Imaginary geometry II: reversibility of SLE_κ(ρ_1;ρ_2) for κ\in (0,4)

1.2 MB·

by  Jason Miller

Imaginary geometry II: reversibility of
SLE (ρ ; ρ ) for κ (0, 4)
κ 1 2
∈
2
1
0 Jason Miller and Scott Sheffield
2

n
a
J
Abstract

6
Given a simply connected planar domain D, distinct points x,y

] ∈
R ∂D, and κ > 0, the Schramm-Loewner evolution SLEκ is a random
P continuous non-self-crossing path in D from x to y. The SLEκ(ρ1;ρ2)
. processes, defined for ρ ,ρ > 2, are in some sense the most natural
h 1 2
−
t generalizations of SLE .
a κ
m When κ 4, we prove that the law of the time-reversal of an
≤
SLE (ρ ;ρ ) from x to y is, up to parameterization, an SLE (ρ ;ρ )
[ κ 1 2 κ 2 1

from y to x. This assumes that the “force points” used to define
1
v SLEκ(ρ1;ρ2) are immediately to the left and right of the SLE seed. A
7 generalization to arbitrary (and arbitrarily many) force points applies
9
whenever the path does not (or is conditioned not to) hit ∂D x,y .
4 \{ }
1 The time-reversal symmetry has a particularly natural interpre-
. tation when the paths are coupled with the Gaussian free field and
1
0 viewed as rays of a random geometry. It allows us to couple two in-
2
stances of the Gaussian free field (with different boundary conditions)
1
so that their difference is almost surely constant on either side of the
:
v
path. In a fairly general sense, adding appropriate constants to the
i
X two sides of a ray reverses its orientation.
r
a
1
Contents
1 Introduction 3
1.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Relation to Previous Work . . . . . . . . . . . . . . . . . . . . 8
1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Preliminaries 10
2.1 SLE (ρ) Processes . . . . . . . . . . . . . . . . . . . . . . . . 10
κ
2.2 Imaginary Geometry of the Gaussian Free Field . . . . . . . . 11
2.3 Naive time-reversal . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Idea for a Gaussian free field reversibility proof . . . . . . . . 22
3 SLE (ρ): conformal Markov characterization 25
κ
4 Bi-Chordal SLE Processes 30
κ
5 Paths conditioned to avoid the boundary 38
5.1 Weighting by martingales . . . . . . . . . . . . . . . . . . . . 38
5.2 A single path avoiding the boundary . . . . . . . . . . . . . . 41
5.3 A pair of paths avoiding each other . . . . . . . . . . . . . . . 49
5.4 Resampling properties and dual flow lines . . . . . . . . . . . 54
6 The time-reversal satisfies conformal Markov property 67
7 Proof of Theorem 1.1 71
7.1 One Boundary Force Point; ρ ( 2,0] . . . . . . . . . . . . . 71
∈ −
7.2 Two Boundary Force Points . . . . . . . . . . . . . . . . . . . 72
7.3 Whole plane SLE and variants . . . . . . . . . . . . . . . . . . 75
8 Multiple Force Points 76
8.1 Time reversal and free field perturbations . . . . . . . . . . . . 76
8.2 Time reversals and shields . . . . . . . . . . . . . . . . . . . . 82
2
Acknowledgments. We thank Oded Schramm, David Wilson and Dapeng
Zhan for helpful discussions.
1 Introduction
For each simply connected Jordan domain D C and distinct pair x,y
⊆ ∈
∂D,theSchramm-Loewner evolutionofparameterκ > 0(SLE )describesthe
κ
law of a random continuous path from x to y in D. This path is almost surely
a continuous, simple curve when κ 4. It is almost surely a continuous, non-
≤
space-filling curve that intersects itself and the boundary when κ (4,8),
∈
and it is almost surely a space-filling curve when κ 8 [23, 13]. We recall
≥
the basic definitions of SLE in Section 2.1.
κ
While SLE curves were introduced by Schramm in [25], the following
κ
fact was proved only much more recently by Zhan in [38]: if η is an SLE
κ
process for κ (0,4] from x to y in D then the law of the time-reversal of η
∈
is, up to monotone reparameterization, that of an SLE process from y to x
κ
in D. This is an extremely natural symmetry.
Since their introduction in [25], it has been widely expected that SLE
κ
curves would exhibit this symmetry for all κ 8.1 One reason to expect
≤
this to be true is that SLE has been conjectured and in some cases proved
κ
to arise as a scaling limit of discrete models that enjoy a discrete analog of
time-reversal symmetry. However, it is far from obvious from the definition
of SLE why such a symmetry should exist.
κ
It was further conjectured by Dub´edat that when κ 4 the property
≤
of time-reversal symmetry is also enjoyed by the so-called SLE (ρ ;ρ ) pro-
κ 1 2
cesses, which depend on parameters ρ ,ρ > 2, and which are in some
1 2
−
sense the most natural generalizations of SLE . We recall the definition
κ
of SLE (ρ ;ρ ) in Section 2.1. (The conjecture assumes the so-called force
κ 1 2
points, whose definition we recall in Section 2.1, are located immediately left
and right of the SLE seed.) Dub´edat’s conjecture was later proved in the
special case of SLE (ρ) processes (obtained by setting one of the ρ to 0 and
κ i
the other to ρ), under the condition that ρ is in the range of values for which
the curve almost surely does not intersect the boundary. This was accom-
plished by Dub´edat [5] and Zhan [39] using a generalization of the technique
used to prove the reversibility of SLE in [38].
κ
1This was the final problem in a series presented by Schramm at ICM 2006 [26].
3
This article will prove Dub´edat’s conjecture in its complete generality
using several new methods that we hope will be of independent interest. In
particular, we will address the following natural questions:
1. Consider a coupling of multiple paths η ,η ,...η in D from x to y. If
1 2 k
we know that the conditional law of each one of these paths given the
others is an SLE (ρ ;ρ ) process (in the appropriate component of the
κ 1 2
complement of the other paths), does this determine the joint law of
all the paths?
2. Is there a natural domain Markov property that characterizes SLE (ρ)
κ
processes,analogoustothepropertyusedtocharacterizeordinarySLE
κ
[25]?
3. When ρ and ρ are in the range for which an SLE (ρ ;ρ ) curve is
1 2 κ 1 2
almost surely boundary intersecting, what is the most natural way to
make sense of an SLE (ρ ;ρ ) curve conditioned not to hit the bound-
κ 1 2
ary (or multiple curves conditioned not to hit each other)?
Although there is some technical work involved in answering these questions,
oncewehavedonesowewillbeabletogivearelativelysimpleandconceptual
proof of Dub´edat’s conjecture. In particular, we will give a new proof of
the time-reversal symmetry of ordinary SLE for κ < 4 (inspired by an
κ
unpublishedargumentofSchrammandsecondauthor), whichisindependent
of the arguments in [38, 5, 39].
One can generalize SLE (ρ ;ρ ) theory to multiple force points, as we
κ 1 2
recall in Section 2.1; when there are more than two, we often use ρ to denote
the corresponding vector of ρ values. The time-reversal of SLE (ρ) with
κ
multiple force points (or a force point not immediately adjacent to the SLE
seed) need not be an SLE (ρ), as illustrated in [5, 39]. However, another
κ
result of the current paper is that in general the time-reversal of an SLE (ρ)
κ
that does not (or is conditioned not to) hit the boundary is also an SLE (ρ)
κ
that does not (or is conditioned not to) hit the boundary, with appropriate
force points. A similar result applies if the SLE (ρ) hits the boundary only
κ
on one of the two boundary arcs connecting x and y (and there are no force
points in the interior of that arc).
This paper is a sequel to and makes heavy use of a recent work of the
authors [17], which in particular proves the almost sure continuity of general
4
SLE (ρ) traces, even those that hit the boundary. The results of the cur-
κ
rent paper have various applications to the theory of “imaginary geometry”
described in [17], to Liouville quantum gravity, and to SLE theory itself.
In particular, they will play a crucial role in a subsequent work by the
authors that will give the first proof of the time-reversal symmetry of SLE
κ
andSLE (ρ ;ρ )processesthatapplieswhenκ (4,8)[18]. Interestingly, we
κ 1 2
∈
will find in [18] that when κ (4,8) the SLE (ρ ;ρ ) processes are reversible
κ 1 2
∈
if and only if ρ κ 4 for i 1,2 . The threshold κ 4 is significant
i ≥ 2 − ∈ { } 2 −
because, when κ (4,8), the SLE (ρ ;ρ ) curves almost surely hit every
κ 1 2
∈
point on the entire left (resp. right) boundary of D if and only if ρ κ 4
1 ≤ 2 −
(resp. ρ κ 4). Thus, aside from the critical cases, the “non-boundary-
2 ≤ 2 −
filling” SLE (ρ ;ρ ) curves are the ones with time-reversal symmetry.
κ 1 2
The time-reversal symmetries that apply when κ 8 will be addressed in
≥
the fourth work of the current series [19]. When κ 8, we will see that one
≥
hastime-reversalsymmetryonlyforonespecialpairofρ ,ρ values; however,
1 2
in the κ 8 context, it is possible to describe time-reversals of SLE (ρ ;ρ )
κ 1 2
≥
processes more generally in termsof SLE (ρ ; ρ ) processes forcertain values
κ 1 2
of ρ and ρ . We will also show in [19] that certain families of “whole-plane”
1 2
variants of SLE have time-reversal symmetry as well, generalizing a recent
κ (cid:98) (cid:98)
work of Zhan on this topic [37].
(cid:98) (cid:98)
1.1 Main Results
The following is our first main result:
Theorem 1.1. Suppose that η is an SLE (ρ ;ρ ) process in a Jordan domain
κ 1 2
D from x to y, with x,y ∂D distinct and weights ρ ,ρ > 2 corresponding
1 2
∈ −
to force points located at x ,x+, respectively. The law of the time-reversal
−
(η) of η is, up to reparameterization, an SLE (ρ ;ρ ) process in D from y
κ 2 1
R
to x with force points located at y ,y+, respectively. Thus, the law of η as a
−
random set is invariant under an anti-conformal map that swaps x and y.
TheproofofTheorem1.1hastwomainparts. Thefirstpartistoestablish
the reversibility of SLE (ρ) processes with a single force point located at the
κ
SLE seed, even when they hit the boundary. This extends the one-sided
result of [5, 39] to the boundary-intersecting regime. The second part is to
extend this result to SLE (ρ ;ρ ) processes.
κ 1 2
This second part of the proof will be accomplished using so-called bi-
chordal SLE processes to reduce the two-force-point problem to the single-
5
force-pointcase. Thebi-chordalprocessesweusewillbeprobabilitymeasures
on pairs of non-crossing paths (η ,η ) in D from x to y with the property
1 2
that the conditional law of η given η is an SLE (ρL) in the left connected
1 2 κ
component of D η and the law of η given η is an SLE (ρR) in the right
2 2 1 κ
\
connected component of D η . We use the superscript “R” to indicate that
1
\
the force points associated with η lie on the counterclockwise arc of ∂D
1
between the initial and terminal points of η . Likewise, the superscript “L”
1
indicates that the force points associated with η lie on the clockwise arc of
2
∂D between the initial and terminal points of η . We will prove in a rather
2
general setting that this information (about the conditional law of each η
i
given the other) completely characterizes the joint law of (η ,η ), a result we
1 2
considerindependentlyinteresting. Onecanthenusetheimaginarygeometry
constructions from [17] to explicitly produce processes in which each η is a
i
one-sided SLE (ρ) when restricted to the complement of the other, but the
κ
marginal law of each path is an SLE (ρ ;ρ ) process in the whole domain.
κ 1 2
The time-reversal symmetry of the individual SLE (ρ) processes can then be
κ
used to prove the time-reversal symmetry of SLE (ρ ;ρ ).
κ 1 2
The first part of the proof of Theorem 1.1, which is the proof of the
reversibility of SLE (ρ) with one force point, involves multiple steps. One
κ
of them is to prove that SLE (ρ) is characterized by the following version of
κ
conformal invariance and the domain Markov property. Let c = (D,x,y;z)
be a configuration which consists of a Jordan domain D C and x,y,z
⊆ ∈
∂D and x = y. We let (resp. ) be the collection of configurations
L R
(cid:54) C C
c = (D,x,y;z) where z lies on the clockwise (resp. counterclockwise) arc of
∂D from x to y.
Definition 1.2 (Conformal Invariance). We say that a family (P : c ),
c q
∈ C
q L,R , where P is a probability measure on continuous paths from x
c
∈ { }
to y in D, is conformally invariant if the following is true. Suppose that
c = (D,x,y;z), c = (D ,x,y ;z ) , and ψ: D D is a conformal map
(cid:48) (cid:48) (cid:48) (cid:48) (cid:48) q (cid:48)
∈ C →
with ψ(x) = x, ψ(y) = y , and ψ(z) = z . Then for η P , we have that
(cid:48) (cid:48) (cid:48) c
∼
ψ(η) P , up to reparameterization.
c(cid:48)
∼
Definition 1.3 (Domain Markov Property). We say that a family (P : c
c
∈
), q L,R , where P is a probability measure on continuous paths
q c
C ∈ { }
from x to y in D, satisfies the domain Markov property if for all c the
q
∈ C
following is true. Suppose η P . Then for every η stopping time τ, the
c
∼
law of η conditional on η([0,τ]) is, up to reparameterization, given by
[τ, )
| ∞
P where c = (D ,η(τ),y;z ). Here, D is the connected component of
cτ τ τ τ τ
6
D η([0,τ]) which contains y. If q = L (resp. q = R), then z is the right
τ
\
(resp. left) most point on the clockwise (resp. counterclockwise) arc of ∂D
from x to y which lies to the left (resp. right) of z and η([0,τ]) ∂D.
∩
Our conformal Markov characterization of SLE (ρ) is the following:
κ
Theorem 1.4. Suppose that (P : c ), q L,R , is a conformally
c q
∈ C ∈ { }
invariant family which satisfies the domain Markov property in the sense of
Definitions 1.2 and 1.3. Assume further that when c = (D,x,y;z) and
q
∈ C
D has smooth boundary and η P , the Lebesgue measure of η ∂D is zero
c
∼ ∩
almost surely. Then there exists ρ > 2 such that for each c = (D,x,y;z)
− ∈
, P is the law of an SLE (ρ) process in D from x to y with a single force
q c κ
C
point at z.
Theorem 1.4 is a generalization of the conformal Markov characterization
of ordinary SLE established by Schramm [25] (and used by Schramm [25]
κ
to characterize SLE processes) but with the addition of one extra marked
κ
point. It is implicit in the hypotheses that η P cannot cross itself and
c
∼
also never enters the loops it creates with segments of the boundary or itself
as it moves from x to y. This combined with the hypothesis that η ∂D has
∩
zero Lebesgue measure almost surely when D is smooth implies that η has a
continuous Loewner driving function [17, Section 6.2] and that the evolution
of the marked point under the uniformizing conformal maps is described by
the Loewner flow (Lemma 3.3). The proof makes use of a characterization of
continuous self-similar Markov processes due to Lamperti [11, Theorem 5.1].
The next step in the proof of the reversibility of SLE (ρ) for ρ > 2 is to
κ
−
show that the time-reversal (η) of η SLE (ρ) for ρ ( 2,0] satisfies the
κ
R ∼ ∈ −
criteria of Theorem 1.4. This is in some sense the heart of the argument. We
will first present a new proof in the case that ρ = 0, which is related to an
argument sketch obtained (but never published) by the second author and
Schramm several years ago. The idea is to try to make sense of conditioning
on a flow line of the Gaussian free field, whose law is an SLE , up to a reverse
κ
stopping time. The conditional law of the initial part of the flow line is then
in some sense a certain SLE (ρ ;ρ ) process conditioned to merge into the
κ 1 2
tip of that flow line. We make this idea (which involves conditioning on an
eventofprobabilityzero)preciseusingcertainbi-chordalSLE constructions.
κ
We will use similar tricks to establish a conformal Markov property for the
time-reversal of SLE , which is then extended to give a similar property for
κ
SLE (ρ).
κ
7
The above arguments will imply that the time-reversal of an SLE (ρ)
κ
is itself an SLE (ρ) for some ρ > 2. This will imply that there exists
κ
−
a function R such that (η) SLE (R(ρ)). One can then easily observe
κ
R ∼
that the function R is continuous and increasing and satisfies R(R(ρ)) =
(cid:101) (cid:101)
ρ which implies R(ρ) = ρ. Using another trick involving bi-chordal SLE
configurations, we can extend the reversibility of SLE (ρ) to all ρ > 2.
κ
−
Using the interpretation of SLE (ρL;ρR) processes as flow lines of the
κ
Gaussian free field with certain boundary data [17], we will also give a de-
scription of the time-reversal (η) of η SLE (ρL;ρR) processes with many
κ
R ∼
force points, provided η is almost surely non-boundary intersecting.
Theorem 1.5. Suppose that η is an SLE (ρL;ρR), in a Jordan domain
κ
D from x to y, with x,y ∂D distinct, that does not (or is conditioned
∈
not to) hit ∂D except at x and y. Then the time-reversal (η) of η is an
R
SLE (ρL;ρR) process (with appropriate force points) that does not (or is con-
κ
ditioned not to) hit ∂D except at x and y.
A more precise discussion and complete description of how to construct
the law of the time-reversal (in particular how to set up the various ρ values)
appears in Section 8. In order to make the theorem precise, we will in
particular have to make sense of what it means to condition a path not to
hit the boundary (which in some cases involves conditioning on a probability
zero event).
1.2 Relation to Previous Work
As we mentioned earlier, the reversibility of SLE for κ (0,4] was first
κ
∈
proved by Zhan [38] but also appears in the work of Dub´edat [5]. Both
proofs are based on a beautiful technique that allows one to construct a
coupling of η SLE from x to y in D with η SLE in D from y to x such
κ κ
∼ ∼
that the two paths commute. In other words, one has a recipe for growing
the paths one at a time, in either order, that produces the same overall joint
(cid:101)
law. In the coupling of [38], the joint law is shown to have the property that
for every η stopping time τ, the law of η given η([0,τ]) is an SLE in the
κ
connected component of D η([0,τ]) containing y from y to η(τ). The same
\
likewise holds when the roles of η and η are reversed. This implies that η
(cid:101)
contains a dense subset of η and vice-versa. Thus the continuity of η and η
implies that η is almost surely the time-reversal (up to reparameterization)
(cid:101)
of η. In particular, the time-reversal (η) of η is an SLE in D from y to x.
(cid:101) κ (cid:101)
R
(cid:101)
8
The approaches of both Dub´edat [5] and Zhan [39] to the reversibility of non-
boundary intersecting SLE (ρ) are also based on considering a commuting
κ
pair of SLE (ρ) processes η,η growing at each other.2 The difference from
κ
the setup of ordinary SLE is that in such a coupling, the conditional law
κ
of η given η([0,τ]), τ an η stopping time, is not an SLE (ρ) process in the
(cid:101) κ
connected component of D η([0,τ]) containing x from x to η(τ). Rather, it
\
is a more complicated variant of SLE , a so-called intermediate SLE.
(cid:101) (cid:101) (cid:101) (cid:101) κ
Because our approach to SLE (ρ) reversibility is somewhat different from
(cid:101) κ(cid:101) (cid:101) (cid:101)
the methods in [5, 39] (and in particular we use a domain Markov property
to characterize the time-reversal), we will not actually need to define inter-
mediate SLE explicitly (e.g., by giving an explicit formula for the Loewner
drift term). We will also avoid the analogous explicit calculations in the
multiple force point cases with tricks involving bi-chordal resampling and
the Gaussian free field.
1.3 Outline
The remainder of this article is structured as follows. In the next section,
we will review the basics of SLE (ρ) processes. We will also give a summary
κ
of how SLE (ρ) processes can be viewed as flow lines of the Gaussian free
κ
field (GFF) — this is the so-called imaginary geometry of the GFF [17]. We
will in particular emphasize how this interpretation can be used to construct
couplings of systems of SLE (ρ) processes and the calculus one uses in order
κ
to compute the conditional law of one such curve given the realizations of the
others. Next, in Section 3, we will prove Theorem 1.4, the conformal Markov
characterization of SLE (ρ) processes. In Section 4, we will show in certain
κ
special cases that the joint law of a system of multiple SLE (ρ) strands is
κ
characterizedbytheconditionallawsof theindividualstrands. Thisprovides
an alternative mechanism for constructing systems of SLE type curves in
κ
which it is easy to compute the conditional law of one of the curves given the
others. It is the key tool for deducing the reversibility of SLE (ρ ;ρ ) from
κ 1 2
the reversibility of SLE (ρ). In Section 5 we discuss how to make sense of
κ
SLE (ρ) processes conditioned not to intersect certain boundary segments.
κ
2The results in [5, 39] only apply if ρ is in the range for which the path avoids the
boundary almost surely. However, it is possible that the arguments could be extended to
the boundary-hitting case of one-force-point SLE (ρ). Zhan told us privately before we
κ
wrote[17]thathebelievedthetechniquesin[39]couldbeextendedtothesingle-force-point
boundary-intersecting case if the continuity result of [17] were known.
9
We will then combine the above elements in Section 6 to show that the
time-reversal of an SLE (ρ) process satisfies the conformal Markov property.
κ
In Section 7 we will complete the proof of Theorem 1.1 by deducing the
reversibility of SLE (ρ ;ρ ) from the reversibility of SLE (ρ). We finish in
κ 1 2 κ
Section 8 by proving Theorem 1.5
2 Preliminaries
The purpose of this section is to review the basic properties of SLE (ρL;ρR)
κ
processesinadditiontogivinganon-technicaloverviewoftheso-calledimag-
inary geometry of the Gaussian free field. The latter provides a mechanism
for constructing couplings of many SLE (ρL;ρR) strands in such a way that
κ
it is easy to compute the conditional law of one of the curves given the
realization of the others [17].
2.1 SLE (ρ) Processes
κ
SLE is a one-parameter family of conformally invariant random curves, in-
κ
troduced by Oded Schramm in [25] as a candidate for (and later proved to
be) the scaling limit of loop erased random walk [13] and the interfaces in
critical percolation [34, 3]. Schramm’s curves have been shown so far also to
arise as the scaling limit of the macroscopic interfaces in several other mod-
els from statistical physics: [28, 35, 4, 27, 16]. More detailed introductions
to SLE can be found in many excellent survey articles of the subject, e.g.,
[36, 12].
An SLE in H from 0 to is defined by the random family of conformal
κ
∞
maps g obtained by solving the Loewner ODE
t
2
∂ g (z) = , g (z) = z (2.1)
t t 0
g (z) W
t t
−
where W = √κB and B is a standard Brownian motion. Write K := z
t
{ ∈
H : τ(z) t . Then g is a conformal map from H := H K to H satisfying
t t t
≤ } \
lim g (z) z = 0.
z t
| |→∞| − |
RohdeandSchrammshowedthattherealmostsurelyexistsacurveη (the
so-calledSLEtrace)suchthatforeacht 0thedomainH istheunbounded
t
≥
connected component of H η([0,t]), in which case the (necessarily simply
\
connected and closed) set K is called the “filling” of η([0,t]) [23]. An SLE
t κ
10