems of commuting multiple SLE curves. In the case of two curves
]
R started from distinct points and growing towards infinity, we use these observables
P to determine rigorously and explicitly the Green’s function and Schramm’s formula.
. As corollaries, we obtain proofs of “fusion” formulas, some of which have been pre-
h
dicted in the physics literature. Our approach does not need a priori information
t
a on the regularity of the SLE observables, but does require a detailed analysis of the
m
regularity and asymptotics of certain Coulomb gas contour integrals. These integrals
[ arenaturalgeneralizationsoftheclassichypergeometricfunctionsandareinteresting
in their own right. We indicate a method for computing the relevant asymptotics of
1
v these integrals to all orders.
8
9
6 Contents
3
0
1 Introduction 3
.
1
1.1 Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0
7 1.2 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1
v: 2 Main results 5
i 2.1 Schramm’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
X
2.2 Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
r
a 2.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Preliminaries 13
3.1 Schramm-Loewner evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.1 SLE (ρ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
κ
3.1.2 Relationship between multiple SLE and SLE (ρ) . . . . . . . . . . 17
κ
3.1.3 Two-sided radial SLE and radial parametrization . . . . . . . . . . 17
1
4 Martingale observables as CFT correlation functions 19
4.1 Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Prediction of Schramm’s formula . . . . . . . . . . . . . . . . . . . . . . . 21
4.3 Prediction of the Green’s function . . . . . . . . . . . . . . . . . . . . . . 22
5 Schramm’s formula 26
5.1 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.2 The special case κ = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6 Green’s function 29
6.1 Existence of the Green’s function: Proof of Proposition 2.12 . . . . . . . . 29
6.2 Probabilistic representation for G : Proof of Proposition 2.13 . . . . . . 34
CFT
6.2.1 The function h(θ1,θ2) . . . . . . . . . . . . . . . . . . . . . . . . . 34
7 Two paths getting near the same point: Proof of Lemma 2.9 42
8 Fusion 47
8.1 Schramm’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
8.2 Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
9 The function G(z,ξ1,ξ2) when α is an integer 53
9.1 A representation for h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
10 Asymptotics of Coulomb gas integrals 61
10.1 Screening integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
10.2 The hypergeometric case of N = 3 . . . . . . . . . . . . . . . . . . . . . . 62
10.3 Asymptotics of F as w → 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2
10.4 Asymptotics of F as w → 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 68
1
10.5 Asymptotics of F as w → 0 and w → 0. . . . . . . . . . . . . . . . . . . 70
1 2
10.6 Asymptotics of F as w → 0 and w → 1. . . . . . . . . . . . . . . . . . . 72
1 2
10.7 Some basic estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
11 Differential equations 75
A Estimates for Schramm’s formula 76
A.1 Properties of the function J(z,ξ) . . . . . . . . . . . . . . . . . . . . . . . 76
A.2 Existence and regularity of P . . . . . . . . . . . . . . . . . . . . . . . . . 85
B Estimates for Green’s function 92
B.1 The asymptotic sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
B.2 Representations for h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
B.3 Proof of Lemma 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
2
1 Introduction
Schramm-Loewner evolution (SLE) processes are universal lattice size scaling limits of
interfaces in critical planar lattice models. By the Markovian property of SLE, geomet-
ric observables determine martingales with respect to the natural SLE filtration. Such
martingale observables satisfy differential equations, which can sometimes be used to find
explicit formulas for the observables, or be used as a basis for estimates, see for instance
[35, 9]. A related problem is that of constructing martingales carrying some specific
geometric information about the SLE process.
The differential equations are usually derived using Itô calculus, assuming sufficient
regularitytoapplyItô’sformula. Ifasolutionwithsuitableboundaryvaluescanbefound,
one way to proceed is to perform a probabilistic argument using the solution’s boundary
behavior to show that it actually represents the desired quantity. In the simplest cases
the differential equation is an ODE, but generically it is a semi-elliptic PDE in several
variables and it is difficult to find solutions with desired boundary data. (But see e.g.
[15, 25] for cases when solutions are available.) Depending on the amount of available
information, non-trivial regularity issues may need to be resolved when analyzing the
PDEs, see, e.g, [16]. Seeking ways to construct solutions and methods for extracting
information from them therefore seems worthwhile.
ThebasicPDEsthatariseinSLEtheoryalsoariseinconformalfieldtheory(CFT),see
e.g. [8,10,5,16]. Ontheotherhand,conformalfieldtheoryprovidesideasandmethodsfor
how to systematically construct solution candidates, see e.g. [6, 7, 23]. We will make use
of the Coulomb gas framework of [23] which models the CFTs using certain Gaussian free
field correlation functions. Very briefly, CFT correlations involving special fields inserted
on the boundary give rise to SLE martingales and thereby PDE solutions. By making
additional, carefully chosen, field insertions, the scaling behavior at the insertion points
can in some cases be prescribed. In this way, using a “calculus of scaling dimensions”,
many SLE martingale observables were recovered in [23] as CFT correlation functions.
The purpose of this paper is to construct one-point martingale observables (equiv-
alently, to solve the corresponding PDEs) related to specific geometric information for
multiple commuting SLEs by exploiting ideas from CFT. In the process we will suggest
an approach for rigorously deriving at least some natural observables in this setting. The
argument proceeds through three steps:
(1) Thefirststepgeneratesanon-rigorouspredictionfortheobservablebyusingascreen-
ingargumentbasedonideasfromCFT[13,23,3]. (Seealso[12,25]andthereferences
in the latter.) The prediction is expressed as a contour integral with an explicit inte-
grand. Wecalltheseintegralsscreening integrals orCoulomb gas integrals, seeSection
4. Themaindifficultyistochoosetheappropriateintegrationcontour, butthischoice
can be simplified by considering appropriate limits.
(2) ThesecondstepistoprovethatthepredictionfromStep1satisfiesthecorrectbound-
aryconditions. Thistechnicalstepinvolvesthecomputationofsomewhatcomplicated
integral asymptotics, but we indicate an approach for computing such asymptotics in
Section 10.
3
(3) The last step is to use probabilistic methods together with the estimates of Step 2 to
rigorously establish that the prediction from Step 1 gives the correct quantity.
We analyze two examples in detail. Both examples involve two curves aiming for ∞
with one marked point in the interior, but we will indicate how the arguments can be
generalized to more complicated configurations.
The first example concerns the probability that the system of SLEs passes to the right
of a given interior point; that is, the analog of Schramm’s formula [35]. This probability
obviously depends only on the behavior of the leftmost curve. (So it is really an SLE (2)
κ
observable.) The main difficulty in this case lies in implementing Steps 1 and 2.
The second example concerns the limiting renormalized probability that the system
of SLEs passes near a given point, that is, the Green’s function. We first give a proof of
the non-trivial fact that the commuting Green’s function actually exists as a limit. The
main step is to verify existence in the case when only one of the two curves grows. (More
precisely, we prove the existence of the SLE (ρ) Green’s function, where ρ is in an interval
κ
andtheforcepointisontheboundary.) Theproofgivesarepresentationformulainterms
of an expectation for two-sided radial SLE stopped at its target point; this is similar to
the main result of [2]. In Step 1, the prediction is made by taking a linear combination
of screening integrals to cancel a leading order term which has the wrong asymptotics
(thereby matching the asymptotics we expect). Then, in Step 2, we carefully analyze the
candidate solution and estimate its boundary behavior. Lastly, given the estimates from
Step2,weshowthatthecandidateobservableenjoysthesameprobabilisticrepresentation
as the Green’s function defined as a limit – so they must agree.
BylettingtheseedpointsoftheSLEscollapse, weobtainrigorousproofsoffusion for-
mulas as corollaries. One can verify a posteriori that these limiting one-point observables
satisfy specific third-order ODEs that can also be obtained from the so-called fusion rules
of conformal field theory, see e.g. [12]. In fact, in the case of the Schramm probability,
the formulas we prove here were predicted using fusion in [20]. The formulas for the fused
Green’s functions appear to be new. The interpretation of fusion in SLE theory as suc-
cessive merging of seeds, and the highly non-trivial fact that fused one-point observables
satisfy higher order ODEs, was rigorously established in [16]. The ODEs for the Schramm
formula for several fused paths were derived rigorously in [16] and the two-path formula
in the special case κ = 8/3 (also allowing for two interior points) was established in [4].
As can be gathered from the length of the paper, there are many details to handle.
In particular, the asymptotic analysis of the screening integrals is quite involved. These
integrals are natural generalizations of hypergeometric functions and are interesting in
theirownright. Similargeneralizedhypergeometricfunctionshavebeenconsideredbefore
in related contexts, see e.g. [13, 15, 25, 26, 22]. However, we have not been able to find
the required analytic estimates in the litterature. In Section 10, we propose a method for
establishing the asymptotics of these integrals to all orders in a rather general setting.
1.1 Outline of the paper
The main results of the paper are stated in Section 2, while we review some aspects of
multiple SLE and SLE (ρ) processes in Section 3.
κ κ
4
In Section 4, we review and use ideas from conformal field theory to generate predic-
tions for Schramm’s formula and Green’s function for commuting SLE with two curves
growing toward infinity as Coulomb gas integrals.
In Section 5, we prove rigorously that the predicted Schramm’s formula indeed gives
the probability that a given point lies to the left of both curves. The proof relies on
a number of technical asymptotic estimates; proofs of these estimates are collected in
Appendix A.
In Section 6, we give a rigorous proof that the predicted Green’s function equals the
renormalized probability that the system passes near a given point. The proof relies both
on pure SLE estimates (established in Section 6) and on asymptotic estimates for contour
integrals (established in Section 10 and Appendix B).
In Section 7, we prove a lemma which expresses the fact that it is very unlikely that
both curves in a commuting system get near a given point.
In Section 8, we consider the special case of two fused curves, i.e., the case when
both curves in the commuting system start at the same point. In the case of Schramm’s
formula, this provides rigorous proofs of some predictions due to Gamsa and Cardy for
Schramm’s formula presented in [20].
In Section 9, we derive formulas for the Green’s function in the special case when 8/κ
is an integer.
In Section 10, we suggest an approach for computing certain asymptotics to all orders
for a class of contour integrals which generalize the classic hypergeometric functions. The
proofs in Appendix B are an application of this approach to the contour integral relevant
for the Green’s function.
In Section 11, we discuss our results from the point of view of differential equations.
1.2 Acknowledgements
Lenells acknowledges support from the European Research Council, Consolidator Grant
No. 682537, the Swedish Research Council, Grant No. 2015-05430, and the Gustafsson
Foundation, Sweden. Viklund acknowledges support from the Knut and Alice Wallenberg
Foundation, the Swedish Research Council, the National Science Foundation, and the
Gustafsson Foundation, Sweden.
It is our pleasure to thank Julien Dubédat and Nam-Gyu Kang for interesting and
useful discussions and Tom Alberts, Nam-Gyu Kang, and Nikolai Makarov for sharing
with us ideas from their preprint [3].
2 Main results
Before stating the main results, we review the definition of a system of two commuting
SLE paths {γ }2 in the upper half-plane H := {Imz > 0} growing toward infinity.
j 1
Let 0 < κ < 8. Let (ξ1,ξ2) ∈ R2 with ξ1 6= ξ2. The Loewner equation corresponding
to two growing curves is
λ (t)dt λ (t)dt
1 2
dg (z) = + , g (z) = z, (2.1)
t g (z)−ξ1 g (z)−ξ2 0
t t t t
5
where ξ1 and ξ2, t (cid:62) 0, are the driving terms for the two curves and the growth speeds
t t
λj(t) satisfy λj(t) (cid:62) 0. The solution of (2.1) is a family of conformal maps (gt(z))t(cid:62)0
called the Loewner chain of (ξ1,ξ2). The system of two commuting SLEs started from
t t
(ξ1,ξ2) is obtained by taking ξ1 and ξ2 as solutions of the system
t t
q
dξt1 = λ1(ξtt1)−+λξt22(t)dt+qκ2λ1(t)dBt1, ξ01 = ξ1, (2.2)
dξt2 = λ1(ξt2)+−ξλ12(t)dt+ κ2λ2(t)dBt2, ξ02 = ξ2,
t t
where B1 and B2 are independent standard Brownian motions with respect to some
t t
measure P = P . The paths are defined by
ξ1,ξ2
γ (t) = limg−1(ξj +iy), γ := γ [0,t], j = 1,2. (2.3)
j t t j,t j
y↓0
For j = 1,2, γ is a continuous curve connecting ξj with ∞ in H. Given z ∈ C\(γ ∪
j,∞ 1,∞
γ ), we let Υ (z) denote 1/2 times the conformal radius of H\(γ ∪γ ) seen from
2,∞ ∞ 1,∞ 2,∞
z. It can be shown that the system (2.1) is commuting in the sense that the order in
which the two curves are grown does not matter [15]. Since our theorems only concern
the distribution of the fully grown curves γ and γ , the choice of growth speeds is
1,∞ 2,∞
irrelevant.
Let us also recall the definition of (chordal) SLE (ρ) for a single path γ in H growing
κ 1
toward infinity. Let ρ ∈ R and let (ξ1,ξ2) ∈ R2 with ξ1 6= ξ2. Let W be a standard
t
Brownian motion with respect to some measure Pρ. Then SLE (ρ) started from (ξ1,ξ2)
κ
is defined by the equations
2/κ
∂ g (z) = , g (z) = z,
t t g (z)−ξ1 0
t t
ρ/κ
dξ1 = dt+dW , ξ1 = ξ1.
t ξ1−g (ξ2) t 0
t t
When referring to SLE (ρ) started from (ξ1,ξ2), we always assume that the curve starts
κ
from the first point of the tuple (ξ1,ξ2) while the second point (in this case ξ2) is the force
point. The SLE (ρ) path γ (t) is defined as in (2.3). Given z ∈ C\γ , we let Υ (z)
κ 1 1,∞ ∞
denote 1/2 times the conformal radius of H\γ seen from z.
1,∞
2.1 Schramm’s formula
Our first result provides an explicit expression for the probability that an SLE (2) path
κ
passes to the right of a given point. The probability is expressed in terms of the function
M(z,ξ) defined for z ∈ H and ξ > 0 by
M(z,ξ) = yα−2z−α2(z−ξ)−α2z¯1−α2(z¯−ξ)1−α2
Z z
× (u−z)α(u−z¯)α−2u−α2(u−ξ)−α2du, (2.4)
z¯
where α = 8/κ > 1 and the integration contour from z¯ to z passes to the right of ξ, see
Figure 1.
6
Imz
z
Rez
0 ξ
z¯
Figure 1. The integration contour used in the definition (2.4) of M(z, ξ) is a path from
z¯ to z which passes to the right of ξ.
Theorem 2.1 (Schramm’s formula for SLE (2)). Let 0 < κ < 8. Let ξ > 0 and consider
κ
chordal SLE (2) started from (0,ξ). Then the probability P(z,ξ) that a given point z =
κ
x+iy ∈ H lies to the left of the curve is given by
1 Z ∞
P(z,ξ) = ReM(x0+iy,ξ)dx0, z ∈ H, ξ > 0, (2.5)
c
α x
where the normalization constant c ∈ R is defined by
α
(cid:16) (cid:17) (cid:16) (cid:17)
2π3/2Γ α−1 Γ 3α −1
2 2
c = − . (2.6)
α Γ(cid:0)α(cid:1)2Γ(α)
2
The proof of Theorem 2.1 will be given in Section 5. The form of the definition (2.5)
of P(z,ξ) is motivated by the CFT and screening considerations of Section 4.
A point z ∈ H lies to the left of both curves in a commuting system iff it lies to
the left of the leftmost curve. Since each of the two curves of a commuting process has
the distribution of an SLE (2) (see Section 3.1.2), Theorem 2.1 immediately implies the
κ
following result for commuting SLE.
Corollary 2.2 (Schramm’s formula for two commuting SLEs). Let 0 < κ < 8. Let ξ > 0
and consider a system of two commuting SLE curves in H started from (0,ξ) and growing
κ
toward infinity. Then the probability P(z,ξ) that a given point z = x+iy ∈ H lies to the
left of both curves is given by (2.5).
Corollary 2.2 together with translation invariance immediately yields an expression
for the probability that a point z lies to the left of a system of two SLEs started from
two arbitrary points (ξ ,ξ ) in R. The probabilities that z lies to the right of or between
1 2
7
the two curves then follow by symmetry. For completeness, we formulate this as another
corollary.
Corollary 2.3. Let 0 < κ < 8. Suppose −∞ < ξ1 < ξ2 < ∞ and consider a system of
two commuting SLE curves in H started from (ξ1,ξ2) and growing toward infinity. Let
κ
P(z,ξ) denote the function in (2.5). Then the probability P (z,ξ1,ξ2) that a given point
left
z = x+iy ∈ H lies to the left of both curves is given by
P (z,ξ1,ξ2) = P(z−ξ1,ξ2−ξ1);
left
the probability P (z,ξ1,ξ2) that a point z ∈ H lies to the right of both curves is
right
P (z,ξ1,ξ2) = P(−z¯+ξ2,ξ2−ξ1);
right
and the probability P (z,ξ1,ξ2) that z lies between the two curves is given by
middle
P (z,ξ1,ξ2) = 1−P (z,ξ1,ξ2)−P (z,ξ1,ξ2).
middle left right
By letting ξ → 0+, we obtain proofs of formulas for “fused” paths. This gives a proof
of the predictions of [20] where these formulas were derived by solving a third order ODE
obtained from so-called fusion rules. We note that even given the explicit predictions of
[20], it is not clear how to proceed to verify them rigorously. Indeed, as soon the evolution
starts, thetipsofthecurvesareseparatedandthesystemleavesthefusedstate. However,
[16] provides a different rigorous approach by exploiting the hypoellipticity of the PDEs
to show that the fused observables satisfy the higher order ODEs. The formula in the
special case κ = 8/3 was proved in [4] using Cardy and Simmons’ prediction [11] for a
two-point Schramm formula. See Theorem 8.1 for a complete statement and details.
Corollary 2.4. Let 0 < κ < 8 and define P (z) = lim P(z,ξ), where P(z,ξ) is as
fusion ξ↓0
in (2.5). Then
Γ(α)Γ(α) Z ∞
P (z) = 2 S(t0)dt0,
fusion 22−απΓ(3α −1) x
2 y
where the real-valued function S(t) is defined by
(cid:26) (cid:18)1 α α 1 (cid:19)
S(t) = (1+t2)1−α F + ,1− , ;−t2
2 1
2 2 2 2
2Γ(1+ α)Γ(α)t (cid:18) α 3 α 3 (cid:19)(cid:27)
− 2 2 F 1+ , − , ;−t2 , t ∈ R.
Γ(1 + α)Γ(−1 + α)2 1 2 2 2 2
2 2 2 2
2.2 Green’s function
OursecondmainresultprovidesanexplicitexpressionfortheGreen’sfunctionforSLE (2).
κ
Let α = 8/κ. Define the function I(z,ξ1,ξ2) for z ∈ H and −∞ < ξ1 < ξ2 < ∞ by
Z (z+,ξ2+,z−,ξ2−)
I(z,ξ1,ξ2) = (u−z)α−1(u−z¯)α−1(u−ξ1)−α2(ξ2−u)−α2du, (2.7)
A
8
z
Rez
ξ1 ξ2 A
z¯
Figure 2. The Pochhammer integration contour in (2.7) is the composition of four loops
based at the point A > ξ .
2
where A > ξ2 is a basepoint and the Pochhammer integration contour is displayed in
Figure 2. More precisely, the integration contour begins at the base point A, encircles the
point z once in the positive sense, returns to A, encircles ξ2 once in the positive sense,
returns to A, and so on. The points z¯ and ξ1 are exterior to all loops. The factors in
the integrand take their principal values at the starting point and are then analytically
continued along the contour.
For α ∈ (1,∞)\Z, we define the function G(z,ξ1,ξ2) by
1
G(z,ξ1,ξ2) = yα+α1−2|z−ξ1|1−α|z−ξ2|1−αIm(cid:0)e−iπαI(z,ξ1,ξ2)(cid:1), z ∈ H, ξ1 < ξ2,
cˆ
(2.8)
where the constant cˆ= cˆ(κ) is given by
4sin2(cid:0)πα(cid:1)sin(πα)Γ(cid:0)1− α(cid:1)Γ(cid:16)3α −1(cid:17) 8
2 2 2
cˆ= with α = . (2.9)
Γ(α) κ
We extend this definition of G(z,ξ1,ξ2) to all α > 1 by continuity. The following lemma
shows that even though cˆ vanishes as α approaches an integer, the function G(z,ξ1,ξ2)
has a continuous extension to integer values of α.
Lemma 2.5. For each z ∈ H and each (ξ1,ξ2) ∈ R2 with ξ1 < ξ2, G(z,ξ1,ξ2) can be
extended to a continuous function of α ∈ (1,∞).
Proof. See Section 9.
The CFT and screening considerations described in Section 4 suggest that G is the
Green’s function for SLE (2) started from (ξ1,ξ2); that is, that G(z,ξ1,ξ2) provides the
κ
normalized probability that an SLE (2) path originating from ξ1 with force point ξ2
κ
passes through an infinitesimal neighborhood of z. The following theorem establishes this
rigorously.
9
Theorem 2.6 (Green’s function for SLE (2)). Let 0 < κ (cid:54) 4. Let −∞ < ξ1 < ξ2 < ∞
κ
and consider chordal SLE (2) started from (ξ1,ξ2). Then, for each z = x+iy ∈ H,
κ
lim(cid:15)d−2P2 (Υ (z) (cid:54) (cid:15)) = c G(z,ξ1,ξ2), (2.10)
ξ1,ξ2 ∞ ∗
(cid:15)→0
where P2 is the SLE (2) measure, the function G is defined in (2.8), and the constant
κ
c = c (κ) is defined by
∗ ∗
2 2Γ(1+2a) 2
c∗ = R0πsin4axdx = √πΓ(cid:16)1 +2a(cid:17) with a = κ. (2.11)
2
The proof of Theorem 2.6 will be presented in Section 6.
Remark 2.7. The function G(z,ξ1,ξ2) can be written as
G(z,ξ1,ξ2) = (Imz)d−2h(θ1,θ2), z ∈ H, −∞ < ξ1 < ξ2 < ∞, (2.12)
where h is a function of θ1 = arg(z−ξ1) and θ2 = arg(z−ξ2). This is consistent with the
expected translation invariance and scale covariance of the Green’s function.
Remark 2.8. Formulas for G(z,ξ1,ξ2) when α is an integer are derived in Section 9.
For α = 2,3,4 (corresponding to κ = 4,8/3,2), the function h(θ1,θ2) in (2.12) is given
explicitly in (9.18), (9.19), and (9.20), respectively.
It is possible to derive an explicit expression for the Green’s function for a system of
two commuting SLEs as a consequence of Theorem 2.6. To this end, we need a correlation
estimate which expresses the fact that it is very unlikely that both curves in a commuting
system pass near a given point z ∈ H.
Lemma 2.9. Let 0 < κ (cid:54) 4. Then,
h i
lim(cid:15)d−2P (Υ (z) (cid:54) (cid:15)) = lim(cid:15)d−2 P2 (Υ (z) (cid:54) (cid:15))+P2 (Υ (z) (cid:54) (cid:15)) ,
ξ1,ξ2 ∞ ξ1,ξ2 ∞ ξ2,ξ1 ∞
(cid:15)↓0 (cid:15)↓0
where P denotes the law of a system of two commuting SLE in H started from
ξ1,ξ2 κ
(ξ1,ξ2) and aiming for ∞, and P2 denotes the law of chordal SLE (2) in H started
ξ1,ξ2 κ
from (ξ1,ξ2).
The proof of Lemma 2.9 will be given in Section 7.
Assuming Lemma 2.9, it follows immediately from Theorem 2.6 that the Green’s
function for a system of commuting SLEs started from (−ξ,ξ) is given by
G (z) = G(z,−ξ,ξ)+G(−z¯,−ξ,ξ), z ∈ H, ξ > 0.
ξ
In other words, given a system of two commuting SLE paths started from −ξ and ξ
κ
respectively, G (z) provides the normalized probability that at least one of the two curves
ξ
passes through an infinitesimal neighborhood of z. We formulate this as a corollary.
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