An HRLC in a planar domain can be described by differential equations involving canonical planar domains. Under reasonable assumptions, HRLC8/3 differs from a constraint measure by a conformal invariant density; for κ HRLCκ differs from a pre-constraint measure by a conformally invariant density.
Background
There are three kinds of SLEs in the literature: radial SLE, chordal SLE, and full plane SLE. The definitions of SLEs use the fact that all simply connected domains with a marked interior point or prime end are conformally equivalent.
Main results
We expect the Loewner random chain to retain observables similar to those of SLE4 and SLE8. Then t 7→ L(u−1(t)) is still a Loewner chain or inner Loewner chain, and is called a time change of L through u.
Review of SLEs in the literature
- Radial SLE
- Chordal SLE
- Full plane SLE
- Equivalence relations
- SLE traces
This means that (cKt) has the same law as (Kc2t) if (Kt) has the chordal standard law SLεκ. We call the law of the full standard plane LE inner hull directed by ξκ the full standard plane SLεκ.
Strip SLE
- Definition
- Equivalence of strip and chordal SLE
- Transience of the strip SLE κ trace
- Cardy’s formula
Similar to the radial and chordal cases, the standard strip SLEκ(D;x → I) is preserved under the self-anticonformal mapping (D;x, I) and has the property. Suppose that the βe law of the strip standard also has the trace SLEκ and is independent of (Kt).
Annulus SLE
Definition
The capacitance is non-negative and is 0 when K =∅. Theorem 2.4.1 The following two statements are equivalent: i) (Kt) is a family of standard modulus p LE hulls; For any p > 0, let µp be the law of the standard modulus p annulus LE hulls driven by ξp.
Equivalence of annulus and radial SLE
The annulus SLεκ(D;w → S) is preserved under the self-anticonformal mapping of (D;w, S) and has the conformally equivalent time homogeneity property. Then {µp} satisfies the above properties and µp is a standard annulus of modulusp SLE if and only if h(t)/t is constant at (0, p].
Disc SLE
- Definition
- Equivalence of disc and full plane SLE
- Finite Riemann surfaces and conformal structure
- Hulls and Loewner chains
- Topology and measure structure
- Positive harmonic functions
A direct consequence of this consequence is that the SLE6 planar hull stopped at the impact time ∂Ω has the same law as the hull produced by the planar Brownian motion starting from 0 and stopping at the exit from Ω. A subdomain D of R is called a finite Riemann surface if R \D is the union of a finite number of disconnected compact contractible subsets of R, each of which contains more than one point.
Definition
Conformally invariant SDE
In fact, from the next corollary we see that the law does not depend on W, so we can omit W. We can define Pt,Wξ to be the harmonic objective function of I inD\W(Ktξ) as long as there is a neighborhood of Iin ∂D that does not intersect W(Ktξ). If there is a quarter of w1 in Db that does not intersect W(Ktξ), then we can define Pt,Wξ to be the minimal function in D\W(Ktξ) with pole w1, normalized by J.
We can also define the (global) HRLCκ(D;w0 →I) or HRLCκ(D;w0 →w1) normalized by J, just like the case that the target is a prime endpoint.
Existence and uniqueness
We can find another Jordan curve γ in Ω disjoint from β, such that the doubly connected domain bounded by γ and ∂D is contained in Ω and contains β. By conformal invariance and the comparison principle of moduli, the modulus of the doubly connected domain is bounded by ϕζt(γ) and ∂D at least m. There is a Jordan curve β in Ω such that the doubly connected domain bounded by β and ∂D is contained in Ω.
Let β be a Jordan curve in Ω disjoint from Kaξ0 such that the doubly connected domain bounded by β and ∂D is contained in Ω.
Interior HRLC
So it is also true if the first ξ in the above sentence is replaced by A ∈Sa. The idea of the proof is a combination of the proof of Theorem 3.2.1 and the proof of Theorem 2.5.1. If we define Qξt,W as the harmonic measure function in D\W(Lξt) of a fixed side arc I of D, we obtain HRLCκ(p0 → I).
HRLC in canonical plane domains
If the half-plane is the upper half-plane H (respectively, right half-plane-iH, or lower half-plane-H, respectively), then it is called a type H (respectively RH or LH) domain. We construct a curve from X, which is the union of all edges of X and the line segment [0, δn]. We construct a curve from X, which is the union of all edges of X and the line segment [0, δ].
We construct a curve from X, which is the union of all edges of X and the distance [0, δn].
Observables
Observables for LERW
On the other hand, if we define g = f +hL(A, B)/(−f(x)∆Gh(x)), then we see from the last section that g satisfies the first group of properties. Then a random walk on (G, A∪B) from v0 hits A with positive probability, and so does LERW on (G, A∪B) from v0. Thus gk(w0) is a discrete martingale until the first time X hits A, or Bk disconnects w0 from A.
Observables for HRLC 2
Choose an analytic Jordan curve β0 in Σ such that the domain bounded by α and β0 is contained in Σ. Since Gt and S(t, w;·) vanish on the non-α sides of D, we can restrict the above integrations to β0 ∪βε. Note that the external unit normal vectors for V and for U at each point ϕt◦W−1(β0) are opposite to each other.
Let Mt be the minimal function in D\L(t) with the pole at w(t), the primary end determined by L at time t, normalized by R.
Resemblance
If there is the law of an interior HRLC aimed at an interior point, a major end, or a side arc, then we can define the MtorMt0 functions in the same way as we define them for HRLC2 starting from a simple end. Similarly, in the case that 0 ∈D, the observables for a LERW onDδ initialized from 0 conditioned to hit the given set of vertices resemble the observables for the corresponding interior HRLC2 inD initialized from 0.
Convergence of LERW to HRLC 2
Convergence of the driving functions
Using the fact that for every 0< s≤r, ReSr reaches its unique maximum and minimum at As,r ate−s and −e−s, respectively, it is not difficult to derive the following Lemma. It is also easy to check that eSr is an odd function, and the principal part of eSer at 0 is 2/z. Let Lδ be the set of paths of simple nets of finite length such that w(0)∈I1,w(n)∈V1δ for all n >0, and there is a path in Dδ from the last vertex P(w ) of w in I2 without passing through I1 or other vertices of w.
2 The following driving process convergence theorem can be derived from Proposition 4.3.2 using Skorokhod's embedding theorem.
Convergence of the curves
So it suffices to prove that when δ and d are small enough, the probability that Xδ will come to a spherical distance d from I1 after a time τδ is less than ε. Since Xδ is obtained by deleting the loops of CRWδδ, it suffices to show that the probability that CRWδδ will come within a spherical distance of I1 after hitting Λ tends to zero asd, δ →0. Since CRWδ is a Markov chain, it suffices to prove that the probability that CRWδv will come within a spherical distance d from I1 tends to zero as d, δ → 0, uniformly in v ∈ δZ2 ∩Λ. According to the Markov property, for each v ∈ δZ2∩Λ, the probability is that CRWδv will arrive at a spherical distance d from I1.
This then means that the diameter βδ[t1, t2] is not greater than 4r, which is less than ε.
Convergence of observables
- The limits of domains and functions
- The existence of some sequences of crosscuts
- Constructing hooks that hold the boundary
- The behaviors of g 0 ◦ J outside any neighborhood of 1
Since g0 is the limit of CEδngn, this implies that g0(z)→ 1 as z ∈D0 and z →I2 in the spherical metric. Suppose that {vn}, chosen at the beginning of this trial, has a subsequence that tends to I2 in the spherical metric. Since γk∩γk±1 = ∅ and γnk±1 converges to γk±1 in the Hausdorff distance, there exists dk > 0 such that the distance between γk and γnk±1 is greater than dk, if n is large enough.
In the second and third cases, since RWnv(τnk−1) ∈ Dn\U(γnk), and the Euclidean distance between γnk and γnk+1 is greater thanδ by construction, therefore.
Some discussions
In the third case, RWnv(τnk) ∈ Dn \U(γnk+1), so gn(RWnv(τnk)) ≤ Mk; and RWnv uses first an edge that intersects γnk−1 and then an edge that intersects γnk at timeτnk. For a standard strip SLE2 trace β, if we let m+πi be the frame point of β atπi+R, then the reversal of β−m has the same law as a strip SLE2(Sπ;πi→R) trace after some time - change. Similarly, the reversal of a standard disk SLE2 trace has the same law as a radial SLE2(D;x→0) trace after a time change.
To construct such a curve, we let Pt,Wξ in the definition of HRLC be the modified Green's function in D\W(Ktξ) with the pole at p0 and with a reflection boundary I.
Preparations
Brownian motion and Brownian excursion
Since we can use groups in S(D) to approximate any closed subset of D, so we have the following lemma. Suppose that w1 6=w2 are the two leading edges of D, and hj maps a neighborhood Uj of wj into Dconformally intob H such that h(Uj∩∂D)ˆ ⊂R. For example, suppose that K has the law of BM(D;p→I) and D0 is a subfield of D containing p and a neighborhood of I.
Brownian bubble
We can find a decreasing sequence Flin S(D) such that F =∩lFlandF occurs in the interior of every Fl. The Brownian bubble defined here is similar to the Brownian bubble defined in [8] for the upper half plane. The main difference is that we do not fill in the holes of a Brownian bubble to make it a skirt, so the information inside will not be lost.
A martingale for HRLC 2
The change of Ht(z) comes from a perturbation in the primary end w(t), which contributes the minimal function in D\W(Kt) with the pole at w(t). Assume that Jt and Qt are some analytic functions in the vicinity of ξ(t) (or without the point ξ(t)) such that Jt = ImJt and Qt = ImQt. Then ξ(t) is a simple pole of Qt and the principal part is −1/(z−ξ(t)) thanks to the normalization of Mt.
Note that LΣ is a time change of LΣ,W, and the choice of gt compensates for the effect of the time change.
Adding Brownian bubbles to HRLC 2
Suppose that γ is an open simple curve in D separating w0 from F in D, and the two ends of γ converge to two different points of I. As t → Tc, every curve in Ut from It± to βt must ∓ cross a ring centered at z0 with modulus tends to. With conformal invariance, the extreme distance between<1,±i >and<∓i, eir∓(t)>inD tends to∞ ast→Tc.
From the discussion of the last chapter, it is reasonable to assume that β(t) converges to an interior point of I ast→∆(L).
Other values of the parameter κ
With this assumption, HRLCκ generates a measure of Γ(D), the space of simple curves in D connecting 0 and∞. Note that two bounded measures on Γ(D) are equal if they agree on the set {⊂ D0} for all simply connected D0 ∈ A. Thus, by adding a Poisson cloud of Brownian bubbles with density α(κ) adding to an arbitrary curve with the law of νκD we obtain a conformal invariant constraint probability measure.
If D = H, then, after all holes are filled, this measure corresponds to the constraint measure constructed in [8] from a chord SLEκ.
